Paper Title: A deterministic proof of Loewner energy reversibility via local reversals.



Paying More Attention to Image: A Training-Free Method for Alleviating Hallucination in LVLMs
    
  Shi Liu1 Kecheng Zheng1,2^† Wei Chen1^†

    
=============================================================================================



§ ABSTRACT 
    We give a new proof of the orientation reversibility of chordal Loewner energy by reversing the orientation of a chord in small increments. This fact was first proved by Yilin Wang <cit.> using the reversibility of chordal Schramm–Loewner evolution (SLE).  Our method is similar in spirit to Dapeng Zhan's proof <cit.> of chordal SLE reversibility, though it is purely deterministic.




§ INTRODUCTION



Let γ be a simple curve from 0 to ∞ in the upper half-plane = {z∈:  z>0}. We parameterize γ by _+ = (0,+∞) so that, if g_t:∖γ(0,t] → is the unique conformal map with the normalization g_t(z) - z → 0 as z→∞, then the expansion g_t(z) = z + 2a_t z^-1 + O(z^-2) as z→∞ satisfies a_t= t for every t>0. By Carathéodory's theorem, we can extend g_t to the prime ends of ∪γ(0,t]. Let λ_t := g_t(γ(t)), which we call the driving function of γ. The Loewner energy of γ is defined as the Dirichlet energy of its driving function: i.e., 

    I_;0,∞(γ) := 1/2∫_0^∞ |λ̇_t|^2 dt.


Loewner energy was introduced in independent works by Friz and Shekar <cit.> and Yilin Wang <cit.>. The latter was in the context of large deviations for a family of random curves called the chordal Schramm–Loewner evolution (_κ) as κ→ 0, which was also considered in <cit.>. Loewner energy has since been identified in terms of various geometric and probabilistic quantities, hinting at a deeper connection between these fields. We refer the reader to the survey articles <cit.> for various perspectives on Loewner energy.

If D⊊ is a simply connected domain and γ is a crosscut (henceforth called a chord) from a prime end a to another prime end b, it is natural given the conformal invariance of chordal SLE_κ to define the Loewner energy of γ as

    I_D,a,b(γ) := I_,0,∞(φ(γ))

where φ:D→ is a conformal transformation with φ(x) = 0 and φ(y) = ∞. It is straightforward to check that the definition does not depend on the choice of such a map φ.

A fundamental property of Loewner energy, called reversibility, is that the Loewner energy of a chord does not depend on the choice of its orientation. It is not obvious a priori that the two quantities are related.
If γ is a chord in  going from 0 to ∞, let γ^ denote the same chord in the reverse orientation, going from ∞ to 0 in . If we denote the inversion map z↦ -1/z by ι, we see from (<ref>) that I_,∞,0(γ^) = I_,0,∞(ι(γ^)) = 1/2∫_0^∞ |λ̇_t^|^2 dt where λ_t^ is the driving function corresponding to the curve ι(γ^). The map (λ_t)_t∈ [0,∞)↦ (λ_t^)_t∈ [0,∞) induced by reversing of the chord is rather intricate and there does not exist an explicit formula except for extremely simple examples. Nevertheless, Wang showed in <cit.> that the Loewner energy of γ^ is equal to that of γ. The purpose of this article is to give another proof of this fact.


    
    Let D be a simply connected domain and let a and b be any two distinct prime ends of D. For a chord γ in D from a to b, let γ^ be the same chord with its orientation reversed to trace from b to a. Then, I_D;a,b(γ) = I_D;b,a(γ^).




 §.§ Proof overview


We first describe previous proofs of Theorem <ref> and explain how ours differs from them. The original proof in <cit.> relied upon the reversibility of chordal SLE_κ for κ∈ (0,4], which was first established in <cit.> and proved subsequently with different methods in <cit.>. By interpreting Loewner energy I_;0,∞ as the large deviation rate function as κ→ 0 for chordal SLE_κ in  from 0 to ∞, Wang showed that the orientation reversal of a finite-energy chord must have the same Loewner energy.

Wang gave a deterministic proof of Loewner energy reversibility in <cit.> through an identification between Loewner energy I_,0,∞(γ) and the Dirichlet energy of log |f'|, where f is the uniformizing map on each component of ∖γ. The reversibility of Loewner energy follows immediately since the uniformizing map f does not depend on the orientation of the chord γ. Based on this result, she revealed in the same work an unexpected link between Loewner chains, conformal welding, and the Kähler geometry of the Weil–Petersson Teichmüller space.

On the other hand, our proof of Theorem <ref> uses directly the definition (<ref>) of Loewner energy in terms of the driving function. The notable difference in our approach is that rather than “lifting" γ to the space of chords without orientation, we consider a “continuous" reversal of its orientation. 

For a Jordan arc η:(0,T]→ with η(0) = 0 and driving function λ, we define the partial Loewner energy of the curve η as I_;0,∞(η) := 1/2∫_0^T |λ̇_t|^2 dt. We extend this definition to other simply connected domains by conformal invariance as in (<ref>). This allows us to consider the energy of a chord with its orientation reversed midway. For instance, If γ = γ_1 ⊕γ_2 is a chord in D from a to b where γ_1 and γ_2 meet at z ∈ D, then we can ask if I_D;a,z(γ_1) + I_D∖γ_1; b,z(γ_2^) equals I_D;a,b(γ). Our heuristic is to show that this identity holds as we gradually move z along γ from b to a.
The following local commutation relation derived from the Loewner equation is at the heart of our approach. 


    Let D be a simply connected domain and a,b be distinct prime ends. Suppose γ:(0,t] → D and η:(0,s]→ D are simple curves with γ(0) = a and η(0) = b. If γ∩η = ∅, then
    
    I_D;a,b(γ) + I_D∖γ ;b,γ(t)(η) = I_D;b,a(η) + I_D ∖η;a,η(s)(γ).



In other words, we can compute the joint Loewner energy of γ and η by computing the energy of either γ or η first, then adding the energy of the other curve in the complement of the first.
The main restriction in Theorem <ref> is that the curves γ and η cannot intersect, even at their endpoints. Suppose that this relation extended to the case where the two curves meet at their endpoints. Then, for two curves γ and η as in the setup of Theorem <ref> and a chord ω in D∖ (γ∪η) running from γ(t) to η(s), we would have 

    I_D;a,b(γ⊕ω) + I_D∖(γ∪ω);b,η(s)(η) 
           = I_D;a,b(γ) + I_D∖γ; γ(t),b(ω) + I_D∖(γ∪ω);b,η(s)(η) 
       = I_D;a,b(γ) + I_D∖γ;b,γ(t)(η) + I_D∖(γ∪η); γ(t),η(s)(ω).

Note that 

    I_D;a,b(γ) + I_D∖γ;b,γ(t)(η) + I_D∖(γ∪η); η(s),γ(t)(ω^) = I_D;a,b(γ) + I_D∖γ;b,γ(t)(η⊕ω^).

If we could furthermore show that I_D∖(γ∪η); γ(t),η(s)(ω) and I_D∖(γ∪η); η(s),γ(t)(ω^) are approximately equal for small ω, then we can reverse a given chord in small increments while keeping the difference in Loewner energy small. This is indeed our approach, except that, instead of ω^, we replace ω with a chord with zero energy in the same direction and endpoints as ω^. Here are further details of our proof strategy. The terms appearing below are standard in the theory of Loewner chains, but they are also defined carefully in Section <ref>.


    
  * Instead of chords in  from 0 to ∞, we consider those in  between two points x,y∈. These chords have finite half-plane capacity. Let γ be such a chord going from x to y parameterized by half-plane capacity on the interval [0,T].

    
  * We define the local reversal operation on γ where we replace the segment γ[t-,t] with the hyperbolic geodesic going from γ(t) to γ(t-) in ∖γ([0,t-] ∪ [t,T]). We apply this operation inductively, starting from the terminal point t = T and reducing t by  in each step, to obtain a chord γ̂^ going from y to x. By the commutation relation (<ref>), the Loewner energy of γ̂^ is less than or equal to that of γ.

    
  * We consider the uniformizing maps g: ∖γ→ and g^: ∖γ̂^→ that satisfy lim_z→∞ (g(z)- z) = lim_z→∞ (g^(z)-z) = 0. Using standard estimates for these maps, we show that g^→ g uniformly on some neighborhood of ∞ as → 0. Equivalently, the unbounded component of ∖γ̂^ converges in the Carathéodory sense to that of ∖γ. Using the lower semicontinuity of I_;y,x with respect to this topology, we show that I_;y,x(γ^) ≤ I_;x,y(γ). The equality between the two energies follows by considering γ^ in place of γ.


The strategy of considering bounded chords and reversing their orientation in small capacity increments appeared in Zhan's proof of SLE reversibility <cit.> (see also <cit.>), which itself was based on the local commutation relations for SLE <cit.>. In fact, our motivation comes from a search for an analogous proof for Loewner energy reversibility as suggested in <cit.>. The local reversal operation in Zhan's proof takes in a small segment of an SLE chord and replaces it with an independently resampled SLE segment in the reverse direction. In contrast, we use hyperbolic geodesics in the reverse direction as replacements. This “surgery" technique where we replace a segment of a Jordan curve with the hyperbolic geodesic in the slit domain appeared previously in <cit.>.
The idea of considering the Carathéodory topology on the hulls generated by the chords (instead of the Hausdorff topology on chords, as in <cit.>) comes from the reformulation of Zhan's SLE reversibility proof by Lawler and Yearwood in <cit.>. 

The final stages of the SLE reversibility proofs in <cit.> consist of checking that their sequence of couplings (γ, γ̂^), whose marginal distributions are SLE curves from x to y and from y to x, converges to (γ, γ^) weakly (Zhan) or in total variation distance (Lawler and Yearwood). This probabilistic technique cannot be adapted directly to our deterministic setting. We overcome this difficulty by exploiting the lower semicontinuity of Loewner energy in the Carathéodory topology. 

Here is an overview for the rest of this paper. In Section <ref>, we introduce the Carathéodory topology on chords and describe its relation to the Hausdorff topology. We show that Loewner energy is lower semicontinuous with respect to Carathéodory topology, and give estimates that can be used to show convergence in Carathéodory topology. Using these tools, we present in Section <ref> the proof of Theorem <ref> using the local reversal operation. 





Acknowledgements. The author is grateful to Yilin Wang for introducing the theory of Loewner energy and the problem during a course given at the Cornell Probability Summer School, and wish to extend the gratitude to the organizers of the program as well. We thank Greg Lawler and Stephen Yearwood for explaining their proof of SLE reversibility, and Yilin Wang and Catherine Wolfram for their helpful comments on an earlier draft of this article. Much of the current work was completed during the Thematic Program on Randomness and Geometry at the Fields Institute, to which the author thanks for their hospitality. The author is partially supported by a fellowship from Kwanjeong Educational Foundation.



§ CARATHÉODORY TOPOLOGY ON BOUNDED CHORDS



Given a simply connected domain D ⊊ = ∪{∞} and distinct prime ends a and b, we define X(D;a,b) as the space of chords in D (i.e., a Jordan arc γ in D with γ⊂∂ D) from a to b. We consider these chords modulo orientation-preserving reparameterizations. In particular, X(D;a,b) and X(D;b,a) differ in the orientations of the chords. For x,y ∈, a natural topology on X(;x,y) from the perspective of Loewner chains is that induced by the Carathéodory topology on the hull created by the chords. 

In this section, we first set up some notations on the mapping-out functions appearing in Loewner chains and give a few estimates on it. We then give a precise definition of the Carathéodory topology on bounded hulls and give a metrization of this topology in terms of the mapping-out functions (Lemma <ref>) to be used in the proof of the main theorem. 

Our main goal in this section is Proposition <ref>, which states that Loewner energy is lower semicontinuous with respect to the Carathéodory topology on X(;x,y). This is based on a stronger statement proved in <cit.> that I_;x,y is lower semicontinuous with respect to the Hausdorff topology on X(;x,y). The link is provided by the observation that the Carathéodory topology is coarser than the Hausdorff topology on bounded hulls (Lemma <ref>). 






 §.§ Mapping-out functions


Let C be the space of non-empty closed subsets of ∪{∞}. We say that K ∈ C is bounded if there exists R<∞ such that |z|≤ R for all z∈ K. The set of compact -hulls is defined as

    K := {K ∈ C: K is bounded, ∖ K is simply connected, and K∩ = K}.

For each K ∈ K, there exists a unique conformal transformation g_K: ∖ K → such that g_K(z) -z → 0 as |z|→∞, which we refer to as the mapping-out function of K. The half-plane capacity of K is defined as the constant (K) appearing in the expansion

    g_K(z) = z + (K)/z + O(1/z^2)   as  z→∞.


We summarize here some of the basic properties of the half-plane capacity drawn from <cit.>.
From the definition (<ref>), we see that for any K∈ K, x∈, and r>0,

    (rK + x) = r^2   (K).

Since Im[g(z)-z] is a bounded harmonic function on ∖ K, we see that 

    (K) = lim_y→∞ y∫ ( z)   dω_∖ K(iy,dz)

where ω_∖ K is the harmonic measure of ∖ K viewed from iy. Therefore, (K)≥ 0 for all K∈ K. The following properties follow directly from the representation (<ref>) using the basic properties of harmonic measure (equivalently, those of complex Brownian motion). For any K,K'∈ K,

    (K')    = (K) + (g_K(K')),
    (K ∪ K')    ≤(K) + (K').

In particular, if K⊂ K', then (K) ≤(K'). Note that 
    ({z∈: |z|≤ r}) = r^2
since z↦ z+r/z is the corresponding mapping-out function.
Combining these two facts, we see that for all K∈ K,

    (K) ≤({z∈: |z| ≤(K)}) = [(K)]^2.

Hence, any chord in  between two boundary points in  has a finite half-plane capacity. As a consequence of the Loewner equation, we have a partial converse

    (K) ≥1/2[max_z∈ K ( z) ]^2.

Equality is achieved when K is a vertical line segment; see <cit.> for a proof. Hence, a compact -hull with a large diameter has small capacity only if it is horizontally spread out while staying close to the real axis. A key input in our proof (Proposition <ref>) shows that this cannot happen for finite-energy chords.

For K∈ K, define the radius of K viewed from x∈ as 

    (K;x):= max_z∈ K |z-x|.
 
We denote (K) := (K;0). We end our discussion of the mapping-out function with the following estimates collected from <cit.>. 

    
    For all K∈ K and z∈∖ K,
    
    |g_K(z) - z| ≤ 3 (K).

    Moreover, there exists a constant c<∞ such that for any K∈ K, if z∈ with |z| ≥ 2 (K), then
    
    |z - g_K(z) - (K)/z|≤  c (K) (K)/|z|^2


In particular, if K_1,K_2∈ K are hulls with (K_1) = (K_2) = h and radii both less than r with respect to the same point x∈, then for every z∈ with |z-x|≥ 2r,
    
    |g_K_1(z) - g_K_2(z)| ≤crh/|z-x|^2.





 §.§ Topology on hulls


We now introduce Carathéodory and Hausdorff topologies on the space of compact -hulls K. We will later obtain corresponding topologies on the space of chords X(;x,y) by considering the hulls generated by the chords. 


Let (K^n)_n∈ N and K be compact -hulls. 
We say that K^n converges to K in the Carathéodory topology if g_K^n^-1 converges to g_K^-1 uniformly away from . This means that for every >0, g_K^n^-1 converges uniformly to g_K^-1 on {z∈:  z > }. We denote the convergence of K^n to K in the Carathéodory topology as K^n K.



The topology of uniform convergence away from  is metrizable by standard methods. It can also be stated geometrically by means of Carathéodory kernel theorem. Given a sequence of unbounded open subsets (U_n) of , let V_n denote the interior of the unbounded connected component of ⋂_k≥ n U_k. The kernel of the sequence (U_n) is defined as V = ⋃_k≥ 1 V_k. We say that the sequence (U_n) converges to its kernel if every subsequence (U_n_j) has the same kernel. Then, K^n  K if and only if (∖ K^n) converges to its kernel and this kernel is ∖ K. Further discussions on the Carathéodory kernel convergence can be found in standard texts on Loewner evolution; see <cit.> for instance.

Define K_r := {K∈ K: (K) ≤ r}, which includes the empty hull. On K_r, Definition <ref> is equivalent to the uniform topology on the mapping-out functions g_K restricted to any small neighborhood of ∞. Let us check this claim carefully since this metrization allows us to use the estimate (<ref>) in our proof of Loewner energy reversibility.


    Fix 0 < r < R and let (K^n) be a sequence of hulls in K_r and K ∈ K. The mapping out functions g_K^n converge uniformly to g_K on {z∈: |z| ≥ R} if and only if K^n  K.


    To show K^n  K given that g_K^n→ g_K uniformly on {z∈:|z|≥ R}, it suffices to show that for each fixed >0, given any subsequence of (K^n), there exists a further subsequence (K^n_j) such that g_K^n_j^-1→ g_K^-1 uniformly on {z∈:  z≥}. Then, by a diagonal argument, we can find given any subsequence of (K^n) a further subsequence (K^n_j) such that g_K^n_j^-1→ g_K^-1 uniformly on {z∈:  z≥} for all >0, thus proving the claim. That K ∈ K_r follows from the fact that ∖ K is the kernel of the sequence of domains (∖ K^n).

    Fix >0. Assume that (K)≤ r, choosing a larger r if necessary. Without loss of generality, assume also that R ≥ 8r. To lighten notations, let us write g_n = g_K^n and g= g_K. If g̃(z) = -1/g(-1/z), note that g̃ extends by Schwarz reflection to a conformal map on {z∈: |z|<1/r} since g(z)-z→ 0 as |z|→∞. By (<ref>), if |w| ≥ 5r, then |g^-1(w)|≥ 2r. Hence,
     
    M:= sup_w∈,  |w|≥ 5r|(g^-1)'(w)| ≤sup_z∈,  |z| ≥ 2rg'(z)^-1 = sup_z∈,  |z|≥ 2r|g̃'(-z^-1)| ·|z/g(z)|^2 < ∞.

    Now, take any w∈ with |w|≥ R-3r and denote z_n = g_n^-1(w) and z = g^-1(w). Again by (<ref>), we have |z_n|≥ 5r for all n and |z|≥ 5r. Hence, recalling g_n(z_n) = w = g(z),
    
    |z_n - z| ≤ M π |g(z_n) - g(z)| = Mπ|g(z_n) - g_n(z_n)|

    where π was multiplied to bound the maximum length of a path connecting g(z_n) and g(z) that stays outside the open ball of radius 5r centered at 0. Since g_n → g converges uniformly on {z∈: |z|≥ R}, given any >0, we can find a positive integer N such that for all w∈ with |w|≥ R - 3r, we have |g_n^-1(w)- g^-1(w)|<. 

    We complete the proof by finding a subsequence (n_j) such that g_n_j^-1→ g^-1 uniformly on  C:={w∈: |w| ≤ R,   w ≥}.  For this, observe that by (<ref>), each g_n^-1 is a holomorphic map taking the compact set C into the closed disk of radius R+3r. By the Schwarz lemma, we see that (g_n^-1) is a uniformly bounded and equicontinuous sequence of holomorphic maps, implying that there exists a subsequence (g_n_j^-1) which converges uniformly to a holomorphic map g̃ on C by the Arzelà–Ascoli theorem and Morera's theorem. Since g_n^-1→ g^-1 on {w∈: |w| ≥ R- 3r} and g^-1 has a unique analytic extension on , we conclude that g̃ = g^-1.    

    The proof that K^n  K implies the uniform convergence of g_K^n→ g_K on {z∈:|z|≥ R} proceeds in an analogous manner.




The second topology we consider on K is the Hausdorff topology, by which we mean the topology induced by the Hausdorff distance d_H(K_1,K_2) = inf{>0: K_1 ⊂ B_(K_2),  K_2 ⊂ B_(K_1)}. If K^n converges to K in the Hausdorff topology, we denote it as K^n  K. 

We note that K (also K_r for any r>0) is not a closed subspace of C under the Hausdorff topology. A classic example is that of the arcs {e^iθ: θ∈ [1/k, π]} converging to the closed unit half-disk in the Carathéodory topology but to the closed unit half-circle in the Hausdorff distance. In fact, Carathéodory topology on K is (strictly) coarser than Hausdorff topology on K. This is an immediate consequence of the following observation given the compactness of K_r under the Carathéodory topology for every r>0 <cit.>. 


    
    Let (K^n) be a sequence in K. If K^n K̃∈ K and K^n  K  ⊂ which is bounded, then ∖K̃ coincides with the unbounded connected component of ∖ K. In particular, if K ∈ K, then K = K̃.






   The identity map from K endowed with the Hausdorff topology to that endowed with the Carathéodory topology is continuous.



    Let (K^n) be a sequence of compact -hulls that converges to K∈ K in the Hausdorff distance. Then r:=sup_n(K^n)<∞. Since K_r is compact in the Carathéodory topology, given an arbitrary subsequence of (K^n), we can find a further subsequence (K^n_j) and K̃∈ K such that K^n_jK̃. Since K∈ K, we conclude K̃ = K by Lemma <ref>.




 §.§ Topology on chords


We remind the reader that a chord in a simply connected domain D is merely an open Jordan arc whose boundary points are in D. We use freely the following consequence of the Jordan curve theorem. See <cit.> for a proof.


    If γ is a chord in a simply connected domain D⊊, then D∖γ has exactly two connected components U_1 and U_2 and these satisfy D∩∂ U_1 = D ∩∂ U_2 = γ.



Given a chord γ∈ X(;x,y) where x,y∈ are distinct boundary points, let us denote by K_γ the compact -hull given by the closure of the bounded connected component of ∖γ. Then, ∩∂ K_γ =γ. However, it is not always the case that any K∈ K is of the form K_γ for some chord γ in . We define the Carathéodory topology on X(;x,y) as the pullback of the Carathéodory topology on the space {K∈ K: ∩∂ K ∈ X(;x,y)}.


    Given a sequence of chords (γ^n) in X(;x,y), we say that γ^n converges to γ∈ X(;x,y) in the Carathéodory topology if K_γ^n converges to K_γ in the Carathéodory topology on K. We denote this convergence by γ^n γ.



    As a point of comparison, the Hausdorff topology on the closures of bounded chords also agree with that on the hulls they generate. For if γ∈ X(;x,y) with x<y (resp. y<x), we have ∂ K_γ = γ∪ [x,y] (resp. [y,x]). Hence, for γ,γ'∈ X(;x,y), we have d_H(γ∪{x,y},γ'∪{x,y}) = d_H(K_γ, K_γ'). That is,  the Hausdorff topology on X(;x,y) coincides with the pullback of the Hausdorff topology on K under the map γ↦ K_γ. We denote this mode of convergence by γ^n γ.


Here is the main result of this section.


    Let x,y∈. The map I_;x,y: X(;x,y)→ [0,∞] is lower semicontinuous when the domain is equipped with the Carathéodory topology. 


    For every ρ∈ [0,∞), the sublevel set {γ∈ X(;x,y): I_;x,y(γ) ≤ρ} is compact in the Hausdorff topology <cit.>. The case for Carathéodory topology follows immediately given Lemma <ref>, Definition <ref>, and Remark <ref>.



We note that the proof of the compactness of the sublevel sets of Loewner energy in <cit.> relies on the classification of chords with uniformly bounded Lowener energy as images of i_+ under a quasiconformal self-map of  with uniformly bounded dilatation <cit.>. In particular, Proposition <ref> is a purely deterministic result.


We remark that for bounded simple curves in  starting from 0 and terminating within , the lower semicontinuity of the partial Loewner energy I_;0,∞(η) with respect to the Carathéodory topology on ∖η follows trivially from the continuity of the Loewner transformation (λ_t)_t∈ [0,T]↦ g_T when we consider the uniform norm on (λ_t)_t∈ [0,T] and the topology of uniform convergence away from  for g_T^-1 <cit.>. The main point of Proposition <ref> is that consider the Loewner energy of the entire chord.

We end this section with a clarification that the Hausdorff topology on the space of chords is equivalent to Carathéodory topology viewed from both sides of the chord.


    Let D be a simply connected domain. Given γ∈ X(D;a,b), denote by U_γ^L and U_γ^R the two connected components of D ∖γ which are to the left and the right of the chord, respectively. Suppose (γ^n) is a sequence in X(D;a,b) such that there is a chord γ∈ X(D;a,b) for which (U_γ^n^L) and (U_γ^n^R) converges to their kernels U_γ^L and U_γ^R, respectively. Then, γ^n γ.


    Without loss of generality, assume D =. Given an arbitrary subsequence of (γ^n), there is further subsequence (γ^n_j) that converges in Hausdorff distance to a compact set K ⊂. By Lemma <ref>, U_γ^L and U_γ^R are each connected components of ∖ K. Since a,b∈γ^n for all n, {a,b}⊂ K ⊂γ∪. 
    
    We claim that if K ∩⊊γ, then ∖ K is connected. Suppose there is a point z∈γ∖ K but ∖ K is not connected. Then there are disjoint sets V^L,V^R ⊂∖ K which are open in ∖ K and V^L ∪ V^R = ∖ K. Since U_γ^L,U_γ^R ⊂∖ K and are connected, they cannot be in both V^L or V^R. Without loss of generality, assume U_γ^L ⊂ V^L and U_γ^R ⊂ V^R as well as z∈ V^L. Since γ = ∩ U_γ^L, there is a neighborhood of z in  which has a nonempty intersection with U_γ^R. Hence, V^L is not open in ∖ K. This is a contradiction, so the claim is proved.
    
    Therefore, K ∩ = γ. But K ∩ = {a,b} since otherwise either U_L^γ or U_R^γ, which are the only connected components of ∖ K, would not be the kernel of (U_γ^n_j^L) or (U_γ^n_j^R), respectively. This implies K = γ∪{a,b} and thus γ^n γ.





§ PROOF OF LOEWNER ENERGY REVERSIBILITY




 §.§ Diameter bound for finite-energy chords




Our strategy for the proof of Theorem <ref> is to reverse the chord in small half-plane capacity increments while keeping the Loewner energy nonincreasing. We wish to show that as we reduce the “hcap step size," the resulting orientation-reversed chord converges in Carathéodory topology to the reversal of the original chord. By Lemma <ref>, it suffices use the estimate (<ref>) to show that the mapping-out functions of these reversed chords converge uniformly near ∞. This requires bounding the diameter of the reversed chord in terms of its half-plane capacity. This is exactly the estimate we obtain from the following lemma. For a chord γ∈ X(;x,y) with x,y ∈, we define (γ) := (K_γ) where K_γ∈ K is the closure of the bounded connected component of ∖γ.


    For each ρ∈ [0,∞), there exist constants 0 < c_ρ < C_ρ < ∞ such that for any x,y∈ and γ∈ X (;x,y) with I_;x,y(γ) ≤ρ, we have 
    
    |x-y| ≤(γ) ≤ C_ρ |x-y|   and   (c_ρ|x-y|)^2 ≤(γ) ≤ (C_ρ |x-y|)^2.



Given the bound (<ref>), the upshot from Proposition <ref> is the existence of a constant c_ρ∈ (1,∞) such that [(γ)]^2 ≤ c_ρ (γ) for any γ∈ X(;x,y) with I_;x,y(γ)≤ρ.

We give a short proof of this lemma using the following restriction on the location of finite-energy chords. This result was first proved in <cit.> using a probabilistic method, but it has been proved again in <cit.> using a purely deterministic methodology. 



    For each θ∈ (0,π), the Loewner energy of all chords in X(;0,∞) passing through the point e^iθ is bounded below by -8log(sinθ). Consequently, any chord γ∈ X(;0,∞) with I_;0,∞(γ) ≤ρ lies in the cone {z∈: θ≤ z ≤π - θ} where θ = arcsin(e^-ρ/8) ∈ (0,π/2].




    Choose a conformal automorphism φ of  that fixes 0 and maps ∞ to 1. Let γ_min and γ_max be the image of the rays z = arcsin(e^-ρ/8) and z = π - arcsin(e^-ρ/8), respectively. Since φ is determined up to pre-composition by the scaling map z↦ r z, the chords γ_min and γ_max do not depend on the choice of φ. Any γ∈ X(;0,1) with I_;0,1(γ) ≤ρ must lie between γ_min and γ_max by Lemma <ref>. That is, K_γ_min⊂ K_γ⊂ K_γ_max in terms of the hulls enclosed by these chords. Then, from (<ref>) and (<ref>), we have 0<1/2 [max_z∈γ_min z]^2 ≤(γ) ≤ [(γ_max)]^2<∞ and these bounds are determined only by ρ. The same bounds apply to γ∈ X(;0,-1) with I_;0,-1(γ)≤ρ by symmetry.
    The conformal invariance of Loewner energy and (<ref>) give the lemma for general x,y∈.


A self-contained proof of Proposition <ref> can also be given using the Loewner equation and the definition of Loewner energy using estimates similar to those appearing in <cit.>.




 §.§ A commutation inequality


For the final step in the preparation of our proof of the Loewner energy reversibility, we observe that the commutation relation (<ref>) is given only for curves that do not intersect each other. We wish to argue that this relation extends to the case of two curves that are joined together at their endpoints. But this requires that as the domain becomes smaller so that a Jordan arc becomes a chord, the partial Loewner energy of the curve converges to the Loewner energy of the chord in the final domain. This difficulty we encounter in joining two curves at their endpoints is analogous to the problem of “lifting" local commutation relations for SLE to global ones discussed by Dubédat in <cit.>.

Short of this fact, we settle with the following “commutation inequality," which is sufficient for our proof thanks to the lower semicontinuity of Loewner energy. A hyperbolic geodesic in a simply connected domain D between prime ends a and b is the chord φ(i_+) where φ:→ D is a conformal transformation with φ(0) = a and φ(∞) = b. For instance, the hyperbolic geodesic in  between x,y∈ is the half-circle with the interval between x and y as its diameter.



    Let D be a simply connected domain. Suppose γ and η are simple curves in D starting from distinct prime ends a and b, respectively. Furthermore, suppose these curves intersect only at their endpoints, labeled z ∈ D. Let η̃ be the hyperbolic geodesic in D ∖γ from b to z. Then,
    
    I_D;a,b(γ) + I_D∖γ;b,z(η̃) ≤ I_D;b,a(η) + I_D∖η;a,z(γ).



    Pick any continuous parameterization γ:(0,1] → D where γ(0) = a and γ(1) = z. Let us denote γ_t := γ(0,t]. For t∈ (0,1), since γ_t and η do not intersect, the local commutation relation (<ref>) implies
    
    I_D;a,b(γ_t) ≤ I_D;a,b(γ_t) + I_D∖γ_t;b,γ(t)(η) = I_D;b,a(η) + I_D∖η ;a,z(γ_t) .

    We obtain the lemma by taking t→ 1 in this inequality and observing that I_D∖γ;b,z(η̃) = 0.





 §.§ The proof


We are now ready to state the proof of the orientation reversibility of Loewner energy. Recall the notation  for convergence in the Carathéodory topology and γ^ for the orientation reversal of a chord γ. If γ∈ X(;x,y) is a chord in  from x to y, then we denote g_γ := g_K_γ in the notation of the mapping-out function (<ref>) where K_γ is the closure of the bounded connected component of ∖γ.



    Using the conformal invariance (<ref>) of Loewner energy, we may assume that we are given a chord γ∈ X(;0,1) with finite Loewner energy ρ := I_;0,1(γ) < ∞. Let T :=  (γ)<∞ and denote γ_t:= γ(0,t]. Suppose γ is parameterized by half-plane capacity: i.e., (γ_t) = t for every t∈ (0,T). Let c_ρ∈ (1,∞) be a constant depending only on ρ given by Proposition <ref> such that for any chord γ̃∈ X(;x,y) with I_;x,y(γ̃) ≤ρ, we have
    
    |x-y| ≤(γ̃;x) ∨(γ̃;y) ≤√(c_ρ (γ̃)).

    
    Here are the induction hypotheses for the local reversal procedure. Let t∈ (0,T] and assume that we have reversed γ[t,T), meaning that we now have a simple curve η starting at 1 joined to γ_t at their endpoints only. In other words, η∈ X(∖γ_t; 1;γ(t)). Assume further that we have (γ_t∪η) ≤ T + c_ρ(T-t) and I_;0,1(γ_t) + I_∖γ_t;1,γ(t)(η) ≤ρ. The base step for the induction is when t=T and η is the empty curve, which trivially satisfy these hypotheses.

    Let us now define the local reversal operation. For ∈ (0,t], define the mapping F_: η↦η̃ in the following way.
    See Figure <ref> for a graphical representation of F_.


    
         
  * Let g: ∖γ_t-→ and h:∖ g(η) → be the mapping-out functions as in (<ref>).
         
         
  * Denote γ^ := (h∘ g)(γ(t-,t)) ∈ X(;x,y) where x = (h∘ g)(γ(t-)) and y = (h∘ g)(γ(t)). 
        
         
  * Let η^ be the hyperbolic geodesic in X(;y,x). That is, η^ is the half-circle centered at x+y/2 with diameter |x-y| oriented from y to x. 
         
         
  * Define F_(η) = η̃:= η⊕ (h∘ g)^-1(η^). Here, ⊕ refers to the concatenation operation.
         


    

    Let us check that F_(η) satisfies the induction hypotheses. 
    By the relations (<ref>) and (<ref>),
        
    (γ^)    = (g(γ[t-,t] ∪η)) - (g(η)) 
       ≤(g(γ[t-,t])) = (γ_t) - (γ_t-) =  .

        By (<ref>), |x-y| ≤(γ^;y) ≤√(c_ρ). This implies (η^) = |x-y|^2 ≤ c_ρ by (<ref>).




  * Since η^∈ X(;y,x), we have (h∘ g)^-1(η^) =  X(∖ (γ_t-∪η);γ(t),γ(t-)). Since η terminates at γ(t), we have η̃∈ X(;∖γ_t-;1,γ(t-)).

      
  * For the half-plane capacity,
        
    (γ_t-∪η̃)    = (γ_t-) + (g(η)) + (η^) 
       ≤(γ_t-) +  (g(η)) + (γ^) + c_ρ
        = (γ∪η) + c_ρ≤ T + c_ρ(T-t+).


        
  * For the Loewner energy, by Lemma <ref>,
        
    I_∖γ_t-;1, γ(t-)(η̃) 
                   = I_∖γ_t-;1, γ(t-)(η) + I_;y,x(η^)
       ≤ I_∖γ_t-;γ(t-),1(γ(t-,t)) + I_∖γ_t;1,γ(t)(η)  =I_∖γ_t;1,γ(t)(η) .

        Hence, I_;0,1(γ_t-) + I_∖γ_t-;1,γ(t-)(η̃) ≤ I_;0,1(γ_t) + I_∖γ_t; 1,γ(t)(η).
    

    Thus, all of the induction hypotheses are satisfied. We now show that the local reversal operation F_ corresponds to a small step in terms of the uniform norm on the mapping-out functions.
    
    Since I_;0,1(γ) = ρ and (γ) = T, we have (γ_t-) ≤(γ) ≤ 2√(c_ρ T) by (<ref>). By (<ref>), we have |g(z) - z| ≤  6√(c_ρ T) for all z∈∖γ_t-. Similarly, I_∖γ_t-;1,γ(t-)(η̃) ≤ρ and (g(η̃)) ≤(γ_t-∪η̃) ≤ (1+c_ρ) T imply (g(η)) ≤(g(η̃)) ≤ 2c_ρ√(2T) and |h(z) - z| ≤ 6c_ρ√(2T) for all z ∈∖ g(η). Combining these, we have 
    
    |(h∘ g)(z) - z| ≤ 15c_ρ√(T)  for all  z ∈∖ (γ_t-∪η).

    Therefore, for all z∈ such that |z|≥ 34c_ρ√(T), we have
    
    |(h∘ g)(z) - (h∘ g)(γ(t))| ≥ |z|-|γ(t)| - |(h∘ g)(z) - z| - |(h∘ g)(γ(t)) - γ(t)|≥ 2 c_ρ√(T).


    
    Recall that (γ^)≤ and (η^)≤ c_ρ. Hence, (γ^;y) ≤√(c_ρ) and (η^;y) ≤ c_ρ√().
    We deduce from (<ref>) that for all w∈ such that |w-y| ≥ 2c_ρ√(), 
        
    |g_γ^(w) - g_η^(w)| ≤ c c_ρ^2^3/2/|w-y|^2
   
        Therefore, for all z∈ such that 
        |z| ≥ 34 c_ρ√(T), we have 
        
    |g_γ_t∪η(z) - g_γ_t-∪η̃(z)| = |(g_γ^( (h∘ g)(z)) - (g_η^( ( h ∘ g)(z))| ≤c/T^3/2.

        Here, the constant c∈ (0,∞) does not depend on any of our parameters nor the chord γ.

    For k∈ N, let γ̂^k:= (F_T/k)^k(γ) ∈ X(;1,0) be the chord in the reverse orientation obtained by repeating the local reversal operation F_T/k through the whole chord γ. Then, by (<ref>), for all z∈ such that |z| ≥ 34 c_ρ√(T),
    
    | g_γ̂^k(z) - g_γ(z)| ≤ c √(T/k) .

    Thus, by Lemma <ref>, γ̂^k γ^ on X(;1,0) as k→∞ where γ^ is the reversal of γ. Also, I_;1,0(γ̂^k) ≤ρ by construction. Hence, by Proposition <ref>,
    
    I_;1,0(γ^) ≤lim inf_k→∞ I_;1,0(γ̂^k) ≤ρ = I_;0,1(γ).
    
    Changing the roles of γ and γ^, we have the reverse inequality. Therefore, I_;1,0(γ^) = I_;0,1(γ). Moreover, if γ∈ X(;0,1) is a chord with I_;0,1(γ)=∞, then we must have I_;1,0(γ^)=∞ since otherwise we would have I_;0,1(γ) ≤ I_;1,0(γ^)<∞ from (<ref>).




alpha




 Paper Title: Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift.






	




Stochastic Domination of Exit Times for Random Walks and Brownian Motion with Drift 

    Xi Geng [Email : xi.geng@unimelb.edu.au]
 School of Mathematics and Statistics

		University of Melbourne
 
 Parkville, VIC Australia

		Greg Markowsky [E-mail :
			greg.markowsky@monash.edu] 

			School of Mathematics 

			Monash University 

			Clayton, VIC Australia
    August 5, 2024
=================================================================================================================================================================================================================================================================================

§ ABSTRACT 
In this note, by an elementary use of Girsanov's transform we show that the exit time for either a biased random walk or a drifted Brownian motion on a symmetric interval is stochastically monotone with respect to the drift parameter. In the random walk case, this gives an alternative proof of a recent result of E. Peköz and R. Righter in 2024. Our arguments in both discrete and continuous cases are parallel to each other. We also outline a simple SDE proof for the Brownian case based on a standard comparison theorem.







§ INTRODUCTION
 
In a series of elegant recent papers, several variants of the following problem were addressed: determine the optimal drift for 1-dimensional simple random walk or Brownian motion starting at 0 to stay for as long as possible in a symmetric interval. To be precise, let {S_n^p:n⩾0}
denote the simple random walk starting at the origin with one-step
distribution 

    ℙ(X=1)=p, ℙ(X=-1)=1-p.

This is often referred to as the biased random walk. 
Given k∈ℤ_+, define 


    σ_p^k≜inf{n:S_n^p=± k}.
 

In <cit.> it was shown that the function p → E[σ_p^k] is increasing for p ∈ (0,1/2) and decreasing for p ∈ (1/2,1), thereby showing that this function is maximized at p=1/2. Later, in <cit.>, it was shown that the stopping time σ_p^k is stochastically maximized at p=1/2, and furthermore if 1/2 ≤ p_1 < p_2 ≤ 1 then σ_p_1^k stochastically dominates σ_p_2^k (Theorem <ref> (i) below). Their proof is based on the use of a clever coupling argument. It was also asserted in <cit.> that an appeal to Donsker's theorem allows one to deduce the analogous result for Brownian motion (Theorem <ref> (ii) below) as a corollary to the result for random walk.

The proofs given in <cit.> are ingenious but combinatorial in nature, and do not seem easy to be modified in order to prove Theorem <ref> (ii) directly (i.e. without proving the result first for simple random walk and then invoking Donsker's theorem). It is natural then to search for a direct proof in the Brownian case. In the end we were able to find two such proofs, one by performing a Girsanov change of measure, and the other by translating the problem into a question about SDE's and applying a standard comparison theorem. Both proofs are relatively short and non-technical, and may be of pedagogical interest for readers unfamiliar with the techniques used. The purpose of this note is to present these two proofs.

The following is the random walk result proved in <cit.>, as well as the analogous Brownian motion result that we will focus on.


(i) [Random walk] 
With notation as above, for any fixed n∈ℕ the function 

    (0,1)∋ p↦ℙ(σ_p^k>n)

is increasing on (0,1/2] and decreasing on [1/2,1). 

(ii) [Brownian motion] Let B_t denote
the standard one-dimensional Brownian motion. Given λ∈ℝ
and b>0, define 

    τ_λ^b≜inf{t:B_t+λ t∉(-b,b)}.

Then for any fixed t>0, the function 

    λ↦ℙ(τ_λ^b>t)

is decreasing in λ on [0,∞). 




2
The Brownian motion case, which is to some extent easier to understand,
is based on a simple application of Girsanov's theorem together with
the crucial observation that the exit location and the exit time are
independent even in the drifted case (Reuter's theorem). The discrete case essentially uses a discrete form of Girsanov's theorem (Lemma <ref> (ii)), which may be of pedagogical interest since it is easy to check "by hand", while the standard Girsanov's theorem for Brownian motion involves an understanding of subtleties of quadratic variation and Levy's characterization of Brownian motion (see <cit.>). 


In the next two sections, we develop the Girsanov proof of the theorem in both the Brownian motion and random walk contexts. Our arguments are entirely parallel in these two cases. In Section <ref>, we outline the SDE proof for the Brownian case based on a well-known comparison theorem. This SDE proof does not seem to be easily adapted to the discrete setting.

Incidentally, as mentioned earlier, part (ii) of the theorem was stated as a corollary to part (i) in <cit.> by invoking the weak convergence of simple random walk to Brownian motion. While this is certainly correct in spirit, it is the opinion of the authors of this paper that making such an argument rigorous would present various unpleasant technicalities. It may be beneficial and simpler to just have a direct proof in the Brownian case, as we have done here.






§ THE GIRSANOV PROOF


In this section, we present the change-of-measure proof for Brownian motion. As indicated earlier, this proof can be directly adapted, mutatis mutandis, to prove the discrete result as well, and we will discuss this in the next section.

Let us begin by setting up the problem on the canonical sample space. We take Ω to be the continuous path
space, B_t(ω)≜ω_t to be the coordinate process
and F_t to be the natural filtration of B_t. 


Let
ℚ_λ denote the probability measure
over Ω under which B_t becomes a Brownian motion with
drift λ t. Note that ℚ_0 is simply the standard Wiener measure. We also set τ(ω)≜inf{t:B_t(ω)∉(-b,b)}
(b>0 is given fixed). The following form of Girsanov's Theorem is well-known (see for instance <cit.> or <cit.>).


 
For each
t>0, one has 

    dℚ_λ/dℚ_0=e^λ B_t- 1/2λ^2 t   on  F_t   and   dℚ_λ/dℚ_0=e^λ B_τ- 1/2λ^2 τ   on  F_τ.

Furthermore, given two drift values λ_1, λ_2 we have 

    dℚ_λ_2/dℚ_λ_1=e^-(λ_1-λ_2)B_t+λ_1^2-λ_2^2/2t   on  F_t   and   dℚ_λ_2/dℚ_λ_1=e^-(λ_1-λ_2)B_τ+λ_1^2-λ_2^2/2τ   on  F_τ.




The part of the first statement is standard and we omit the proof. The second part of the first statement
follows from the fact that if we set M_t ≜ e^λ B_t- 1/2λ^2 t, then the stopped martingale {M_τ∧ t:n ≥ 0} is bounded and thus uniformly integrable. 

The second statement follows from the first by the chain rule.



(cf. <cit.>]) Under ℚ_λ,
the random variables B_τ and τ are independent. 


The result is obvious if λ=0. For general H_s ≥ 0, we let f,g be arbitrary test functions. Using Lemma <ref> and the independence of B_τ and τ in the λ=0 case (several times), 
one has 

    𝔼^ℚ_λ[f(B_τ)g(τ)]
        =𝔼^ℚ_0[e^λ B_τ- 1/2λ^2 τf(B_τ)g(τ)]=𝔼^ℚ_0[e^λ B_τf(B_τ)]𝔼^ℚ_0[e^-1/2λ^2 τg(τ)]
        =𝔼^ℚ_0[e^λ B_τf(B_τ)]𝔼^ℚ_0[e^-1/2λ^2 τg(τ)] ×𝔼^ℚ_0[e^λ B_τe^- 1/2λ^2 τ]   (the last quantity is just 1)
        =𝔼^ℚ_0[e^λ B_τe^- 1/2λ^2 τf(B_τ)]𝔼^ℚ_0[e^λ B_τe^-1/2λ^2 τg(τ)]=𝔼^ℚ_λ[f(B_τ)]𝔼^ℚ_λ[g(τ)].

The result thus follows. 


We also need the following elementary estimate.

Let X be a random variable, and M any real number. Then
𝔼[X|X⩽ M]⩽𝔼[X].


Let ν be the law of X. Then 

    𝔼[X 1_{X⩽ M}]     =∫_(-∞,M]xν(dx) = (∫_(-∞,M]xν(dx)) (∫_ν(dy)) 
        = ∫_^2x 1_(-∞,M]×(x,y)ν(dx)ν(dy)

and 

    ℙ(X⩽ M)·𝔼[X]     =(∫_(-∞,M]ν(dx))(∫_yν(dy))
        =∫_^2y 1_(-∞,M]×(x,y)ν(dx)ν(dy).

The result follows by noting that 

    ∫_(-∞,M]×(-∞,M]xν(dx)ν(dy)=∫_(-∞,M]×(-∞,M]yν(dx)ν(dy)

and 

    x 1_(-∞,M]×(M,∞)(x,y)⩽ y 1_(-∞,M]×(M,∞)(x,y).



 

Let 0⩽λ_1<λ_2 be given fixed.
The starting observation is that 

    ℚ_λ_2(τ>t)=𝔼_λ_2[ 1_{τ>t}]=𝔼_λ_1[e^-(λ_1-λ_2)B_τ+λ_1^2-λ_2^2/2τ 1_{τ>t}],

by Lemma <ref>. The independence
property given by Lemma <ref> implies that 

    𝔼_λ_1[e^-(λ_1-λ_2)B_τ+λ_1^2-λ_2^2/2τ 1_{τ>t}]
        =𝔼_λ_1[e^-(λ_1-λ_2)B_τ]𝔼_λ_1[e^λ_1^2-λ_2^2/2τ 1_{τ>t}]=𝔼_λ_1[e^λ_1^2-λ_2^2/2τ 1_{τ>t}]/𝔼_λ_1[e^λ_1^2-λ_2^2/2τ]
        = 𝔼_λ_1[e^λ_1^2-λ_2^2/2τ|τ>t]/𝔼_λ_1[e^λ_1^2-λ_2^2/2τ]ℚ_λ_1(τ>t)

It follows that 

    ℚ_λ_2(τ>t)⩽ℚ_λ_1(τ>t)𝔼_λ_1[e^λ_1^2-λ_2^2/2τ|τ>t]⩽𝔼_λ_1[e^λ_1^2-λ_2^2/2τ].

The latter inequality is a direct consequence of Lemma <ref>
since λ_1^2<λ_2^2. 




§ THE DISCRETE CASE
 

We now discuss the proof of Theorem <ref> (i), the discrete case. Let S_n denote unbiased simple random walk with respect to a measure ℚ_1/2; that is, the simple random walk with equal probability of moving to the right and to the left. We need to be able to change measure in order to transform one random walk into another with a different bias. This requires a discrete analog of Lemma <ref>, i.e. Girsanov's Theorem, and for this we need the discrete analog of the exponential martingale e^λ B_t- 1/2λ^2 t. This is contained the following lemma.

 
(i) For p,q ∈ (0,1) with p+q=1, the process M_n = (2√(pq))^n (√(p/q))^S_n is a martingale with respect to ℚ_1/2.


(ii) If we define a new measure ℚ_p by


    dℚ_p/dℚ_1/2= (2√(pq))^n (√(p/q))^S_non  F_n,

then with respect to ℚ_p the process S_n is biased random walk. To be precise, ℚ_p(S_n+1 = r | S_n = r-1) = p and ℚ_p(S_n+1 = r | S_n = r+1) = q. Furthermore,


    dℚ_p/dℚ_1/2= (2√(pq))^σ(√(p/q))^S_σon  F_σ.

(iii) Given two biases p_1, p_2, we have


    dℚ_p_2/dℚ_p_1= (√(p_2q_2/p_1q_1))^n (√(p_2q_1/p_1q_2))^S_n   on  F_n   and   dℚ_p_2/dℚ_p_1= (√(p_2q_2/p_1q_1))^σ(√(p_2q_1/p_1q_2))^S_σ   on  F_σ.





For part (i), we note that S_n can be taken as S_n = ∑_j=1^n X_j, where the X_j's are independent random variables equal to ± 1 with equal probabilities. We note that 


    𝔼_ℚ_1/2[(√(q/p))^X_j] = 1/2(√(q/p) + √(p/q)) = 1/2√(pq).

The process M_n is therefore revealed to be a product martingale with respect to ℚ_1/2.

For part (ii), we note first that part (i) of this lemma was required to show that dℚ_p/dℚ_1/2 defines a genuine change of measure. We may calculate as follows (assuming ℚ_p(S_n = r-1)>0).


    ℚ_p     (S_n+1 = r | S_n = r-1) = 𝔼_ℚ_p[1_{S_n+1 = r} 1_{S_n = r-1}]/𝔼_ℚ_p[1_{S_n = r-1}]
        = 𝔼_ℚ_1/2[(2√(pq))^n+1(√(p/q))^S_n+11_{S_n+1 = r} 1_{S_n = r-1}]/𝔼_ℚ_1/2[(2√(pq))^n (√(p/q))^S_n 1_{S_n = r-1}]
       = (2√(pq))^n+1(√(p/q))^r𝔼_ℚ_1/2[1_{S_n+1 = r} 1_{S_n = r-1}]/(2√(pq))^n(√(p/q))^r-1𝔼_ℚ_1/2[1_{S_n = r-1}]
        = 2p ℚ_1/2(S_n+1 = r | S_n = r-1) = p.


A similar calculation shows ℚ_p(S_n+1 = r | S_n = r+1)=q. The second part of part (ii) holds because M_n is uniformly integrable.

As in the proof of Lemma <ref>, part (iii) follows from (ii) by the chain rule.



 Under ℚ_p,
the random variables S_σ and σ are independent. 



    The proof is identical, line for line, with that of Lemma <ref>, provided one replaces B with S, τ with σ, ℚ_λ with ℚ_p, and the exponential martingale of Brownian motion with the discrete-time martingale M_n from Lemma <ref>.


 We could argue here, as we have done in the previous lemma, that the proof of this result is the same as for part (ii) of this theorem with appropriate substitutions, and with Lemmas <ref> and <ref> in place of Lemmas <ref> and <ref> (Lemma <ref> needs no modification). However, we include the proof here, for completeness.

Let 1/2⩽ p_1<p_2 be given fixed.
We note that 

    ℚ_p_2(σ>n)=𝔼_p_2[ 1_{σ>n}]=𝔼_p_1[(√(p_2q_2/p_1q_1))^σ(√(p_2q_1/p_1q_2))^S_σ 1_{σ>n}],

by Lemma <ref>. The independence
property given by Lemma <ref> implies that 

    𝔼_p_1[(√(p_2q_2/p_1q_1))^σ(√(p_2q_1/p_1q_2))^S_σ 1_{σ>n}]
        =𝔼_p_1[(√(p_2q_1/p_1q_2))^S_σ]𝔼_p_1[(√(p_2q_2/p_1q_1))^σ 1_{σ>n}]=𝔼_p_1[(√(p_2q_2/p_1q_1))^σ 1_{σ>n}]/𝔼_p_1[(√(p_2q_2/p_1q_1))^σ]
        = 𝔼_p_1[(√(p_2q_2/p_1q_1))^σ|σ>n]/𝔼_p_1[(√(p_2q_2/p_1q_1))^σ]ℚ_p_1(σ>n)

It follows that 

    ℚ_p_2(σ>n)⩽ℚ_p_1(σ>n)𝔼_p_1[(√(p_2q_2/p_1q_1))^σ|σ>n]⩽𝔼_p_1[(√(p_2q_2/p_1q_1))^σ].

Note that p_2q_2/p_1q_1 < 1, since p_2 > p_1. Lemma <ref>
 therefore completes the proof. 

 

 


§ THE SDE PROOF


In this section, we outline the SDE proof of Theorem <ref> (ii).

A natural idea is to try to represent the modulus of B^λ_t ≜ B_t+λ t as an Itô process, namely the solution to an SDE of the form


    dX_t=σ(X_t)dB_t+b(X_t)dt.
 The reason for doing this is that it allows us to invoke a well-known comparison theorem due to Ikeda and Watanabe <cit.>. This theorem states essentially that if X^1 and X^2 are diffusions on the same probability space, starting at the same value and each satisfying equations of the form (<ref>) with the same σ but with different drift terms b_1,b_2 with b_1(x) ≤ b_2(x) for all x, then X^1_t ≤ X^2_t for all t a.s. (naturally there are conditions on σ and the b's, but we do not worry about them for the moment). This allows us to deduce the result, as the exit time from the symmetric interval in question is simply the hitting time of the corresponding level by the process |B^λ_t|.

We must first remark, however, that the process |B_t^λ| does not satisfy
an SDE of the form (<ref>). This is because the semimartingale
decomposition of |B_t^λ| involves its local time at zero,
which is not absolutely continuous with respect to dt (when λ=0,
Tanaka's formula gives |B_t|=dW_t+dL_t where W_t is
a Brownian motion and L_t is the local time of B at zero). 

On the other hand, the process Y_t ≜ |B_t^λ|^2=(B_t^λ)^2
does satisfy an SDE of the required form, which we shall derive below. The case when λ=0
is straightforward; here Y_t = B_t^2, and one has 

    dY_t=2B_tdB_t+dt=2√(Y_t)·B_t/√(Y_t)dB_t+dt=2√(Y_t)dW_t+dt,

where W_t≜∫_0^tB_t/√(ρ_t)dB_t
is a Brownian motion by Lévy's characterisation. The case λ≠0 requires
extra care since the same calculation does not lead to a meaningful
SDE for Y_t (the drift part contains B_t^λ which
cannot be expressed in terms of Y_t). 

We take a Markovian perspective to write down the intrinsic SDE for
Y_t. It is well known that the process |B_t| can be equivalently
viewed as a Markov process with generator 

    Af=1/2f”;  D( A)={ f∈ C_b^2([0,∞)):f'(0+)=0}.

To derive the generator for the process X_t≜|B_t^λ|,
let x>0 be given fixed (it represents the current state X_t=x).
Explicit calculation shows that 

    ℙ(B_t^λ=x|X_t=x)=e^λ x/e^λ x+e^-λ x, ℙ(B_t^λ=-x|X_t=x)=e^-λ x/e^λ x+e^-λ x.

Given X_t=x, if B_t^λ=x the process evolves like
B_s+λ s, while if B_t^λ=-x the process evolves
like -B_s-λ s (s∈[t,t+δ t]). Since the Brownian
motion B is symmetric, it is clear that the diffusive part (the
second order term) of the generator of X_t is also 1/2d^2/dx^2.
Its drift part (the first order term) is given by 


    ℙ(B_t^λ=x|X_t=x)×λ+ℙ(B_t^λ=-x|X_t=x)×(-λ)=λtanhλ x.

In other words, X_t is a Markov process with generator

    L^X=1/2d^2/dx^2+λtanh(λ x)d/dx;  D( L^X)={ f∈ C_b^2([0,∞)):f'(0+)=0}.

A simple change of variables x=√(y) shows that the generator
of Y_t is given by 

    L^Y=2yd^2/dy^2+(1+2λ√(y)tanh(λ√(y)))d/dy,

and the corresponding SDE for Y_t is 

    dY_t=2√(Y_t)dW_t+(1+2λ√(Y_t)tanh(λ√(Y_t)))dt.


Note that this agrees with (<ref>) when λ=0. We now observe that the function

    λ→ 1+2λ√(y)tanh(λ√(y))


is increasing for λ∈ [0,∞) for all y,
and therefore the comparison theorem of Ikeda-Watanabe easily gives our desired result. 




Remarks: Invoking the theorem of Ikeda-Watanabe requires checking certain conditions on the functions comprising the SDE (<ref>). These conditions are slightly technical to state but straightforward to verify for the SDE in question, so we have chosen not to include them. 

Unlike X_t itself, the process Y_t is indeed an Itô diffusion (it does satisfy the SDE (<ref>)) because the local time term will not appear. In fact, in the decomposition

    dY_t=2X_tdX_t+dX_t· dX_t,

the local time term X_tdL_t (coming from the first term in
(<ref>)) vanishes identically due to the fact that L_t
only increases when X_t=0 (this is a basic property of the local
time). 

From general theory, the SDE (<ref>) has a unique strong
solution for every initial condition Y_0=y⩾0. One can
then show that its solution (with Y_0=0) has the same distribution
as the process |B_t+λ t|^2. This can be taken as a more
direct approach to justify the above considerations. 




§ CONCLUDING REMARKS


Theorem <ref> seems suprisingly resistant to generalizations. For instance, we believe that the following should be true.


(i) [Random walk] Let P = { p_n }_n=1^∞ and P̂ = {p̂_n}_n=1^∞, where p, p̂ are sequences such that 1/2 ≤ p_n ≤p̂_n<1 for all n. Let ℚ_P denote the measure such that S_n becomes a time-inhomogeneous Markov process with transitional probabilities 
    ℚ_P(S_n+1 = r | S_n = r-1) = p_n,
    ℚ_P(S_n+1 = r | S_n = r+1) = q_n,
 and similarly for ℚ_P̂. Let σ be defined as before. Then ℚ_P(σ > n) ≥ℚ_P̂(σ > n) for all n ∈ℕ.

(ii) [Brownian motion] Let B_t denote
the standard one-dimensional Brownian motion. Given suitable functions λ(t), λ̂(t) with 0 ≤λ(t) ≤λ̂(t)
and b>0, define 

    τ_λ^b≜inf{t:B_t+λ (t)∉(-b,b)},

and similarly for τ_λ̂^b.
Then for any fixed t>0, we have 

    ℙ(τ_λ^b>t) ≥ℙ(τ_λ̂^b>t).



The two parts of the conjecture are stated in somewhat different ways, the first in terms of probabilities while the second defined pathwisely. However, in spirit they are the same (and could be reformulated to match): in both cases we are dealing with a process that has a positive drift, but the drift is allowed to vary in time. 

Remarkably, neither of the two approaches we presented here can be used to prove this conjecture, at least not without some significant new ideas brought in. The SDE approach to the second part of the conjecture seems difficult, as deriving the required SDE is a significant obstacle. The change of measure approach seems like it should work, as it is not difficult to change measure in either case to change the symmetric process (simple random walk or Brownian motion) into the drifted one (see for instance <cit.>). However, what is lost is Lemmas <ref> and <ref>, the independence of the exit distribution and exit time. These facts, quite remarkable in their own way, seem only to hold for constant drifts.

2 
(Xi: I would probably remove this paragraph because it is a bit arbitrary and not precise.) The intuition behind Lemmas <ref> and <ref> is a bit mysterious. It would seem that positive drift would mean an exit to the right would happen faster than to the left, and this would violate independence, but on the other hand paths would be unlikely to exit to the left after a long time because the drift would eventually take over and push the process to the right. We will therefore get some longer exits to the right. Apparently these influences essentially cancel, and we achieve independence. It would be nice if a more concrete interpretation for this fact could be developed.


Returning to the question of generalizing Theorem <ref>, one might also try allowing the drift to depend on the position of the path. However, in this case we believe that the conjecture would be false, since we could make a large positive drift when the process goes negative, and this would push it back to the center, probably resulting in longer stays in the interval.



§ ACKNOWLEDGEMENTS


XG gratefully acknowledges the ARC support DE210101352. Both authors are thankful to Nathan Ross for valuable discussions and suggestions on the problem. 

plainurl




 Paper Title: Spectral gap for random Schottky surfaces.


Å𝔸
𝔹
 
ℂ
 
𝔻
 
𝔼
 
𝔽
 
𝔾
 
ℍ
 
𝕀

𝕁
 
𝕂
 
𝕃
 
𝕄
 
ℕ
 
𝕆
 
ℙ
 
ℚ
 
ℝ
 
𝕊
 
𝕋
 
𝕌
 
𝕍
 
𝕎
 
𝕏
 
𝕐
 
ℤ

𝒜
 
ℬ
 
𝒞
 
𝒟
ℰ
ℱ
 
𝒢
ℋ
 
ℐ
 
𝒥
 

ℒ
 
ℳ
 
𝒩
 
𝒪
𝒫
 
𝒬
 
ℛ
𝒮
 
𝒯
 

 
𝒱
 
𝒲
 
𝒳
 
𝒴
𝒵
 

𝒜
ℬ
 
𝒞
 
𝒟
 
ℰ
 
ℱ
𝒢
 
ℋ
 
ℐ
 
𝒥
 
𝒦
ℒ
 
ℳ
 
𝒩
 
𝒪
 
𝒫
𝒬
 
ℛ
 
𝒮
 
𝒯
 
𝒰
𝒱
 
𝒲
 
𝒳
 
𝒴
 
𝒵

𝐀
 
𝐁
 
𝐂
 
𝐃
 
𝐄
 
𝐅
 
𝐆
 
𝐇
 
𝐈
 
𝐉
 
𝐊
 
𝐋
 
𝐍
 
𝐀
 
𝐎
 
𝐏
 
𝐐
 
𝐑
 
𝐒
 
𝐓
 
𝐔
 
𝐕
 
𝐗
 
𝐘
 
𝐙

𝙰
 
𝙱
 
𝙲
 
𝙳
 
𝙴
 
𝙵
 
𝙶
 
𝙷
 
𝙸
 
𝙹
 
𝙺
 
𝙻
 
𝙼
 
𝙽
 
𝙾
 
𝙿
 
𝚀
 
𝚁
 
𝚂
 
𝚃
 
𝚄
 
𝚅
 
𝚆
 
𝚇
 
𝚈
 
𝚉
 

ℍ^2


PSL

Ø

PSO
 
PO

 d

ε

^-1
 
tr 
Im 
 
Re 
 
cl 

#1||#1||
 
#1#1_tr
 
#1#1_HS
 
#1#1_op

#1#2⟨#1,#2⟩

δ_Γ
 
≪_Γ
 
≫_Γ
 
≍_Γ
 
#1Υ_#1

𝒲
𝒲^∘
 
#1#1
#1S(#1)
 
#1E(#1)
 
#1|#1|_w
 
#1#1

def
=



§ ABSTRACT 
We establish a spectral gap for resonances of the Laplacian of random
Schottky surfaces which is optimal, according to a conjecture of Jakobson
and Naud.


Spectral gap for random Schottky surfaces
    Irving Calderón, Michael Magee and Frédéric Naud
    August 5, 2024
====================================================





§ INTRODUCTION


This paper addresses the question of whether typical hyperbolic Schottky
surfaces have almost optimal spectral gaps. By hyperbolic Schottky
surface we mean an infinite area Riemannian surface X which is
connected, orientable, complete, of constant curvature -1, with
finitely generated fundamental group and without cusps. The spectral
gap refers either to a gap in the spectrum of the Laplace-Beltrami
operator Δ_X or, more generally, to a region in the complex
plane where there are no resonances of X. 

Let us explain the background on the spectral theory of Schottky surfaces
needed for this work[Borthwick's book <cit.> is a good reference
for the spectral theory of infinite area hyperbolic surfaces.]. One can obtain X as a quotient Γ\ of the
hyperbolic plane  by a Γ discrete, finitely generated,
non-abelian free subgroup of (2,). We will consider a basis
={γ_1,…,γ_N} of Γ having the geometric
description presented in Section <ref>.
Any Γ-orbit in  accumulates on a compact subset Λ_Γ
of ∂ called the limit set of Γ. We denote
by  the Hausdorff dimension of Λ_Γ. The assumption
that Γ is not abelian and X has infinite area implies that
∈(0,1).

We consider Δ_X as an unbounded operator on L^2(X). It is essentially self-adjoint,
positive semidefinite. By results of Lax-Phillips in
<cit.>, the continuous spectrum of Δ_X
is the interval [1/4,∞), and the discrete
spectrum consists of finitely many eigenvalues—counted with multiplicity—in
(0,1/4). A result of Patterson in <cit.>
says that when >1/2, the bottom of the spectrum of
Δ_X is the simple eigenvalue λ_0(X)(1-).
In this case, we define the L^2-spectral gap of X as λ_1(X)-λ_0(X),
where λ_1(X) is the minimum element of the spectrum other
than λ_0(X). This definition is not well suited for the
case ≤1/2, since the spectrum of Δ_X
is precisely [1/4,∞). For this reason, we will work
with the following more general notion of spectral gap, which applies
regardless of the value of : The resolvent operator

    R_X(s)(Δ_X-s(1-s))^-1:C_c^∞(X)→ C^∞(X)

is initially defined on the halfplane { s∈| s>1/2}.
Mazzeo and Melrose show in <cit.> that R_X
has a  meromorphic continuation to the entire complex plane. See also the papers of Guillopé and Zworski <cit.>
for the more specific case of surfaces.

The poles
of this family of operators are called resonances of X and form a discrete subset of . Resonances are the natural replacement data to the missing L^2
eigenvalues in non-compact situations. They corresponds to non-L^2 eigenfunctions satisfying an outgoing condition at infinity, see <cit.> for details.
Let _X be the set of resonances of X. The multiplicity
of a resonance s is defined by rank(∫__sR_X),
where _s is a circle enclosing s and no other resonance
of X, with anticlockwise orientation. Resonances s with s>1/2
yield L^2-eigenvalues s(1-s) of Δ_X with the same multiplicity.
Hence the gaps in the discrete spectrum of Δ_X can be thought
as regions of  without resonances. This is the point of view
we adopt. Here are some important facts about resonances: 
is the only resonance with real part ≥ and it is simple.
Moreover, there is always an >0 such that

    _X∩{s∈| s>-}={}.

This follows from <cit.> when >1/2
and from <cit.> when ≤1/2.
We define the spectral gap of X as the maximum value (Γ)
of such . On the one hand, it was shown in <cit.>
that (Γ) can be arbitrarily small. On the other hand,
a conjecture of Jakobson and Naud <cit.>
predicts, among other things, that for any ϵ>0 there are
infinitely many resonances of X with s>/2-ϵ,
so (Γ)≤/2. It is a pressing question as
to whether a typical X has spectral gap close to this conjectural
optimal value. This question has famous analogs in graph theory that
discuss below.

We address this question from a probabilistic perspective, using the
model of hyperbolic Schottky surfaces introduced in <cit.>:
consider a Schottky surface X=Γ\ and let ϕ_n
be a random homomorphism Γ→ S_n with uniform distribution.
We obtain a random diagonal action of Γ on ^2×{1,2…,n},
by setting for all γ∈Γ,

    γ.(z,j):=(γ(z),ϕ_n(γ)(j)).

The quotient 

    X_nΓ\(^2×{1,2…,n})

is then a random n-sheeted Riemannian covering of X. Note that if Y is a
finite cover of X, the value of δ is the same for both
surfaces and any resonance of X is a resonance of Y with multiplicity
at least as large as that of X. Hence (Y) is at most (X).
The main theorem of this paper is the following.

Let X be a nonelementary, hyperbolic
Schottky surface and let X_n be a random n-sheeted covering
of X with uniform distribution. For any compact  contained
in the halfplane {s∈| s>/2}, the probability
that X and X_n have the same resonances in  with the
same multiplicities tends to 1 as n→∞.


This theorem says that a typical finite covering of X has the maximal
possible spectral gap provided we restrict to resonances of
bounded frequency (imaginary part). A weaker version of Theorem
<ref> with /2 replaced
by 3/4 was proved in <cit.>.

Theorem <ref> together with <cit.>
implies the following corollary on L^2 eigenvalues.

Let Γ be a Fuchsian Schottky group such that >1/2
and the surface[In this case X is either a pair of pants with three funnels or
a torus with one funnel.] X=Γ\ has Euler characteristic -1. Let X_n
be a uniformly random degree n covering space of X. With probability
tending to one as n→∞, the only eigenvalue of Δ_X_n
is (1-).


The proof of our spectral gap for random Schottky surfaces relies
on the next result of independent interest. We explain in Section
<ref> how these statements are related. See
Section <ref> for the definition of
the transfer operators _s,ρ_j.

Let Γ be a Fuchsian Schottky
group. Suppose that (
is a sequence of random, finite dimensional unitary representations
of Γ that a.a.s. strongly converge to the left regular representation
(ρ_Γ,ℓ^2(Γ)) in the sense of Section <ref>.
Then, for any compact subset  of the halfplane {s∈| s>/2}
there is an integer ℓ=ℓ(,Γ)>0 such that


    lim_j→∞(_s,ρ_j^ℓ<1 for all s∈)=1.




Following the proof of Theorem <ref> one can
see that if we assume that (ρ_j)_j≥1 a.s. strongly converge
to ρ_Γ, then there is an ℓ=ℓ(,Γ) such
that

    ∑_j=1^∞[1-(_s,ρ_j^ℓ<1 for all )s∈)]<∞.



Theorem <ref> shows that the process of `inducing
strong convergence', first uncovered in <cit.>
for finite-area surfaces, is also valid in the infinite volume Schottky
setting in a stronger sense than that of L^2-eigenvalues.



 §.§ Spectral gaps of random graphs


We now explain the analogy with random graphs—this is also discussed
in detail in the introduction of <cit.>.
We need some notation: If G is a graph with n vertices, we denote
λ_0(G)≥λ_1(G)≥⋯≥λ_n-1(G)
the eigenvalues of its adjacency operator. Note that λ_0(G)=d
when G is d-regular. Let _d,n be a random d-regular
graph of size n with uniform distribution. A celebrated conjecture
of Alon <cit.> predicted that for any ϵ>0,
λ_1(_d,n)<2√(d-1)+ϵ with probability
tending to one as n→∞. The number 2√(d-1) is relevant
for two reasons: it is the spectral radius of the adjacency operator
of the d-regular tree—which is the universal cover of any d-regular
graph—and also, a result of Alon-Boppana <cit.>
says that for any d-regular graph on n vertices G_d,n, 


    λ_1(G_d,n)≥2√(d-1)-o(1)  as n→∞.

Hence one cannot replace 2√(d-1) by a smaller number in Alon's
conjeture. This means that, for d-regular graphs, 2√(d-1)
plays the role of /2 for Schottky surfaces. Even though
the Alon-Boppana bound is not very hard to prove for graphs, we point out that the analog conjecture
of Jakobson-Naud for Schottky surfaces has not been established yet and seems much harder to reach. Alon's
conjecture was proved by Friedman in <cit.>.
See also <cit.> for a new proof and <cit.>
for a proof of a slightly weaker result.

Friedman conjectured in <cit.> a variant of
Alon's conjecture for random covering spaces of degree n[In graph-theoretic literature, these are called n-lifts.]
of any initial finite graph G provided that:


  * One replaces 2√(d-1) by the spectral radius r_G
of the adjacency operator of the universal cover of G;

  * One allows eigenvalues of the adjacency operator that already belonged
to G (as one must).

Friedman's conjecture was proved in the breakthrough work <cit.>
of Bordenave and Collins. In fact, Bordenave-Collins proved a vast
generalization of Friedman's conjecture where one twists a random
Hecke operator, formed from random permutations, by any fixed finite
dimensional matrices, assuming the matrices satisfy a symmetry condition
that forces the resulting operator to be self-adjoint. In fact, the
<cit.> is a vital ingredient
in the proof of our spectral gap for random Schottky surfaces, as
is explained in Section <ref>. 



 §.§ Some related results


The first result on spectral gap of random hyperbolic surfaces is
due to Brooks and Makover <cit.> who prove,
for a combinatorial model of random closed hyperbolic surfaces depending
on a parameter n that influences the genus (non-deterministically),
that there exists a constant C>0 such that λ_1≥ C
with probability tending to one as n→∞, where λ_1
is the first non-zero eigenvalue of the Laplacian of the surface.
Mirzakhani proved in <cit.> that for a Weil-Petersson
random closed hyperbolic surface of genus g, λ_1≥0.0024
with probability tending to one as g→∞. In <cit.>,
the second and third named authors of the current work proved that
Theorem <ref> holds with /2
replaced by 3/4.

Returning to closed surfaces, by building on <cit.>,
Magee-Naud-Puder proved in <cit.>
that for a uniformly random degree n cover Y_n of a fixed
closed hyperbolic surface Y and all ϵ>0, the probability
that Y_n has no new eigenvalues below 3/16-ϵ
tends to 1 as n→∞. This result was adapted to Weil-Petersson
random surfaces independently by Wu-Xue <cit.>
and Lipnowski-Wright <cit.>;
here the corresponding statement is that there are no eigenvalues
between 0 and 3/16-ϵ. When it comes solely
to L^2 eigenvalues, these `3/16' results are at the
strength of the main result of <cit.> giving
resonance-free regions in terms of 3/4: for compact
surfaces, =1 and 3/4(1-3/4)=3/16.
On the other hand, closed surfaces involve additional difficulties
due to their non-free fundamental groups.

Some works on Weil-Petersson random surfaces, like <cit.>,
do not imply spectral gaps but offer instead spectral delocalization
results .

Uniform spectral gaps for deterministic covering spaces of infinite
area hyperbolic surfaces has also been of interest in number theoretic
settings; see <cit.>
for a selection of results. The quantitative results here are those
by Gamburd and Calderón-Magee. Much of the motivation of these works
came from the `thin groups' research program; see Sarnak's article
<cit.> for an overview. 

Another closely related concept is that of essential spectral
gap. We say that t∈ is an essential spectral gap of a hyperbolic
Schottky surface X=Γ\ if the half-plane {s∈| s>t}
has finitely many resonances of X. Two important results here are
due to Bourgain and Dyatlov: The first, proved in <cit.>,
says that there is ϵ>0, depending only on ,
such that -ϵ is an essential spectral gap of X.
This result is relevant if ≤1/2. On the other hand, it
is proved in <cit.> that
X has an essential spectral gap 1/2-η for some η=η(Γ)>0.
This result is relevant if >1/2. One can reformulate the conjecture
of Jakobson-Naud <cit.> mentioned
earlier by saying that the optimal essential spectral gap of X
is /2+ for any ε>0. We do not know
yet if our probabilistic techniques can be used to address high frequency
problems (i.e. resonances with large imaginary parts) and have not
attempted to do so in the present paper. However, our main result
says that this conjecture holds in the bounded frequency, large cover
regime For a broader perspective on resonances of hyperbolic surfaces
than we are able to offer here, we recommend Zworski's survey article
<cit.>.



 §.§ Overview of the main proof and structure of the article


The proof of our main result involves tools from three areas: thermodynamic
formalism, random matrices and operator algebras. Let us highlight
the main steps.

We start by recalling how one can study the resonances of deterministic
coverings of a Schottky surface with tools from the thermodynamic
formalism. Consider first a single Schottky surface X=Γ\.
Building on the symbolic coding of the—recurrent part of—geodesic
flow of X, the thermodynamic formalism attaches to Γ a
family (_Γ,s)_s∈ of transfer operators.
These act on -valued functions of a union [See Section <ref> for the definition
of .] of disks in . By choosing an appropriate functional space,
the (_Γ,s)_s∈ are trace-class operators on an
infinite dimensional Hilbert space _Γ, in particular
compact. For us, the key link between resonances and transfer operators
is the following: The resonances of X on the halfplane { s>0}
coincide with the parameters s for which 1 is an eigenvalue of
_Γ,s. Moreover, the multiplicities are the same.

Consider now the cover X_ϕ of X associated to a homomorphism
ϕ:Γ→ S_n, and let ρ_0 be the corresponding
unitary representation of Γ on V_n^0. Any resonance
s of X, say of multiplicity m_X(s), is also a resonance
of X_ϕ and m_X_ϕ(s)≥ m_X(s). We say that s
is a new resonance of X_ϕ if m_X_ϕ(s)>m_X(s)[This definition includes the case where m_X(s)=0 or, in other
words, s is not a resonance of X]. Twisting the classical transfer operators by ρ_0, we obtain
a family (_s,ρ_0)_s∈ of trace-class operators,
on a Hilbert space _Γ,n of maps → V_n^0,
that detect new resonances. Namely, on the halfplane { s>0},
s_0 is a new resonance of X_ϕ if and only if 1 is
an eigenvalue of _s_0,ρ_0. There is a natural identification
of the space _Γ,n with _Γ⊗ V_n^0.
Under it, the powers of the transfer operators take the form


    _s,ρ_0^ℓ=∑_γ∈Γ_ℓT_γ,s⊗ρ_0(γ),

where the sum is over the set Γ_ℓ of elements of Γ
of word length ℓ and the T_γ,s are certain trace-class
operators of _Γ. To understand why random matrices enter
naturally into the proof, it is useful to think of (<ref>)
as saying that _s,ρ_0^ℓ is a noncommutative polynomial
with operator coefficients in the variables x_γρ_0(γ),γ∈Γ_1.

Let us now add the randomness to the picture: Consider the random,
degree n cover X_n of X associated to a uniformly random
homomorphism ϕ_n:Γ→ S_n. We now have a random unitary
representation (ρ_n^0,V_n^0) of Γ and random
transfer operators _s,ρ_n^0. To obtain the “polynomial
expression” of _s,ρ_n^0^ℓ one replaces ρ_0
by ρ_n^0 in (<ref>); the
coefficients T_γ,s remain the same. Our strategy to show
that, most likely, a single s_0 is not a new resonance of X_n
is finding an integer ℓ>0 such that _s,ρ_n^0^ℓ<1
with high probability. We achieve this is by feeding into Theorem
<ref> the main theorem of <cit.>,
a deep result on random matrices—which is also a vital tool in
<cit.>—saying that the random representations
ρ_n^0 almost surely tend, in a suitable sense, to the left
regular representation ρ_Γ of Γ. This implies,
from the “polynomial expressions” of _s,ρ_n^0^ℓ,
that _s_0,ρ_n^0^ℓ tends in probability,
as n→∞, to the norm of the deterministic operator _s_0,ρ_Γ^ℓ
on _Γ⊗ℓ^2(Γ), defined as in (<ref>),
and which we denote by _ℓ,s_0 in the article. 

Applying an upper bound of Buchholz for the norm of an operator of
the form

    ∑_γ∈Γ_ℓT_γ⊗ρ_Γ(γ),

we show that whenever s_0>/2, _ℓ,s_0→0
as ℓ→∞ with a rate of convergence. Hence _s_0,ρ_n^0^ℓ<1
with high probability for any big enough ℓ. To improve the result
from a single s_0 to s_0 in a fixed compact subset of the
halfplane { s>/2}, we apply a rough estimate of the dependence
of the transfer operators on the parameter s. 

The rest of the article is divided into five sections. Section <ref>
covers the background on hyperbolic Schottky surfaces needed for our
main result: a geometric definition of Fuchsian Schottky group, the
model we use of random Schottky surfaces and the link between transfer
operators and resonances. Section <ref>
gathers several technical estimates that we use later in the article.
The goal of Section <ref> is to prove Theorem
<ref>: we define strong convergence, we justify
why the norm of the random transfer operators of X_n are controlled
by that of the limit operators _ℓ,s and we state our main
norm bound for the limit operators. This last bound is Proposition
<ref>, and is proved in Section <ref>.
Finally, in Section <ref> we state the
result of Bordenave-Collins and we complete the proof of Theorem <ref>.



§ PRELIMINARIES


This section contains the background on Schottky groups and surfaces
needed for this work: the basic definitions and notation, the description
of the model of random Schottky surfaces we consider, and the link
between resonances and transfer operators.



 §.§ Fuchsian Schottky groups


Here we give a geometric definition of Fuchsian Schottky groups, we
recall their basic properties and fix the notation we use to work
with them. 

We consider the action of (2,) by Möbius transformations
on the extended complex plane =∪{∞} given
by 

    [ a b; c d ]z=az+b/cz+d.

This action preserves the upper halfplane 

    ={z∈| z>0},

which we endow with the Riemannian metric dx^2+dy^2/y^2
in coordinates z=x+iy—this is the upper halfplane model of the
hyperbolic plane—. Moreover, (2,)↷
is isometric, and it identifies the group of orientation preserving
isometries of  with (2,). If Γ is a discrete
subgroup of (2,) without torsion, the quotient Γ\
is a complete, connected and orientable hyperbolic surface[By hyperbolic surface we mean a Riemannian surface of constant curvature
-1.]. Conversely, any such surface can be represented as Γ\,
for some Γ as before.

The focus of this work are the hyperbolic surfaces associated to Fuchsian
Schottky groups, which are defined as follows: Consider an integer
N≥2 and let ={1,…,2N}. If a∈,
we denote a the unique element of  such that a≡ a+N2N.
Consider open disks (D_a)_a∈ in  centered in 
and elements (γ_a)_a∈ of (2,) such that
γ_ a=γ_a^-1, and verifying the following
conditions: 


  * The closures of the (D_a)_a∈ are pairwise disjoint;

  * The image of \ D_a under γ_a
is the closure of D_a.

We write  for the union ∪_a∈D_a, and I_a
for the open interval D_a∩. The group Γ generated
by (γ_a)_a∈ is free with basis γ_1,…,γ_N
by Klein's Ping-Pong Lemma, since the (D_a)_a∈ are disjoint.
It is also a discrete subgroup of (2,) since -
is a fundamental domain of Γ↷,
and condition (1) above guarantees that Γ is convex cocompact. 

For us, a Fuchsian Schottky group is a subgroup of (2,)
admitting a description as the Γ above. As proved by Button
in <cit.>, any convex cocompact, non cocompact Fuchsian
group is a Fuchsian Schottky group.

Let Γ be a Fuchsian Schottky group. We denote by N_Γ
the rank of Γ. We assume always that Γ comes equipped
with a fixed choice of Schottky data (D_a)_a∈ and (γ_a)_a∈.
Let ={γ_1,…,γ_N_Γ}. From now on
we identify each a∈ with the corresponding γ_a.
So,  is a basis of Γ and =⊔. Let
· be the word length on Γ with respect to
. We respectively denote Γ_n and Γ_≥ n
the sets of γ∈Γ of word length =n and ≥ n.
The notation γ_1→γ_2 means that γ_1γ_2=γ_1+γ_2. 

Consider γ∈Γ_n with n≥1 and its expression
a_1⋯ a_n with the a_j∈. Sometimes we refer to
the a_j as the letters of γ, and to γ
as a word in the alphabet . The first and last letters
a_1 and a_n of γ will be respectively denoted γ
and γ, while γ stands for the word
a_1⋯ a_n-1 obtained by erasing the last letter of γ.
We associate to each γ an open disk and an interval:

    D_γ=γD_γ,   I_γ=D_γ∩.

We denote by γ the length of I_γ.



 §.§ Random covering spaces


Here we recall the model of random Schottky surface we work with,
which was introduced in <cit.>.

Consider a Fuchsian Schottky group Γ, the corresponding hyperbolic
surface X=Γ\ and let S_n be the group of
permutations of [n]{1,2,…,n}. For any homomorphism
ϕ:Γ→ S_n, we consider the diagonal action γ(z,j)=(γ z,ϕ(γ)j)
of Γ on ×[n]. The quotient

    X_ϕΓ\(×[n])

is a hyperbolic surface, and the projection ×[n]→
induces an n-sheeted covering map X_ϕ→ X. We denote by
V_n the Hilbert space ℓ^2([n])[This is the Hilbert space of maps [n]→ with the inner product
fg=∑_jf(j)g(j).], and V_n^0 the subspace of maps f:[n]→ such that ∑_j∈[n]f(j)=0.
Note that ϕ induces unitary representations of Γ on
V_n and V_n^0.

Since a homomorphism Γ→ S_n is completely determined by
its restriction to the basis , Hom(Γ,S_n)
is finite and hence we can talk about the uniform probability measure
on it. Throughout the article, ϕ_n is a uniformly random
homomorphism Γ→ S_n—in other words, a random variable
on Hom(Γ,S_n) with uniform distribution—and
X_n is the corresponding random n-sheeted covering space of
X. We write ρ_n and ρ_n^0 for the random unitary
representations of Γ on V_n and V_n^0 coming from
ϕ_n.



 §.§ Conventions and notation for Hilbert spaces


Here fix the notation and conventions on Hilbert spaces we use throughout
the article. 

All the Hilbert spaces we consider are assumed to be complex. Let
V and W be Hilbert spaces. We denote ··_V
the inner product of V—with the convention that it is -linear
in the first coordinate and -antilinear in the second.—,
and ·_V is the norm coming from the inner product.Let
B(V) be the group of bounded linear operators on V. The unitary
group U(V) of V is the group of operators on V preserving
··_V. We denote by T the operator
norm of any linear map T:V→ W with respect to the norms of the
inner products. When V=W, sometimes we write T_V instead
of T to emphasize the space on which T acts. We write
T^* for the adjoint of T. The tensor product V⊗ W—suitably
completed—is also a Hilbert space with the inner product


    v_1⊗ w_1v_2⊗ w_2_V⊗ Wv_1v_2_Vw_1w_2_W.

Recall that a unitary representation of a group Γ on V
is a homomorphism ρ:Γ→ U(V). We often write the unitary
representations as a pair (ρ,V).

For any r≥1, we consider ^r as Hilbert space with the
standard inner product 

    (z_1,…,z_r)(w_1,…,w_r)=∑_jz_jw_j.

We usually write M_r() instead of B(^r).



 §.§ Transfer operators and their relation to resonances


Here we introduce transfer operators, the main tool we use to detect
resonances of Schottky surfaces. These are a family (_s,ρ)_s∈
of operators associated to any unitary representation (ρ,V)
of a Fuchsian Schottky group Γ, acting on a certain space
_ρ of functions → V. The relation between transfer
operators and resonances we use to prove our main result is stated
in Proposition <ref>: when ρ
comes from a homomorphism ϕ:Γ→ S_n, we will use the
_s,ρ to study the resonances of X_ϕ. In order to
do so, one has to choose a functional space _ρ where the
transfer operators behave well. Like in <cit.>,
our choice of _ρ is a Bergman space. We give also the “polynomial
expression” of the powers of _s,ρ with operator coefficients,
which is key for the proof of our main result. 

Let Γ be a Fuchsian Schottky group and let V be a Hilbert
space of finite dimension. For any unitary representation (ρ,V)
of Γ and all s∈, we define the operator _s,ρ
on measurable functions f:→ V by 

    _s,ρf(z_a)∑_w∈Γ_2
     w=a
     w'(z_a)^sρ( w)f( wz_a).

We clarify two points about this definition: the subindex in z_a
means that z_a belongs to D_a, and w'(z_a)^sexp(sτ( w'(z_a))),
where τ is the branch of the logarithm

    -(-∞,0]→⊕ i(-π,π).


The Bergman space (,V) is the Hilbert space of square integrable[With respect to the Lebesgue measure of .],
holomorphic maps → V with the inner product

    f_1f_2∫_f_1(z)f_2(z)_V z.

Note that (,V) is stable under _s,ρ. In this
work, we take (,V) as the domain of the transfer operators.
This guarantees that _s,ρ is trace-class for any s∈—see
<cit.>—, so compact in
particular.

Here is the relation between resonances and transfer operators that
we use to establish our main result. It follows from the correspondence
between resonances of a hyperbolic Schottky surface and the zeros
of its Selberg zeta function—see <cit.>—,
and the factorization <cit.>
of the Selberg zeta function of X_ϕ.

Let ϕ be a homomorphism
from a Fuchsian Schottky group Γ to S_n, and let ρ_0
be the corresponding unitary representation of Γ on V_n^0.
If s∈ is a resonance of X_ϕ with greater multiplicity
than in X, then 1 is an eigenvalue of _s,ρ_0.


Now we give a convenient formula for the powers of _s,ρ,
and then the “polynomial expression” of _s,ρ^ℓ
with operator coefficients. 

We claim that for any integer ℓ>0 and all f∈(,V)
we have

    _s,ρ^ℓf(z_a)=∑_w∈Γ_ℓ+1
     w=a
     w'(z_a)^sρ( w)f( wz_a).

Indeed, consider w∈Γ_ℓ+1, say w=a_1⋯ a_ℓ+1
with a_j∈. We denote a_ja_j+1⋯ a_ℓ by w_j→.
For any z∈ D_a_ℓ+1, let

    θ(w,z)∑_j=1^ℓτ(a_j'(w_j+1→z)).

Iterating (<ref>) one obtains

    _s,ρ^ℓf(z_a)=∑_w∈Γ_ℓ+1
     w=a
    exp(sθ(w,z_a))ρ( w)f( wz_a),

for any a∈. When z_a is real, the terms a_j'(w_j+1→z_a)
in (<ref>) are real as well, so

    θ(w,z_a)=τ(∏_j=1^ℓ-1a_j'(w_j+1→z_a))=τ( w'z_a).

Since θ(w,·) and τ∘ w' coincide in
I_a and are holomorphic on D_a, in fact they coincide on
D_a. Hence exp(sθ(w,z_a))= w'(z_a)^s
for all s∈, and (<ref>) follows.

The “polynomial expression” of _s,ρ and its powers
is essentially due to the following isomorphism of Hilbert spaces:
For any φ∈(,V) and any v∈ V, the map φ_v(z)φ(z)v
lies in (). Consider an orthonormal basis (v_i)_i
of V. The map

    φ↦∑_iφ_v_i⊗ v_i

is an isomorphism (,V)→()⊗ V, and it does
not depend on the choice of orthonormal basis of V. Here are the
coefficients of the “polynomial expression” of _s,ρ^ℓ:
For any w∈Γ_≥2 and any s∈ we define M_w,s:(D_S(w))→(D_E(w))
by

    M_w,sψ_S(w)(z_E(w)) w'(z_ w)^sψ_S(w)( wz_E(w)),

with the s-th power defined as in (<ref>). Let
_a be the orthogonal projection ()→(D_a).
The operators 

    M_w,s_ w^*M_w,s_ w

are trace-class, as is shown in <cit.>.

Let (ρ,V) be a unitary
representation of Γ. For any integer ℓ≥1, under the
isomorphism (𝐃,V)≃()⊗ V given by
(<ref>), we have

    _s,ρ^ℓ=∑_w∈Γ_ℓ+1M_w,s⊗ρ( w).




We identify (,V) and ()⊗ V via (<ref>).
Let us denote by _s,ρ^(ℓ) the operator on the right-hand-side
of (<ref>). It suffices to show that
_s,ρ^ℓ and _s,ρ^(ℓ) agree on all the
f∈(,V) of the form φ⊗ v, with φ∈()
and v any unit vector in V. For any such f and all b∈,
by (<ref>) we have


    [_s,ρ^ℓf](z_b)     =∑_w∈Γ_ℓ+1
     w=b
     w'(z_b)^sρ( w)f( wz_b)
        =∑_w∈Γ_ℓ+1
     w=b
     w'(z_b)^sφ( wz_b)ρ( w)v
        =(∑_w∈Γ_ℓ+1
     w=b
    M_w,s⊗ρ( w))f(z_b)
        =[_b_s,ρ^(ℓ)f](z_b),

which proves our claim.




§ DETERMINISTIC A PRIORI BOUNDS


In this section we gather technical lemmas about Schottky groups,
transfer operators and Bergman spaces. Some of them are well known,
while others are specifically tailored for our needs. We suggest the
reader to skim the statements in a first reading, and come back to
the details when the later sections demand it. 

We start with four properties of w↦Υ_w, whose proofs
can be respectively found in <cit.>.

Let Γ be a Fuchsian Schottky
group. The following holds:


  (Rough multiplicativity) For all w_1,w_2∈Γ_≥1
with w_1→ w_2 we have

    w_1w_2w_1w_2.


  (Mirror estimate) For all w∈Γ_≥1 we have

    ww.


  (Derivatives) For all w∈Γ_≥2 and any z∈ D_ w
we have

    | w'(z)| w.


  (Exponential Bound) There are constants θ_Γ,τ_Γ∈(0,1)
such that for all γ∈Γ_≥1 we have 

    θ_Γ^γγτ_Γ^γ.




In the next lemma we consider τ_Γ as in (<ref>).


Let Γ be a Fuchsian
Schottky group. For any ∈(0,1) and any integer k≥1
we have

    ∑_w∈Γ_k w^+τ_Γ^k.




Consider w∈Γ_k and ∈(0,1). By (<ref>)
we have wτ_Γ^k. <cit.>
tells us that w^μ(I_w), where μ is a
Patterson-Sullivan probability measure on ∂ associated
to Γ. Then

    ∑_w∈Γ_k w^+   τ_Γ^k∑_w∈Γ_kμ(I_w)=τ_Γ^k,

which completes the proof. In the last step we used the fact that
∑_w∈Γ_kμ(I_w)=1 [The probability measure μ is supported on the limit set of Γ,
which is covered by the pairwise disjoint intervals (I_w)_w∈Γ_k.]. 

We move on to two auxiliary results about Bergman spaces that we use
later. Both follow easily form the fact that the Bergman space of
an open domain Ω in  has a reproducing kernel
B_Ω. Namely, B_Ω is a map Ω×Ω→
respectively holomorphic and antiholomorphic in the first and second
coordinates, such that for any f∈(Ω) and all z_0∈Ω
we have

    f(z_0)=∫_ΩB_Ω(z_0,z)f(z) z.

For an open disk D⊆, say of radius r and center c,
there is the explicit formula


    B_D(z_1,z_2)=r^2/π[r^2-(z_1-c)(z_2-c)]^2.

See <cit.>.

Let D and D'
be open disks in , respectively with centers c,c'∈ and
radii r,r'. Suppose that D contains the closure of D'. For
any holomorphic function ψ:D→ we have

    ψ_(D')≤r'/π^1/2(r-r'-|c-c'|)^2ψ_(D).




Without loss of generality we assume that c=0. We compute the norm
of ψ in (D') with the aid of B_D: For any z_0∈ D
we have

    ψ(z_0)=∫_DB_D(z_0,z)ψ(z) z.

Any z'∈ D' verifies |z'|=|z'|≤ r'+|c'|, so from
(<ref>) we readily see that

    |B_D(z',z)|≤1/π(r-r'-|c'|)^2

for any z∈ D. Hence,


    ∫_D'|ψ(z')|^2 z'     =∫_D'|∫_DB_D(z',z)ψ(z) z|^2 z'
       ≤1/π^2(r-r'-|c'|)^4∫_D'(∫_D|ψ(z)| z)^2 z'
       ≤1/π^2(r-r'-|c'|)^4∫_D'ψ_(D)^2 z'
        =(r')^2/π(r-r'-|c'|)^4ψ_(D)^2,

and the result follows. From line 2 to line 3 in the preceding computation
we used Jensen's inequality.

For ease of reference we record the following consequence of Lemma
<ref>.


Let Γ be a Fuchsian Schottky
group. For any w∈Γ_≥2 and any ψ∈(D_S(w))
we have

    ψ_(D_w) wψ_(D_S(w)).




For any γ∈Γ-{I}, we denote respectively the center
and the radius of D_γ by c_γ and r_γ.
Let a=S(w). Since D_w is contained in D_a, by Lemma <ref>
we have


    ψ_(D_w)≤r_w/π^1/2(r_a-r_w-|c_a-c_w|)^2ψ_(D_a).

Note that r_w+|c_a-c_w| is the radius of the smallest circle
with center c_a containing D_w, hence r_a-r_w-|c_a-c_w|1.
Also, 2r_w= w. The claim follows from these observations
and (<ref>).


Let Γ be a Fuchsian Schottky
group and let V be a Hilbert space of finite dimension. For any
f∈(,V), any w∈Γ_≥2 and all z_w∈ D_w
we have

    f(z_w)_V f_(,V).




We start with the case V=. Consider φ∈() and
let a= w. Let c_a,r_a and B_a be respectively
the center, radius and the Bergman kernel of D_a. Since z_w
lies in D_a, we have φ(z_w)=_aφB_a(z_w,·)_(D_a). 

Let

    _a⋃_γ∈Γ_2
    γ=a
    D_γ.

Note that there is _Γ>0 such that for any z∈_a
we have |z-c_a|≤ r_a-_Γ[Since the closure of _a is contained in D_a.].
It follows then from (<ref>) that |B_a(z_w,z_a)|1
for any z_a∈ D_a. The Cauchy-Schwarz inequality on (D_a)
gives


    |φ(z_w)|    ≤B_a(z_w,·)_(D_a)_aφ_(D_a)
       φ_(),

which completes the proof in this case.

For a general V and f∈(,V), we consider an orthonormal
basis v_1,…,v_d of V and we express f as ∑_jφ_j⊗ v_j
for some φ_j∈(). To complete the proof we apply
(<ref>) as follows:

    f(z_w)_V    =(∑_j=1^d|φ_j(z_w)|^2)^1/2
       (∑_j=1^dφ_j_()^2)^1/2= f_(,V).


We close this section with an estimate of the norm variation of a
transfer operator with respect to the parameter s. Recall that
N_Γ is the rank of a Schottky group Γ. 


Let Γ be a Fuchsian
Schottky group and let  be a compact subset of . There
are positive constants B_,J_Γ
and C_ such that for any ℓ1, any finite dimensional
unitary representation (ρ,V) of Γ and any s_1,s_2∈
we have

    _s_1,ρ^ℓ-_s_2,ρ^ℓ≤ J_Γ|s_1-s_2|(ℓ+1)[(2N_Γ-1)C_^B_Γ]^ℓ+1.




Consider f∈(,V), b∈ and z_b∈ D_b. From
(<ref>) we see that 

    (_s_1,ρ^ℓ-_s_2,ρ^ℓ)f(z_b)=∑_w∈Γ_ℓ+1
     w=b
    ( w'(z_b)^s_1- w'(z_b)^s_2)ρ( w)f( wz_b).

In all the proof, w is an element of Γ_ℓ+1 with w=b.
By (<ref>) and (<ref>),
both | w'(z_b)| and w are <1 whenever ℓ1.
We assume this for the rest of the proof.

To estimate the norm of (_s_1,ρ^ℓ-_s_2,ρ^ℓ)f(z_b)
we will use the next elementary bound, valid for all z_1,z_2∈:

    |e^z_1-e^z_2|≤| z_1-z_2| e^max{ z_1, z_2}.

Thus

    | w'(z_b)^s_1- w'(z_b)^s_2|≤|s_1-s_2||τ( w'z_b)|e^max{[s_1τ( w'z_b)],[s_2τ( w'z_b)]},

where τ is the branch of the logarithm as in (<ref>).
Since | w'z_b|<1, then |τ( w'z_b)|≤π+log(| w'z_b|),which
by (<ref>) and (<ref>)
implies that

    |τ( w'z_b)|≤ B_Γ(ℓ+1)

for some B_Γ>0. Let C_max_z∈e^|z|.
By (<ref>), we have the following for any s∈:

    exp[(sτ( w'z_b))]    ≤exp|sτ( w'z_b)|
       ≤ C_^B_Γ(ℓ+1).

Plugging (<ref>) and (<ref>) into (<ref>)
yields

    | w'(z_b)^s_1- w'(z_b)^s_2||s_1-s_2|(ℓ+1)C_^B_Γ(ℓ+1)


Applying the triangle inequality in (<ref>) followed by (<ref>)
and Lemma <ref>, we get


    (_s_1,ρ^ℓ-_s_2,ρ^ℓ)f(z_b)_V   |s_1-s_2|(ℓ+1)C_^B_Γ(ℓ+1) f_(,V)#Γ_ℓ+1/2N_Γ
       |s_1-s_2|(ℓ+1)[C_^B_Γ(2N_Γ-1)]^ℓ+1 f_(,V),

and since

    (_s_1,ρ^ℓ-_s_2,ρ^ℓ)f_(,V)=(∫_(_s_1,ρ^ℓ-_s_2,ρ^ℓ)f(z)_V^2dz)^1/2,

we are done.




§ STRONG CONVERGENCE AND THE PROOF OF THEOREM <REF>



This section has two parts: in Section <ref>
we introduce the notion of strong convergence for random unitary representations
of a free group Γ, which is in fact a condition on the associated
representations of the group algebra [Γ]. We also explain
how to upgrade the strong convergence via the matrix amplification
trick in order to handle the random transfer operators of Theorem
<ref>. The proof of this last result is completed
in Section <ref> by using a norm bound that
we prove in Section <ref>. 



 §.§ Strong convergence and matrix amplification


Suppose in this section that Γ is a free group of rank N
and (ρ_j,W_j)_j≥1 is a sequence of (possibly random)
unitary representations of Γ. 

Let us recall two notions of convergence for random variables: We
say that a sequence of complex random variables (Z_j)_j≥1
converges asymptotically almost surely (a.a.s.) to a constant
z∈ if for all ϵ>0,

    lim_j→∞(|Z_j-z|>ϵ)=0.

This is the same as convergence in probability. We say additionally
that the convergence is almost sure (a.s.) if for all ϵ>0
we have

    ∑_j=1^∞(|Z_j-z|>ϵ)<∞.



A sequence of unitary representations (ρ_j,W_j)_j≥1
of Γ strongly converges to (ρ_∞,W_∞)
if for all f∈[Γ]—the group algebra of Γ—we
have

    lim_j→∞ρ_j(f)_W_j=ρ_∞(f)_W_∞.

When the unitary representations (ρ_j)_j≥1 are random,
we say that they a.a.s. (a.s.) strongly converge to ρ_∞
if the convergence in (<ref>) is a.a.s.
(a.s.).


In the rest of this section, we assume that the random unitary representations
(ρ_j,W_j)_j≥1 of Γ are finite dimensional and
that they a.a.s. or a.s. strongly converge to the left regular representation
(ρ_Γ,ℓ^2(Γ)) of Γ. 

By matrix amplification (see for example <cit.>),
the strong convergence of the sequence (ρ_j)_j≥1 implies
the following more general statement. 

 For any r≥1 and
any finitely supported map γ↦ F_γ with values
in M_r() we have

    lim_j→∞∑_γ∈ΓF_γ⊗ρ_j(γ)_^r⊗ W_j=∑_γ∈ΓF_γ⊗ρ_Γ(γ)_^r⊗ℓ^2(Γ)

 (a.s or a.a.s., respectively, if the representations are random).


We need a mild extension of Proposition <ref>
allowing the F_γ to be compact operators an infinite dimensional
Hilbert space, which is obtained via a standard approximation argument
(see, for example, <cit.>). 

For any finitely supported
map γ↦ T_γ on Γ taking values in the
compact operators of a separable Hilbert space , we have

    lim_i→∞∑_γ∈ΓT_γ⊗ρ_i(γ)_W⊗^n_i=∑_γ∈ΓT_γ⊗ρ_Γ(γ)_W⊗ℓ^2(Γ)

 (a.s or a.a.s., respectively, if the representations are random).





 §.§ The proof of Theorem <ref>


In this section we consider a Fuchsian Schottky group Γ and
a sequence (ρ_j)_j≥1 of random unitary representations
of Γ a.a.s. strongly converging to ρ_Γ. Recall—see
(<ref>)—that the powers of the transfer
operators can be written as

    _s,ρ_j^ℓ=∑_w∈Γ_ℓ+1M_w,s⊗ρ_j( w).

By Corollary <ref> we know that,
for big n, the size of _s,ρ_i^ℓ is governed a.a.s.
by that of the limit operator

    _ℓ,s∑_w∈Γ_ℓ+1M_w,s⊗ρ_Γ( w)

on ()⊗ℓ^2(Γ). We will prove Theorem <ref>
by showing that we can choose ℓ=ℓ(Γ,) such that _ℓ,s<1
for any s∈.

The key in the next norm bound of _ℓ,s, whose proof is
deferred to Section <ref>.

Let Γ be a Fuchsian Schottky group.
There are constants A_Γ>0,C_Γ>1 and τ_Γ∈(0,1)
with the next property: for any integer ℓ1, any >0
and any s∈ with s≥/2+ we have

    _ℓ,s≤ A_ΓC_Γ^ se^π| s|(ℓ+1)τ_Γ^ℓ.



Taking Proposition <ref> momentarily for granted, we
establish now Theorem <ref>.


Consider A_Γ>0,C_Γ>1 and τ_Γ∈(0,1)
as in Proposition <ref>. It suffices to prove the result
when  is a rectangle of the form

    ={ s∈|/2+≤ s≤ M_r,| s|≤ M_i} ,

with ∈(0,1) and M_r,M_i>1. We fix these
three parameters for the rest of the proof, as well as ℓ=ℓ(,Γ)
big enough so that Lemma <ref> and
Proposition <ref> apply to it, and

    A_ΓC_Γ^M_re^π M_i(ℓ+1)τ_Γ^ℓ<1-2.

Consider also 

    _1=_1(Γ,)/2J_Γ(ℓ+1)[(2N_Γ-1)C_^B_Γ]^ℓ+1,

where J_Γ,B_Γ and C_ are as in Lemma <ref>. 

Consider, as in the statement, a sequence (ρ_j,W_j)_j≥1
a sequence of finite dimensional unitary representations of Γ
that a.a.s. strongly converges to ρ_Γ. Recall that under
the isomorphism (,W_j)→()⊗ W_j given
by (<ref>) we have

    _s,ρ_j^ℓ=∑_w∈Γ_ℓT_w,s⊗ρ_j(w),

with T_w,s as in (<ref>)—see Lemma <ref>—. 

We denote by ℰ_s,j the event that 

    _s,ρ_j^ℓ≤_ℓ,s+,

where _ℓ,s is the limiting operator on ()⊗ℓ^2(Γ)
defined in (<ref>). Each T_w,s is a compact operator
of () since it is a sum of M_w b,s, which are compact
as is explained right after (<ref>).

The bound of the norm of _ℓ,s of Proposition <ref>
and (<ref>) imply that for any s∈, on ℰ_s,j
we have

    _s,ρ_j^ℓ<1-.

Let 𝚂 be a finite subset of  such that the open
disks with center in 𝚂 and radius _1 cover ,
and let ℰ_j=∩_𝚜∈𝚂ℰ_𝚜,j. 

Since 𝚂 is finite, then (ℰ_j)→1 as
j→∞. Consider any s∈. Take 𝚜∈𝚂
such that |s-𝚜|<_1. By Lemma <ref>
and (<ref>), on ℰ_j we have

    _s,ρ_j^ℓ   ≤_s,ρ_j^ℓ-_𝚜,ρ_𝚓^ℓ+_𝚜,ρ_𝚓^ℓ
        <_1J_Γ(ℓ+1)[(2N_Γ-1)C_^B_Γ]^ℓ+1+(1-)=1-/2,

which completes the proof.






§ NORM ESTIMATION OF THE LIMIT OPERATORS


The goal of this section is to prove the norm estimate of the limit
operators _ℓ,s—Proposition <ref>—,
which was used to establish Theorem <ref>. By
definition, these operators are finite sums of the form ∑_γT_γ⊗ρ_Γ(γ)∈ B(⊗ℓ^2(Γ)),
where ρ_Γ is the left regular representation of a free
group Γ and  is a Hilbert space. Let  be any operator
of this form. Here are the ingredients of our proof: the first and
key one is a bound due to Buchholz of _⊗ℓ^2(Γ)
in terms of the norms of certain matrices _j with operator
coefficients, determined by the (T_γ)_γ∈Γ.
This result is presented in Section <ref>.
Then, in Section <ref> we prove a lemma
allowing us to bound _j by its “Hilbert-Schmidt
norm”, and we estimate the size of the (T_γ)_γ∈Γ
corresponding to the limit operators. We combine these results in
Section <ref> to prove Proposition <ref>.



 §.§ Buchholz inequality 


In this section, Γ is a free group of rank N and we use
the following notation: Let us fix a basis ={γ_1,…,γ_N}
of Γ and work with the word length · on Γ
with respect to ∪. As before, Γ_k is the
set of γ∈Γ with γ=k. Recall that
B() is the group of bounded linear operators on a Hilbert space
. For any subset Λ of Γ, we denote by L^2(Λ,)
the Hilbert space of L^2-maps Λ→ with respect to
the counting measure on Λ. 

We start with some motivation: It was shown in <cit.>
that for any integer m≥1 and any α∈ L^2(Γ_m,)
one has

    ∑_w∈Γ_mα_wρ_Γ(w)_ℓ^2(Γ)≤(m+1)α_L^2.

This is commonly known as Haagerup's inequality. It has various
applications and generalizations—see <cit.>
for details—. What we need here is a version of (<ref>)
where the α_w are operators. This was established by Buchholz
in <cit.>.

Consider a map R:Γ→ B(),w↦ R_w. For any nonnegative
integers m,n we define the map (m,n):L^2(Γ_m,)→ L^2(Γ_n,)
by

    [(m,n)f](w_n)=∑_w_m∈Γ_mR_w_mw_n(f(w_m)).

We set (m,n)_n^*R(m,n)_m,
where 𝚃_j is the orthogonal projection L^2(Γ,)→ L^2(Γ_j,).
We state below only the inequality from <cit.>
that we need for our purposes.

Let Γ be a free group of finite
rank N≥2. Consider a Hilbert space  and a map Γ→ B(),γ↦ R_γ
supported on Γ_ℓ, for some integer ℓ≥1. Then

    ∑_γ∈Γ_ℓR_γ⊗ρ_Γ(γ)_⊗ℓ^2(Γ)≤(ℓ+1)max_0≤ j≤ℓ(j,ℓ-j).






 §.§ Auxiliary results


Consider two Hilbert spaces  and . Let 

    ^⊕ j L^2({1,2,…,j},).

For any linear map T:^⊕ m→^⊕ n, we define
T_i,j as 𝚀_iT𝚁_j^*, where 𝚀_i
and 𝚁_j are respectively the orthogonal projections
to the i-th and j-th factors of ^⊕ m and ^⊕ n. 

The next lemma is a simple extension of a well-known inequality for
complex matrices.

For any bounded operator T:^⊕ m→^⊕ n
we have

    T≤(∑_i=1^n∑_j=1^mT_i,jT_i,j^*)^1/2.




We denote 𝚀_iT by T_i. For all v∈^⊕ m
we have


    Tv=(∑_i=1^nT_iv^2)^1/2≤(∑_i=1^nT_i^2)^1/2 v.

Thus, T≤(∑_iT_i^2)^1/2.
To conclude, note that T_i^2=T_iT_i^*
and

    T_iT_i^*=∑_j=1^mT_i,jT_i,j^*.


We also need an estimate on the norm of the coefficients M_w,s—defined
in (<ref>)—of the limit operators.

Let Γ be a Fuchsian Schottky group.
There is C_Γ>1 with the next property: For any w∈Γ_ℓ+1
with ℓ≥1 and all s∈ with s>0 we have

    M_w,s e^π| s|C_Γ^ s w^ s.




Let a=S(w) and b=E(w), so that M_w,s:(D_a)→(D_b).
Note that M_w,s=M_w,s[Both operators have the same norm by the submultiplicativity of ·
since M_w,s=_b^*M_w,s_a and M_w,s=_bM_w,s_a^*.]. We work with the latter operator. Consider any z_b∈ D_b
and a complex number s=σ+it with σ>0. From the definition
of w'(z_b)^s we have


    | w'(z_b)^s|=e^-α t| w'(z_b)|^σ

for some α∈(-π,π). 

Now we compute the norm of M_w,sψ_a, for any
ψ_a∈(D_a), using the change of variable z_w= wz_b:

    M_w,sψ_a_(D_b)^2    =∫_D_b| w'(z_b)^sψ_a( wz_b)|^2 z_b
        =∫_D_w| w'(z_b)^s|^2/| w'(z_b)|^2|ψ_a(z_w)|^2 z_w
       ≤ e^2π|t|∫_D_w| w'(z_b)|^2σ-2|ψ_a(z_w)|^2 z_w.

Since σ>0, applying (<ref>) and Lemma
<ref> we get


    M_w,sψ_a_(D_b)^2   ≤ e^2π|t|C_Γ^2σ-2 w^2σ-2ψ_a_(D_w)^2
        e^2π|t|C_Γ^2σ w^2σψ_a_(D_a)^2,

from where our claim follows. 

Let us write the limit operators _ℓ,s as a sum indexed
by Γ_ℓ instead of Γ_ℓ+1:


    _ℓ,s    =∑_w∈Γ_ℓ+1M_w,s⊗ρ_Γ( w)
        =∑_w∈Γ_ℓ(∑_b∈
    
    w→ b
    M_w b,s)⊗ρ_Γ(w)
        =∑_w∈Γ_ℓT_w,s⊗ρ_Γ(w),

where

    T_w,s∑_b∈
    
    w→ b
    M_w b,s.

These verify essentially the same bound as the M_w,s. Here are
the details:


 Let Γ be a Fuchsian Schottky group.
There is a constant C_Γ>1 such that, for any w∈Γ_ℓ
with ℓ≥1 and all s∈ with positive real part we have

    T_w,s e^π| s|C_Γ^ s w^ s.




Let us fix w∈Γ_ℓ. Note that the operators M_w b,s
with b∈ such that w→ b have pairwise orthogonal images,
so 


    T_w,s≤√(2N_Γ-1)max_b∈
    
    w→ b
    M_w b,s.

The concatenation (<ref>) and mirror (<ref>)
estimates imply that w b w. Thus, for all s∈
with s>0, Lemma <ref> yields

    M_w b,s e^π| s|C_Γ^ s w^ s,

which combined with (<ref>) proves the bound on T_w,s.




 §.§ The proof of the norm bound of the limit operators 

In all the proof we fix
ℓ≥1,>0 and s∈ with s≥/2+.
Consider the following map R_∙:Γ→ B(()):

    R_w=
    T_w,s   if  w=ℓ,
    
    0    otherwise.

Recall that

    _ℓ,s=∑_w∈Γ_ℓR_w⊗ρ_Γ(w)

by (<ref>). Since R_∙ is supported on Γ_ℓ,
by Theorem <ref> we have

    _ℓ,s≤(ℓ+1)max_0≤ j≤ℓ(j,ℓ-j),

with (j,ℓ-j):L^2(Γ_j,())→ L^2(Γ_ℓ-j,())
as in (<ref>). We bound the norm of (j,ℓ-j) using
Lemma <ref>, the triangle inequality for ·
and the identity[Which holds for any bounded operator T on a Hilbert space.]
T^*T= T^2 as follows:


    (j,ℓ-j)   ≤(∑_w_ℓ-j∈Γ_ℓ-j∑_w_j∈Γ_j
    
    w_j→ w_ℓ-jR_w_jw_ℓ-j^2)^1/2
        =(∑_w∈Γ_ℓT_w,s^2)^1/2.

By (<ref>), w<1 for any w∈Γ_ℓ_0
provided ℓ_01. Assuming this holds for our initial ℓ,
we have w^2 s≤ w^+2 for any w∈Γ_ℓ.
From the bound for T_w,s of Lemma <ref>,
and Lemma <ref> we get


    (j,ℓ-j)    e^π| s|C_Γ^ s(∑_w_ℓ∈Γ_ℓw_ℓ^2 s)^1/2
       ≤ e^π| s|C_Γ^ s(∑_w_ℓ∈Γ_ℓw_ℓ^+2)^1/2
       ≤ e^π| s|C_Γ^ sτ_Γ^ℓ.

The claimed upper bound for _ℓ,s follows from
(<ref>) and (<ref>).




§ PROOF OF THE SPECTRAL GAP FOR RANDOM SCHOTTKY SURFACES 


In the situation of Theorem <ref>, our
strategy to prove that the random cover X_n has no new resonances
in  is to show that, with high probability, some power _s,ρ_n^0^ℓ
of the relevant random transfer operator has norm <1 for all s∈.
This is shown by feeding into Theorem <ref>
a deep result on random matrices of Bordenave–Collins that we recall
in Section <ref>. Having this, we complete
the proof of our spectral gap for random Schottky surfaces in Section
<ref>.



 §.§ Background on random matrices


The result of Bordenave-Collins is a contribution to a general problem
on random matrices that we now explain intuitively: The broad goal
is to study the joint behavior of finite sequences of independent,
random n× n matrices X_n,1,…,X_n,N in some fixed
model , as n→∞. It is known that for many important
models , the matrices X_n,1,…,X_n,N jointly converge to
deterministic operators X_∞,1,…,X_∞,N of some
Hilbert space in the sense that, for any noncommutative polynomial
P in 2N variables, the operators P_n a.s. “tend” to
P_∞, where P_j=P(X_j,1,…,X_j,N,X_j,1^*,…,X_j,N^*).
This series of results started with the breakthrough work <cit.>
of Haagerup and Thorbjørnsen, where they treat the case of the Gaussian
Unitary Ensemble. The analog results for the Gaussian Orthogonal and
Symplectic Ensembles where established by Schulz in <cit.>;
<cit.> and <cit.>
treat some Wigner matrices under certain conditions on the distribution,
while <cit.> consider random unitary
matrices with Haar distribution. Bordenave and Collins handle in <cit.>
the case relevant for random covers of Schottky surfaces, namely,
when X_n,1,…,X_n,N are the random unitary operators on
V_n^0 coming from a sequence φ_n,1,…,φ_n,N
of independent random permutations of [n] with uniform distribution.

Let us now turn the random matrix intuition into a rigorous statement
in terms of representation theory. Besides being better suited for
our purposes, this language allows us to describe the limiting operators
in a natural and straightforward way: Let Γ be a free group
of rank N and consider a basis ={γ_1,…,γ_N}
of it. A sequence φ_n,1,…,φ_n,N as above is
the same thing as a uniformly random homomorphism ϕ_n:Γ→ S_n
simply by setting φ_n,j=ϕ_n(γ_j). Thus, the
goal is to understand the random unitary representations (ρ_n^0,V_n^0)_n≥1
of Γ. Below we reformulate <cit.>
in terms of these representations. Recall that ρ_Γ denotes
the left regular representation of Γ.

Let Γ be a free group of finite rank. The
sequence (ρ_n^0,V_n^0)_n≥1 of random representations
a.a.s. strongly converges to (ρ_Γ,ℓ^2(Γ)).





 §.§ Proof of Theorem <ref>


We are finally ready to prove our main result.

Suppose we are given a Schottky surface X=Γ\
and 𝒦 as in the statement of Theorem <ref>.

By plugging Theorem <ref> into Theorem <ref>
we see that there is ℓ=ℓ(,Γ) such that the probability
that

    _s,ρ_n^0^ℓ<1 for all s∈

tends to 1 as n→∞. When (<ref>) holds for a deterministic
homomorphism ϕ:Γ→ S_n and a fixed s_0∈ ,
1 is not an eigenvalue of the corresponding _s_0,ρ. Hence
by Proposition <ref>, s_0
is not a new resonance of X_ϕ. In our probabilistic setting,
we conclude that a.a.s. there are no new resonances of the random
cover X_n in .

amsalpha




 Paper Title: The Aldous--Lyons Conjecture I: Subgroup Tests.












verbose,tmargin=2.5cm,bmargin=2.5cm,lmargin=1.5cm,rmargin=1.5cm,footskip=1cm























algorithmAlgorithm





[1]
#1







ruled
algorithmtbploa

algorithm


equationsection
figuresection
plain
thm[section]



conj[thm]Conjecture
cor[thm]Corollary
*thm*
definition
problem[thm]
*problem*Problem
remark
*rem*
remark
rem[thm]
lyxcode
	
		
		
		
		
		
	 
  
	
definition
defn[thm]
plain
prop[thm]
plain
fact[thm]
plain
lem[thm]
definition
example[thm]
plain
claim[thm]Claim
plain











algorithmAlgorithm













 

 
 
 




   
   


























ε









































































L. Bowen]Lewis Bowen
Lewis BowenDepartment of Mathematics 1 University Station C1200 University of Texas at AustinAustin, TX, 78712 USA.
lpbowen@math.utexas.edu

M. Chapman]Michael Chapman
Michael ChapmanCourant Institute of Mathematical SciencesNew York University,251 Mercer St, New York, NY 10012, USA.
mc9578@nyu.edu

A. Lubotzky]Alexander Lubotzky
Alexander LubotzkyWeizmann institute of ScienceRehovot, Israel.
alex.lubotzky@mail.huji.ac.il

T. Vidick]Thomas Vidick
Thomas VidickWeizmann institute of ScienceRehovot, Israel.
thomas.vidick@weizmann.ac.il


§ ABSTRACT 
    This paper, and its companion <cit.>, are devoted to  a negative resolution of the Aldous–Lyons Conjecture Aldous_Lyons_Conj,Aldous–Lyons_conj_blogpost. This conjecture, originated in probability theory, is well known (cf. <cit.>) to be equivalent to the statement that every invariant random subgroup of the free group is co-sofic. We disprove this last statement.
    

    
   In this part we introduce subgroup tests. These tests are finite distributions over continuous functions from the space of subgroups of the free group to {0,1}. Subgroup tests provide a general framework in which one can study invariant random subgroups of the free group. Classical notions such as group soficity and  group stability arise naturally in this framework. By the correspondence between subgroups of the free group and Schreier graphs, one can view subgroup tests as a property testing model for certain edge-labeled graphs. This correspondence  also provides the connection to random networks.
    
    Subgroup tests have values, which are their asymptotic optimal expectations when integrated against co-sofic invariant random subgroups.  Our first main result is that, if every invariant random subgroup of the free group is co-sofic, then one can  approximate the value of a subgroup test up to any positive additive constant. 

    Our second main result is an essentially value preserving correspondence between certain non-local games and subgroup tests. By composing this correspondence with a stronger variant of the reduction in ^*= <cit.>,  proved in the companion paper <cit.>, we deduce that approximating the sofic value of a subgroup test  is as hard as the Halting Problem, and in particular, undecidable. The combination of our two main results proves the existence of non co-sofic invariant random subgroups of the free group. 

    

The Aldous–Lyons Conjecture I: Subgroup Tests
    [
    August 5, 2024
=============================================





§ INTRODUCTION

In their seminal paper <cit.>, Aldous and Lyons ask whether every unimodular network is sofic. Conditional on a certain result which will be addressed in a companion paper <cit.>, we prove the answer is `no'.
This conjecture, which originated in probability theory, can also be expressed in the language of invariant random subgroups, and this is the one we will use in this paper.
We explain and motivate this conjecture before discussing our approach. The formal definitions and results will start in  Section <ref>.



 §.§ The Aldous–Lyons Conjecture




  §.§.§ What is soficity?

Roughly speaking, a countable group Γ is sofic if it admits a sequence of partial actions on finite sets which approximates the action of Γ on itself by left-translations. If Γ=⟨ S | R⟩ is finitely presented then this is equivalent to the existence of a sequence of finite graphs which converge to the Cayley graph of Γ in the sense of Benjamini–Schramm, which is a kind of local-on-average convergence.

To be more precise, a rooted graph is a pair (G,o) where G=(V,E) is a graph and o ∈ V is a distinguished vertex called the root. A random rooted graph (G,o) is sofic if there exists a sequence (G_i)_i ∈ of finite graphs such that, if o_i is a uniformly random vertex of G_i, then (G_i,o_i) converges to (G,o) in distribution (this means: for every r>0, the radius r neighborhood of the root in G_i converges in distribution to the radius r neighborhood of the root in G). This notion of convergence is due to Benjamini and Schramm <cit.>: They proved that if each finite graph G_i is planar and there is a uniform degree bound, then the limit (G,o) is almost surely recurrent. 

The notions of Benjamini–Schramm convergence and soficity naturally generalize from graphs to labeled graphs or networks <cit.>, simplicial complexes <cit.>, manifolds <cit.>, measured equivalence relations <cit.>, measured groupoids 
<cit.> and most generally, measured metric spaces <cit.>. 

Again, a finitely presented group is sofic if one of its Cayley graphs is sofic.[If one of the Cayley graphs is sofic, then all those that use finitely many generators are.]  For example, amenable groups are sofic. In fact, the Cayley graph of an amenable group admits a sequence of finite sub-graphs in which the isoperimetric ratio (number of boundary vertices to number of vertices) tends to zero, which implies this sequence Benjamini–Schramm converges to the Cayley graph.  Also, residually finite groups are sofic because they admit sequences of finite quotient groups whose Cayley graphs approximate the Cayley graph of the given group. 

Gromov implicitly introduced sofic groups in  <cit.>. He was motivated by Gottshalk's Conjecture: If Γ is a countable group and k ∈, then any Γ-equivariant continuous map ϕ:[k]^Γ→ [k]^Γ which is injective is necessarily surjective. This is obviously true when Γ is finite. Gromov proved that it holds when Γ is sofic. Benjy Weiss gave another proof and coined the term `sofic' from the Hebrew word for finite <cit.>.



      The soficity property has been useful in obtaining positive results about L^2 and spectral theory invariants <cit.>, group rings MR2089244, MR2417890, invariant couplings of random fields <cit.>, and in constructing dynamical invariants of groups actions MR2552252, MR2854085, MR3077882, MR3132735, MR3993930. See Cap_Lup_Sofic_Hyperlinear_book, MR3821628 for more background on sofic groups. The basic idea is roughly the same in each case: One transfers properties of the finite approximating objects to properties of the limit.  

      It is a major open problem to determine whether all countable groups are sofic. We do not directly address this problem because the class of objects we study, described next, is more general than the class of groups.

 





  §.§.§ What is unimodularity?






The main tool for analyzing Benjamini–Schramm limits is called the Mass Transport Principle which makes precise the intuitive notion of statistical homogeneity. To explain, a doubly-rooted graph is an ordered triple (G,o_1,o_2), where o_1,o_2 are vertices of G. A transport function is a function f, satisfying some measurability condition, which takes as input a doubly rooted graph and outputs a non-negative real number. The interpretation is that f(G,o_1,o_2) is the amount of mass the first root o_1 sends to the second root o_2. So if (G,o) is a random rooted graph then _(G,o)[∑_x ∈ V(G) f(G,o,x)] is the average amount of mass sent out of the root. Symmetrically, _(G,o)[∑_x ∈ V(G) f(G,x,o)] is the average amount of mass sent into the root. If these two quantities are equal for every transport function f, then we say (G,o) satisfies the Mass Transport Principle. Equivalently, we say it is unimodular (this is the term used by Aldous and Lyons in <cit.>). Alternatively, unimodularity can be formulated in terms of graphings or as invariance with respect to the root-changing equivalence relation on the space of rooted graphs. The former arises from the ergodic theory of measured equivalence relations (e.g. <cit.>). The latter point-of-view was introduced in <cit.> and further developed in <cit.>.


The term unimodular comes from the following special case. Suppose G is a connected transitive graph; this means that its automorphism group Aut(G) acts transitively on its vertex set. Then (G,o) is unimodular if and only if the automorphism group Aut(G) is unimodular in the sense that its left and right Haar measures agree.











The Mass Transport Principle arose in Häggström's study of percolation clusters on trees <cit.>. In fact, it is an exercise to show that if (G,o) is unimodular and p ∈ [0,1] is given, then the Bernoulli p-percolation cluster containing o is also unimodular. Unimodularity was further developed in <cit.> (see also <cit.> where it was used to construct a version of the Euler characteristic for unimodular planar maps).

It is straightforward to verify that (1) unimodularity is closed under weak limits and (2) if G is a finite graph and o is a uniformly random vertex of G then (G,o) is unimodular. It follows from these observations that soficity implies unimodularity. Question 10.1 of <cit.> asks whether unimodularity implies soficity.

As above, unimodularity has been generalized to random rooted labeled graphs, simplicial complexes, manifolds and so on. Benjamini and Schramm note that many results which are known to hold in the deterministic setting of unimodular transitive graphs can be generalized to  unimodular random graphs <cit.>. Aldous and Lyons demonstrate this with results about random walks, amenability, ends of graphs, spanning forests, percolation, and so on <cit.>.



  §.§.§ What is an invariant random subgroup?


As a first step towards our goal, let us describe the algebraic formulation of the Aldous–Lyons Conjecture in terms of invariant random subgroups, which were introduced in <cit.>, <cit.>, <cit.> (and implicitly in <cit.>). The concept of invariant random subgroups has been profitably studied in the context of locally compact groups (e.g. stuck1994stabilizers, MR3664810,Gelander_ICM2018). 
However, we will restrict our focus to the case of most relevance to this paper: finitely generated groups. 
Already in this discrete setup, invariant random subgroups play an important role in graph convergence <cit.>, stability properties of groups <cit.> and the analysis of dynamical systems stuck1994stabilizers, MR3193754 for example.

So, fix a finitely generated group Γ. An invariant random subgroup (or IRS) of Γ is a random variable taking values in the space of all subgroups of Γ, whose law is invariant under the conjugation action of Γ. For example, every normal subgroup of Γ is an IRS. More generally, if H ≤Γ has only finitely many conjugates, then a uniformly random sample of its conjugates is an IRS. As finite index subgroups have only finitely many conjugates, they induce  IRSs which are called elementary. To generate more examples, one can take convex combinations and weak* limits of elementary IRSs, and the IRSs that arise this way are called co-sofic. It turns out that for some groups every IRS is co-sofic, and for others there are non co-sofic IRSs (cf. <cit.>). The Aldous–Lyons Conjecture can be formulated as follows: 


    Are all IRSs of a (non-commutative) free group co-sofic?

 
For the rest of the paper, this is the formulation that we tackle, and it is known — and we explain it in the following paragraphs — to be equivalent to the probability theoretic formulation on Benjamini–Schramm limits of finite graphs.

The law (or distribution) of an IRS is a Borel probability measure μ on 𝔰𝔲𝔟(Γ), the space of subgroups of Γ. We let (Γ) denote the space of all such measures. By abuse of language, we say that μ is an IRS if μ∈(Γ).
To see the relation between the two formulations of the Aldous–Lyons Conjecture, note that if H is a subgroup of Γ, then the Schreier coset graph of _H\^Γ with respect to some generating set S is a rooted (directed, edge-labeled) graph, with the root being the coset of the identity.  This gives a map from the space  of subgroups 𝔰𝔲𝔟(Γ) to the space of isomorphism classes of rooted (edge-labeled, directed) graphs. 

If μ∈(Γ) then μ pushes forward under this map to a unimodular measure <cit.>. 
 Moreover, if μ is elementary, then its pushforward is induced by uniformly choosing a root of some finite connected graph. Therefore, if μ is co-sofic, then its pushforward is sofic.

Now, consider the special case in which Γ=(S) is the free group generated by a finite set S. Suppose μ is an invariant random subgroup of Γ, and that the (random) Schreier coset graph of μ is sofic (in the directed, edge-labeled rooted graph category). This means there exists a sequence of finite directed, edge-labeled graphs G_i which Benjamini–Schramm converge to the (random) Schreier coset graph of μ. After perturbing the given sequence if necessary, we may assume that each G_i is itself a Schreier coset graph with respect to some finite-index subgroup H_i of (S) — the existence of such perturbations is exactly where the freeness of the group (S) plays a role. Moreover, if H_i is a uniformly random conjugate of H_i, then the distribution of H_i converges to μ. Therefore, a positive solution to the Aldous–Lyons Conjecture implies that every IRS of (S) is co-sofic.[There is a standard “decoration of graphs” technique that allows one to show that the directed, edge labeled (with finitely many labels) version of the Aldous–Lyons conjecture is equivalent to the non-directed, non-edge labeled version. As Bálint Virág pointed to us, the same technique is used, e.g., to show that every group is the automorphism group of some graph. As this is standard, we do not elaborate on it anymore.]







 §.§ Our approach



  §.§.§ What is a subgroup test?

The set () of all IRSs of the free group and the set _() of co-sofic IRSs of the free group are both convex. Thus, the Aldous–Lyons Conjecture is asking whether certain convex sets can be separated. A natural way to distinguish between two convex sets is to use (continuous) linear functionals. So, in order to separate the above, one should search for a rich enough collection of functionals. Subgroup tests and their values will play this exact role.  

A challenge is a continuous map from () to {0,1}. A subgroup test  is a probability distribution on a finite set of challenges. One can integrate any subgroup test against any probability measure over (). If π is in (), then this integral is called the value of the strategy π against the test .

 Many natural and well studied properties of groups, specifically soficity and pointwise permutation stability, can be formulated as properties of subgroup tests. 

The terminology of “tests” and “values” originates in the field of interactive proofs in computer science. 
An interactive proof designates an interaction between two entities, a “verifier” and a “prover”. The goal of the prover is to convince the verifier of the validity of a certain claim, and the goal of the verifier is to test the prover so that it  accepts the interaction only if the claim  is indeed correct. 
Examples of “claims” studied in computer science are the 3-colorability of a given input graph, or that a Turing machine whose description is passed as input halts.
In our approach, the methods of the verifier and the prover are restricted. The prover is required to sample an element of (). 

The sampling method itself is restricted to some sub-class of IRSs  (e.g., elementary IRSs); the choice of distribution according which to sample is the only degree of freedom the prover has, and is called its strategy. The verifier gets access to (the indicator function of) the prover's sampled subgroup H, and it makes a decision whether to accept or reject this subgroup. 
To decide, the verifier holds some finite set K ⊆ — known beforehand to the prover — and according to which combinations of elements of K are contained in H, it accepts or rejects; the rules that control which combinations of elements result in acceptance and which combinations result in rejection are also know  to the prover beforehand. The value of a strategy in a game is the probability, taken over the prover's sampling and the verifier's probabilistic choices, that the verifier makes the decision to accept when interacting with a prover using this strategy. 

In spite of these restrictions, it is possible to verify hard computational problems in this setup.[For the expert, we note that restrictions on the  prover and verifier can affect both the completeness and soundness properties of a proof system. Hence, a priori, they may either lower or raise the complexity of the associated class of interactive proofs compared to classical single-prover interactive proofs.] For example, it is not difficult to design a subgroup test such that there is a value-1 strategy for this test if and only if a given graph is 3-colorable. Jumping ahead, the correct analogy is with the theory of multiprover interactive proofs, and even more specifically interactive proofs with quantum provers sharing entanglement, also known as non-local games. We explain below this connection, which plays an essential role in the second part of this paper.





  §.§.§ What is a non-local game?

Morally, non-local games originated in Bell's resolution <cit.> of the Einstein–Podolsky–Rosen paradox <cit.>. Einstein was famously concerned by the ability of space-time separated particles to correlate in ways that might  violate relativity, a possibility that seemed to be suggested by the mathematical modeling of quantum mechanics. Einstein argued for a local hidden variable theory — namely, that the particles did share some information beforehand,  which allowed them to correlate — and that physicists should search for this hidden information.
Bell provided a thought experiment, that in modern jargon is an instance of a non-local game, which proves that the kinds of correlations that arise from isolated quantum mechanical systems are intrinsically different from those that could be generated in any hidden variable model. 
Bell's work lay the foundation for the subsequent design, and execution, of experiments which verify the existence of quantum entanglement (and thus refute Einstein's approach; see also <cit.>). The 2022 Nobel prize in physics was awarded to Aspect, Clauser and Zeilinger, partly for performing non-local games as experiments and verifying that the winning statistics in the games exceeds what local hidden variable models allow.


A correlation is a function p A× A× X× X→ [0,1], where X and A are finite sets, such that for every pair x,y∈ X the function p(·,·|x,y) is a probability distribution over A× A — the quantity p(a,b|x,y) should be thought of as the answer to “what is the probability a and b are provided as answers given that x and y were asked as questions?”; this cryptic phrase will soon be clarified.  A correlation  p is said to be deterministic if there are two functions f_1,f_2 X→ A such that p(a,b|x,y)=1 only if f_1(x)=a and f_2(y)=b.
The convex hull   C_c of  deterministic correlations is the set of local or  classical correlations.[An example of a non-local correlation is p(a,b|x,y)=1 only when a=y and b=x. Intuitively, this is because this correlation implies signaling from one system to the other.] 
On the other hand, the  quantum correlations  C_q are those that can be generated by performing quantum measurements on a (finite-dimensional) bipartite physical system, where the measurement on one part depends only on x and generates a, while  the measurement on the other part depends only on y and generates b. For a formal description, see Remark <ref>. The set C_q is convex, but interestingly it is not closed <cit.>.

A non-local game  is specified by a probability distribution μ on the set X×X, and a decision function D:X× X× A× A→{0,1}.[As can be seen in Section <ref>, our definition of non-local games is slightly different. This difference is mainly cosmetic, and is driven by our motivation to relate non-local games to subgroup tests.] 
The interpretation of this combinatorial object  as a “game” comes from thinking of it (similarly to subgroup tests) as an interactive proof, but this time with two provers, a la <cit.>. 
The common dramatization goes as follows: Two provers are spatially separated. The verifier samples a pair of “questions” x,y∼μ, and “asks”  one prover x and   the other y. The provers then apply some local procedure to choose their “answers” a and b respectively. Finally, the verifier accepts if D(x,y,a,b)=1 and rejects otherwise.  

Since the process by which the provers generate their answers is hidden to us, we can only “observe” (samples from) the distribution of answers given questions, namely, the underlying correlation. So, the correlation can be thought of as the “strategy” used by the provers (similar to the way IRSs are seen as strategies for the prover in a subgroup test).  Bell's separation of C_c from C_q amounts to devising a game that is “easy” for players that use quantum correlations, yet “hard” for players that use only classical ones. 

Now, there is more than one candidate mathematical model for entanglement in quantum mechanics. A slightly generalized model, suggested by Tsirelson <cit.>,  gives birth to a set of correlations known as quantum commuting correlations C_qc. This set is closed and contains the quantum correlations.  Tsirelson famously asked whether the closure of C_q is equal to C_qc <cit.>. At this point, we hope the type of problem already resonates with the reader, as this is again a separation of convex sets type of problem.[Connes' embedding problem (CEP) is also a separation of convex sets type of problem, specifically, separating all  characters of the free group from those that are limits of finite dimensional characters. This problem was shown <cit.> to be equivalent to Tsirelson's problem, and also has close connections with invariant random subgroups — as every IRS induces a character. We omit discussions on the relations between this work and CEP from the text, as they are not particularly helpful for understanding our approach.]



  §.§.§ Harnessing undecidability

It was known that if  the quantum correlations are dense in the quantum commuting correlations, then the complexity class of multiprover interactive proofs with entangled provers (^*) contains only decidable languages.
By proving that the Halting Problem, which is undecidable, is in ^*, the authors of <cit.> were able to resolve Tsirelson's problem in the negative.


In this paper a similar path is followed: Rather than directly find a functional that separates the co-sofic and general IRSs of the free group, we describe a computational problem — approximating the sofic value of a subgroup test — whose undecidability implies a negative resolution of the Aldous–Lyons conjecture.  Our approach can be described in two steps: First, show that a positive solution to the Aldous–Lyons Conjecture implies that approximating the sofic value of a subgroup test is decidable. Second, prove that this computational task is undecidable.

Elaborating on the first,  given a subgroup test , we prove that its optimal value against co-sofic IRSs — which we call the sofic value of  — can be approximated from below, and its optimal value against any IRS — which we call the ergodic value of  — can be approximated from above.  Thus, if the Aldous–Lyons Conjecture has a positive answer, then the sofic and ergodic value of a test always agree, and using the aforementioned approximation procedures they can be calculated to any  predetermined accuracy. 

Elaborating on the second, we provide a mechanism for translating certain non-local games (see Section <ref>), which we call tailored non-local games, to subgroup tests. This translation is essentially value preserving. Namely, if the non-local game had a (certain kind) of perfect quantum strategy — which we call a Z-aligned permutation strategy that commutes along edges — then the analogous subgroup test has a perfect (co-sofic) strategy. In the other direction, if the subgroup test has an almost perfect (co-sofic) strategy, then the non-local game has an almost perfect strategy. Hence, a certain strengthening of the reduction in <cit.>, which states that small additive constant approximations to the value of tailored non-local game are undecidable (Theorem <ref>), implies that the sofic value of a game cannot be approximated to arbitrary precision. Thus, the Aldous–Lyons Conjecture has a negative solution.




The rest of the introduction provides a deeper dive into the  definitions, results and ideas   of this paper.




    

 §.§ Our approach





§ INTRODUCTION

Probabilistic methods for studying algebraic, geometric and combinatorial objects have proved themselves extremely powerful (cf. alon2016probabilistic,tenenbaum2015introduction). To better understand a collection of objects, you define a sampling scheme on it and analyze the expected behaviours of the samples. Invariant random subgroups, defined in <cit.>, are a flexible tool of this exact kind: They allow one to study lattices in Lie groups <cit.>, graph convergence <cit.>, stability properties of groups <cit.> and various dynamical systems <cit.>.

Given a countable group Γ, let (Γ) be the collection of all subgroups of Γ. An invariant random subgroup (IRS) is a Borel probability distribution over (Γ) which is invariant under conjugation. The most straightforward example of an IRS is the Dirac measure concentrated on a normal subgroup. Another basic example is the uniform distribution over the conjugation orbit of a finite index subgroup. We call such an IRS simple. To generate more examples, one can take convex combinations and weak* limits of the simple IRSs. The IRSs that arise this way are called co-sofic. It turns out that for some groups every IRS is co-sofic, and for others there are non co-sofic IRSs (cf. <cit.>). A major open problem in this field, due to Aldous and Lyons (Conjecture <ref>), asks  which is true for a (non-commutative) free group? In this paper (and its companion <cit.>), we prove that non-commutative free  groups have non co-sofic IRSs. 

The set of all IRSs of the free group, and the set of co-sofic IRSs of the free group, are both convex. Thus, the Aldous–Lyons conjecture is asking whether certain convex sets can be separated. The natural way to distinguish between convex sets is by using linear functionals. Thus, one should search for a rich collection of functionals if they seek to separate the above. Subgroup tests and their values will play this role.

Though it is tempting to try and find a specific functional that separates the co-sofic and general IRSs  of a free group, we suggest another approach in the spirit of the recent resolution of Tsirelson's problem <cit.>.  Tsirelson's problem is also a separation problem of convex sets, the quantum and quantum commuting correlations. 
It was known that if  the quantum correlations are dense in the quantum commuting correlations, then the complexity class of Multi-prover Interactive Proofs with Entangled Provers (MIP^*) contains only decidable languages.
Thus, by proving that the Halting Problem is in MIP*, they were able to resolve Tsirelson's problem.
In this paper we construct a similar setup, namely, define a complexity class whose containment of undecidable languages will imply the existence of non co-sofic IRSs in the free group. 
For those familiar with multi prover interactive proofs, we remark that  subgroup tests  are analogous to non-local games. They are combinatorial objects that test probability distributions over subsets of the free group. Many natural and well studied properties of groups, specifically soficity and pointwise permutation stability, can be formulated as properties of subgroup tests. 


In the rest of the introduction we give brief definitions of the relevant notions, and overview the main results of this paper.





 §.§ Subgroup Tests
 
Let S be a finite set, =(S) be the free group with basis S, () the collection of subgroups of  and Prob(sub(ℱ)) the set of all Borel probability measures on sub(ℱ). 
A challenge  is a pair (K;D) such that K is a finite subset of , and D is a function from subsets of K to {0,1}. A subset A⊆ passes the challenge (K;D) if D(A∩ K)=1, and fails it otherwise.
A test  is a finite collection of challenges {(K_i;D_i)}_i∈ Q, where Q is a finite index set, together with a probability distribution μ over the set Q. 
A strategy is a (Borel) probability distribution over subgroups of the free group, namely an element π∈( ()). 
One can run the test  against π, as follows. 

First the verifier samples i∈ Q according to μ. Next, the prover samples a subgroup  H according to π. Finally, the verifier makes the decision to accept if H passes the i^ th challenge (K_i;D_i), and reject if H fails the i^ th challenge.






The value of using the strategy π against the test  is its acceptance probability, namely

    (,π)=_H∼π_i∼μ[D_i(H∩ K_i)].

For a fixed , the value is a continuous linear functional from probability distributions over subgroups of  to ℝ. 
Hence, it enables us to study convex subsets of (()) by their optimal value against a given test. 


    As mentioned earlier, a challenge is, essentially, a continuous map from the space of subgroups of  to {0,1}. One can extract from any such continuous map  a challenge (K;D) that describes it and vice versa. Therefore, a test is a convex combination of continuous maps from () to {0,1}, and the value is integration of this map against a (probability) measure. 







 §.§ Invariant random subgroups


As sub(ℱ) inherits the product topology
from the power set {0,1}^ℱ,  it is compact.
Every action of a group Γ on a set X extends naturally to an action of Γ on the power set {0,1}^X by 

    ∀γ∈Γ,A⊆ X   γ.A={γ.a | a∈ A} .

Hence, the conjugation action of ℱ on itself is inherited by {0,1}^. Also, ()⊆{0,1}^ is preserved by this action. Hence, the conjugation action extends  to its power  set {0,1}^() by 

    ∀ w∈ℱ, A⊆sub(ℱ)    w. A={wHw^-1| H∈ A} .

 The conjugation action of ℱ further extends to Prob(sub(ℱ)):
For every Borel set B⊆() and μ∈ Prob(sub(ℱ)), we have

    (w.μ)(B)=μ(w^-1.B) .
 



The set of invariant random subgroups IRS(ℱ)
consists of all probability measures in Prob(sub(ℱ))
that are invariant under the action of ℱ. 


The ergodic value of a test  is the best performance of a strategy π∈() against it, namely

    _()=sup{(,π) |π∈()} .


The extremal points in the convex set  (Γ) are commonly referred to as ergodic IRSs.
    As we take the supremum of a linear functional over a convex compact set, the optimum is obtained on an extremal point, which explains the phrase ergodic value. 




 §.§ Finitely described invariant random subgroups
 
Let X be a finite set, and let (X) be the symmetric group acting on X. Every map σ S→(X) extends uniquely to an action of =ℱ(S) on X. 
For a vertex x∈ X, we can define its stabilizer to be 
    Stab(σ,x)={ w∈ℱ|σ(w).x=x} ,

which is a subgroup of ℱ.
Hence, we can associate with σ S→(X) an IRS of  via the following sampling procedure: 
(1) Choose x∈ X uniformly at random. (2)
  Output Stab(σ,x).
We denote by Φ the map from finite actions of  to () defined by this procedure, namely

    Φ(σ)=_x∈ X[ 1_ Stab(σ,x)] ,

where 1_H is the Dirac measure concentrated on the subgroup H and _x∈ X is the expectation according to the uniform measure over X. The set of finitely described invariant random subgroups _() is the image of the map Φ in (). 

The sofic value of a test  is the (asymptotic) best performance of a strategy π∈_() against it, namely

    _()=sup {(,Φ(σ)) | |X|<∞, σ S→(X)} .




The name sofic value is appropriate, as demonstrated in Section <ref>. Furthermore, we often abuse notation and use (,σ) instead of (,Φ(σ)), even though the action itself is not an IRS.



Since _()⊆(), we have _()≤_() for every test . The Aldous–Lyons conjecture <cit.> can be formulated as follows:
  Is  IRS_(ℱ)
 weak* dense in IRS(ℱ)?


In their paper <cit.>, Aldous and Lyons ask this as a question. In the passing of time, the positive form of this problem, namely that _() is dense in (), received the name The Aldous–Lyons Conjecture (cf. <cit.>). Thus, we use  the more common “Conjecture” phrasing instead of Problem. 

The algebraic formulation we  provided earlier, which can also be found  in Section 6 of <cit.>, may seem different from the above. The question was  whether every invariant random subgroup of  is co-sofic, where co-sofic means an IRS which is the weak* limit of convex combinations of uniform distributions over finite index subgroups. Since the finitely described IRSs are dense in the co-sofic ones, the formulation in Conjecture <ref> is equivalent to this standard (algebraic) formulation.



Lastly, the term co-sofic is closely related to the better known term of a sofic group (cf. Definition <ref>).
A finitely presented group ℱ/N is sofic if and only if the Dirac measure concentrated on N is a co-sofic IRS (cf. Proposition 6.1 in <cit.>).
      Thus, a positive solution to the Aldous–Lyons Conjecture <ref> implies in particular that every group is sofic.[The way we formulate it, a positive solution to the Aldous–Lyons Conjecture only implies that all finitely presented groups are sofic. It is standard to deduce it for all groups from the finitely presented ones <cit.>.  ] 


      The soficity property has been useful in obtaining positive results about group rings <cit.>, invariant couplings of random fields <cit.>, and invariants of groups actions <cit.>. See <cit.> for more background on sofic groups.

    




 
Since (,·) is a continuous functional, a positive solution to the Aldous–Lyons Conjecture <ref> implies that  for every test ,  _()=_().



    Due to the relation between subgroups of the free group and Schreier graphs, one can view subgroup tests as a new property testing model for (edge-labeled) graphs:
Let X be a finite set and σ S→ X  a transitive action (the non-transitive case is a convex combination of transitive ones). For x∈ X, the stabilizer Stab(σ,x) is exactly the set of labeled paths in the Schreier graph of (S) with respect to Stab(σ,x) and S that begin and end at x, namely, closed paths originating from x. Thus, when the   finitely described IRS Φ(σ) is put against a test , we actually test the aforementioned Schreier graph  in the following way:
First, choose a vertex x∈ X uniformly at random and sample i∼μ.
  List the paths from K_i and for each one check whether it is closed or open.  According to this check,  decide using D_i whether to accept or reject.





 §.§ Decidability of approximating the sofic value
 

    We briefly recall basic notions from computability and complexity theory.
Let Σ be a finite set that will play the role of an alphabet.
Let Σ^* be the free monoid generated by Σ, namely
the set of all finite words in Σ. 
A language is a subset L⊆Σ^*. Every language
L defines a decision problem: given w∈Σ^*, decide
whether w∈ L.
A promise language is a pair L_yes,L_no⊆Σ^*
such that L_yes∩ L_no=∅. A promise language (L_yes,L_no)
defines a promise decision problem: given w∈ L_yes∪ L_no,
decide whether w∈ L_yes. A promise language (L_yes,L_no) is decidable if there exists a Turing Machine (TM for short) M satisfying the following: If w∈ L_yes∪ L_no is given to M as input, then M halts and outputs a single bit M(w) satisfying

    M(w)=1     w∈ L_yes,
    
        M(w)=0     w∈ L_no.

Let L^1 and L^2 be two (promise) languages.  We say that L^1 reduces to L^2 if there exists a time Turing machine M for which: If σ∈ L^1_yes (respectively,  σ∈ L^1_no) is taken as the input of M, then M halts and the output M(σ) is in L^2_yes (respectively, M halts and the output  M(σ) in L^2_no).
Thus, if L^2 is decidable then L^1 is also decidable.







If the  challenge distribution μ in a subgroup test  is rational, then  can be encoded as a finite bit string. There are many (non-equivalent) ways of doing so.  
Assume we fixed some encoding for tests with rational challenge distributions such that, given the encoding, all underlying combinatorial data of the test can be calculated in finite time. 
Namely, the  set of generators S, the indexing set of challenges Q, the collection of challenges {(K_i;D_i)}_i∈ Q and the distribution μ can all be read from the encoding in finite time.
 

For every fixed encoding of tests such that all combinatorial data of the test can be read from it in finite time:

    
  * There is a Turing machine M_1 that takes as an input (the encoding of) a test , and outputs an infinite sequence of non-decreasing numbers (α_t)_t=1^∞⊆ [0,1]  such that lim_t→∞α_t=_().
    
  * There is a Turing machine M_2 that takes as an input (the encoding of) a test , and outputs an infinite sequence of non-increasing numbers (β_t)_t=1^∞⊆ [0,1]  such that lim_t→∞β_t=_().



    Clause (2) of Theorem <ref> should be compared to the Navascués–Pironio–Acín (NPA) hierarchy in the study of non-local games <cit.>. See also the introduction of <cit.>.



    If the Aldous–Lyons Conjecture <ref> has a positive solution, then for every θ>0, there is a Turing machine which accepts a subgroup test  as input and outputs its sofic value _() up to an additive error of at most θ.






























 §.§ Tailored non-local games and their associated subgroup tests





As described before, the standard definition of  a (non-local) game consists of two finite sets, a questions set X and an answers set A, together with a probability distribution μ over X× X and a decision function D X× X× A× A→{0,1}. In this paper, we use a slightly modified definition, that makes the connection with subgroup tests clearer. Before diving into our alternative definition, note first that the distribution μ defines a graph structure on X, by associating edges with its support. Furthermore, as the exact choice of A is irrelevant, we can assume it is the set of all bit strings up to some length Λ.

A (synchronous, non-local) game consists of a graph  G=(V,E), a distribution μ over E,
a length function ℓ V→ℕ,  formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} — the union of which is denoted by S=⋃_x∈ VS_x — and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y.[So, the main difference in our definition, is that the decision function D rejects automatically answers that are not of the appropriate length according to ℓ.]
     A (synchronous, quantum) strategy for  is a map ρ⋃_x∈ VS_x→ U(n), where the images are involutions, and ρ(S_x) commute for every fixed x∈ V.[This is again not the standard definition of a quantum strategy. Unpacking our definition, ρ associates an observable (Hermitian operator on a finite-dimensional Hilbert space that squares to the identity) with each bit of the answer when asked for x∈ V. Since the answer must be totally measured, all the observables associated with a specific vertex must commute. The fact there is a single  map ρ, and not one for each prover, is the synchronicity of this strategy. ]
Such a strategy defines for every xy∈ E a probability distribution over maps γ S_xy→{0,1} in the following way: 
For every ∈ S, let  ^_0 be the projection on the (+1)-eigenspace of ρ(), and ^_1  the projection on its (-1)-eigenspace. 
Furthermore, let τ be the dimension normalized trace on n× n matrices.  
Then, the probability that γ S_xy→{0,1} is sampled according to ρ is by definition 

    τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()) .

A map γ sampled that way is said to be sampled according to ρ, and we denote it by γ∼ρ.[Note that this sampling depends on the chosen edge xy∈ E. Namely, though implicit in the notation, γ is defined from S_xy for some edge xy∈ E. It is important to note that γ is not globally defined on all of S.]
    Relating to standard notations, we remark that by letting a=γ|_S_x and b=γ|_S_y, the correlation p(a,b|x,y) induced by the quantum strategy  ρ (as in Remark <ref>) agrees with   formula (<ref>).

    In a similar manner to subgroup tests, one can run the game  against the strategy ρ: 
    
    First, the verifier samples an edge xy∈ E according to μ. Then, the prover samples γ S_xy→{0,1} according to ρ. Finally, the verifier accepts if D_xy(γ)=1, and rejects otherwise.
   
   
   
   
   
 
  As for subgroup tests, the  value of the strategy ρ against the non-local game  is its acceptance probability when ran against the game. Namely,

    (,ρ)=_xy∼μ_γ∼ρ[D_xy(γ)] .

 The (synchronous) quantum value of , ^*(), is the supremum of (,ρ) over all possible quantum strategies against .


The main result in <cit.> (recalled in detail in Theorem <ref> herein) is a reduction from the Halting Problem to approximating  the quantum value of a game.
Our main contribution in the second half of this paper (Sections <ref>,<ref> and <ref>) is a mapping from a specific sub-class of non-local games — which we term tailored non-local games

and describe further below — to subgroup tests, that  (almost) preserves their values.


    There is a transformation that takes as input a tailored non-local game (see Definition <ref>) and outputs a subgroup test (which we call the associated synchronous subgroup test, see Definition  <ref>), such that:
    
        
  * If the game had a perfect (i.e., a value 1) Z-aligned, commuting along edges, permutation strategy (see Definitions <ref> and <ref>), then the associated subgroup test has a perfect strategy. 
        
  * Every almost perfect finitely described strategy for the associated subgroup test can be transformed into an almost perfect quantum strategy to the original tailored non-local game. 
    




Informally, a tailored game is one where the decision function D_xy takes the following form. First, D_xy examines the value taken by γ on a subset of the generators S_x∪ S_y (we refer to these as readable variables). Based on this, D_xy determines a collection of linear equations on the entire string γ. Finally, D_xy accepts if and only if all these equations are satisfied. This form for the game can be seen as a generalization of binary linear constraint system games (Example <ref>), which corresponds to the case where the set of readable variables is empty. 

A permutation strategy, as its name suggests, is a quantum strategy which is induced by permutations. Such a strategy is said Z-aligned when the observables associated with the readable variables (i.e. the bits of the answer γ that control the parity checks in the tailored game) are all diagonal matrices (with respect to some natural basis). The restriction to Z-aligned strategies is crucial for item (1) in Theorem <ref> above to hold. 

In a companion paper <cit.>,  we prove that the reduction of the Halting Problem to approximating the quantum value of a game presented in <cit.> can be modified such that the resulting non-local game is tailored and satisfies the required stronger assumptions (Theorem <ref>): If the input Turing machine halts, then the game has a perfect Z-aligned permutation strategy that commutes along edges, and if the input Turing machine does not halt, then the value of the game is bounded from above by 1/2. Therefore, together with our mapping from non-local games to subgroup tests, we deduce that approximating the sofic value of a game is undecidable. In turn, by Corollary <ref>, 
  the Aldous–Lyons Conjecture <ref> has a negative solution.


    In <cit.>, the authors were able to  specify a non-local game whose quantum value is bounded above by 1/2 but has a perfect quantum commuting strategy. Using the results of <cit.>, which we elaborate on further in the next remark,  this implies the existence of a finitely presented *-algebra which can be endowed with a tracial state and cannot be embedded into an ultrapower of the hyperfinite II_1 factor. This is a somewhat constructive counter example, but the state itself is only proven to exist and does not have a constructive form (at this point). By the various formulations of Connes' embedding problem, this proves that there is a non-hyperlinear character of the free group, which is the analogue of a non co-sofic IRS of the free group, but it is provided in a non-constructive form.

By applying the “same” construction as in <cit.> with the algorithm constructed in Theorem <ref> replacing the NPA hierarchy algorithm,  one can pin down an explicit subgroup test with perfect ergodic strategy but all sofic strategies bounded above by some constant (which can be approximated by following the proofs). This is the most constructive we can get in this case, namely, the specific non co-sofic IRS which is the perfect strategy for this subgroup test is proven only to exist, and is not constructed. Thus, we decided not to specify this test. 
    
    It would be very interesting to find an explicit non co-sofic IRS, namely separate between () and _() in a constructive manner.


  We comment on a natural attempt to extend the construction of <cit.> so as to directly deduce the existence of a non-sofic group, a result which would imply a negative answer to the Aldous–Lyons Conjecture. 

The idea (which is folklore, and explicitly mentioned in <cit.>) is as follows. It is well-known that to every synchronous game 𝒢 one can associate an algebra 𝒜(𝒢) such that there is a tight connection between the C^*-representation theory of this algebra and perfect  values for the game in various correlation sets <cit.>. In the special case where 𝒢 is a linear constraint system (LCS) game, there is a group  Γ such that 𝒜()  is (isomorphic to the completion of) the complex group algebra ℂ[Γ] <cit.>.[Actually, it is isomorphic to the completion of  ℂ[Γ]/⟨+1⟩, where  is a special generator in Γ and ⟨+1⟩ is the two-sided ideal generated by +1 in ℂ[Γ].] 

Now, if the reduction given in <cit.> returned only LCS games, then one would obtain the following result: there exists an LCS game with a perfect quantum commuting strategy, but no (asymptotically) perfect quantum strategy. By the connection mentioned above this implies that the group Γ associated with 𝒜()  is non-hyperlinear. As every sofic group is hyperlinear, one would thus obtain an example of a non-sofic group.

Since the reduction in <cit.> does not return LCS games, a workaround could be to find an embedding from the algebra 𝒜(𝒢), where 𝒢 is returned by <cit.>, into 𝒜(𝒢') where 𝒢' is an LCS game. Obstacles to this “black-box” approach are discussed in <cit.>. 

One can view our approach as finding a “sweet spot” that is in some sense half-way between synchronous games, which were used in <cit.>, and LCS games. This is because tailored games are a natural generalization of LCS games. In the companion paper we show that <cit.> can be adapted to return tailored games, and in the present paper we show that this suffices to resolve the Aldous–Lyons Conjecture. The existence of non-sofic groups (and non-hyperlinear groups) remains an  elusive open problem. 


  

    
Recall that a (synchronous 2-player) non-local game  consists of a (finite) graph G=(V,E), a length function ℓ V→ℕ, formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} for every vertex x∈ V, a distribution μ over the edges E, and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y. Namely, the decision function D_xy takes as input a pair of functions α S_x→{0,1} and β S_y→{0,1} and outputs a single bit. A (synchronous quantum) strategy for  is a map ρ⋃_x∈ VS_x→ U(n), where the images are of order 2, and ρ(S_x) commute for every fixed x∈ V.   Since ρ() is of order 2, we denote by ^_0 the projection on the (+1)-eigenspace of ρ(), while ^_1 the projection on its (-1)-eigenspace. Then, the  value of  ρ against  is

    (,ρ)=_xy∼μ∑_α S_x→_2 
    β S_y→_2 D_xy(α,β)τ(∏_∈ S_x∏_∈ S_y^_α()^_β()),

where τ is the dimension normalized trace on n× n matrices. The (synchronous) quantum value of , ^*(), is the supremum over all possible strategies against . The main result of MIP*=RE <cit.> is a reduction from the halting problem to approximating  the quantum value of a game. Given a Turing machine M, they construct a game _M such that: If M halts, then _M has a strategy ρ with (_M,ρ)=1 (and in particular, ^*(_M)=1).
    And if M never halts, then ^*(_M)≤1/2.
We now define a family of tests that will allow us to leverage this result, up to some caveats, to our needs.



A synchronous test  consists of a (finite) graph G=(V,E), a length function ℓ V→ℕ, a distribution μ over the edges E, and a challenge (K_xy;D_xy) associated to every edge xy∈ E. While keeping the S_x notation from before, we assume K_xy consists of only words in S_xy∪, where  is a global generator which plays the role of negation. Regardless of the specific test, the challenges always include the following checks:

    
  * All generators are order 2, namely ^2∈ H and
    
    ∀∈ S_xy   ^2∈ H.

    
  * The special generator  is non-trivial and central, namely ∉ H and 
    
    ∀∈ S_xy   [,]∈ H,

    where [·,·] is the (multiplicative) commutator.

The similarities between the two definitions are clear. One would hope that there is a straightforward way of transforming non-local games into synchronous tests, such that the values are preserved, or can  at least be bounded by one another in an appropriate manner. If this is possible, then one can take the transformation from Turing machines M to games _M a la MIP*=RE,  transform _M to the associated synchronous test (_M), and obtain a reduction from the halting problem to SOFVAL_c,s. While this is not true in general, by restricting the types of non-local games we focus on, we are able to do just that. The terms tailored form, perfect Z-aligned permutation strategy and augmented synchronoys test will be explained in Section <ref>.

There is a map from non-local games  in tailored form to (augmented) synchronous tests (()) such that:

    
  * If  has a perfect Z-aligned permutation strategy, then the associated  test (()) has a perfect strategy.
    
  * If ^*()≤1/2, then _((()))≤ 1-δ, for some universal constant δ>0.
 


    If the reduction from Turing machines to non-local games in MIP*=RE can be done such that all games are in tailored form, and in the completeness case the outcoming game has a perfect Z-aligned permutation strategy, then SOFVAL_1,1-δ is undecidable, and thus Conjecture <ref> is false.




 §.§ Structure of the paper

 In Section <ref>, we prove  structural results about invariant random subgroups, some of which are standard and well known to experts. In Section <ref>, we  prove our first main theorem, 
 Theorem <ref>,  and draw corollaries from it. 
 In Section <ref>, we provide  several examples of subgroup tests,  inspired  by  both group theory and complexity theory.  In Section <ref>, we discuss a rigidity property of tests that generalizes group stability and which is analogous to robustness of non-local games. 
In Section <ref>, we define tailored non-local games and analyze their properties. 
In Section <ref>,  we describe the (almost) value preserving correspondence between tailored non-local games and synchronous subgroup tests, informally outlined in Theorem <ref>. This allows us in Corollary <ref> to provide a negative solution to the Aldous–Lyons Conjecture <ref>. The proof of the aforementioned value preserving correspondence, Theorem <ref>, is proved in Section <ref>. Theorem <ref>, which is the other main ingredient in our refutation, is proved in  a companion paper <cit.>. There is no open problems section, but  further research directions can be found in Remarks  <ref>, <ref> and <ref>.

For a reader who seeks the main ideas of the paper without diving too deep into the proofs, we suggest reading, in addition to the introduction, only Sections <ref> and <ref>.



 §.§ Acknowledgements



We thank Alon Dogon for reading an early draft of this paper and providing many useful improvements to the presentation. We would also like to thank Mikael de la Salle for various discussions along the way. Finally, we want to thank Peter Sarnak for organizing a seminar on “strong convergence” in Princeton and asking the second author to give a talk, as it organized his thoughts  and led to the current presentation of ideas.

Lewis Bowen is supported by NSF grant DMS-2154680.
Michael Chapman acknowledges with gratitude the Simons Society of Fellows and is supported by a grant from the Simons Foundation (N. 965535).
Alex Lubotzky is supported by the European Research Council (ERC)
under the European Union's Horizon 2020 (N. 882751), and by a research grant from the Center for New Scientists at the Weizmann Institute of Science.
Thomas Vidick is supported by a research grant from the Center for New Scientists at the
Weizmann Institute of Science, a Simons Investigator award, AFOSR Grant No. FA9550-22-1-0391, and
ERC Consolidator Grant VerNisQDevS (101086733).
 


§ GROUP THEORETIC PRELIMINARIES




 §.§ Topology (and other properties) of invariant random subgroups
[The content of this section is standard. For example, it appears in Section 3 of <cit.>.]
We first go through some properties of the space {0,1}^ (induced with the product topology). For every  K,L⊆ℱ, let 

    𝒞(K,L)={A⊆ℱ| K⊆ A, L∩ A=∅}⊆{0,1}^.
 
Then,

    
  * When K and L are finite, the set (K,L) is open and closed. Moreover, the collection 
    {(K,L)| K,L⊆, |K|,|L|<∞}
 is a basis for the topology of {0,1}^. In addition, {0,1}^ is compact.
    
  * For every w∈ℱ and subsets K,L⊆ℱ we have
    
    w.𝒞(K,L)=𝒞(wKw^-1,wLw^-1).


The following is a standard fact. A proof is added because elements of it are later used in Lemma <ref>.
 
    The set of subgroups () is closed in the collection of all subsets {0,1}^. 



    A subset A⊆ is a subgroup if it satisfies the following three conditions: 
  
    (1) _∈ A ,  
           (2)  w∈ A w^-1∈ A ,  
           (3) w,w'∈ A w· w'∈ A.

For every w,w'∈, define the following open sets:  The subsets not containing _, C_=(∅,{_}). The subsets containing w∈ but not w^-1, C_w=({w},{w^-1}). The subsets containing w,w'∈ but not w· w',
    C_w,w'=({w,w'},{w· w'}).
     Then,
     
    (⋃_w,w'∈  C_w,w')∪(⋃_w∈ C_w ) ∪ C_={0,1}^∖(),
 
     which proves that () is a closed set.






Recall the definition of (), the invariant random subgroups of the free group  (Definition <ref>).


 
Let μ,ν∈ Prob({0,1}^ℱ). Then,

    
    
  * μ=ν if and only if μ(𝒞(K,L))=ν(𝒞(K,L)) for every finite K,L⊆ℱ. 
    
  * μ is conjugation invariant if and only if for every s∈ S we have s.μ=μ. Hence, μ∈() if and only if for all s∈ S and finite K,L⊆ℱ we have μ(𝒞(K,L))=μ(s.𝒞(K,L)).



A sequence of measures μ_n∈ Prob({0,1}^ℱ) is said to weak* converge to μ if for all continuous f{0,1}^ℱ→ℝ, 

    _H∼μ_n[f(H)] _H∼μ[f(H)].

We denote weak convergence by μ_n μ. 

    In our context, there is a more “down to earth” way of viewing weak* convergence. For every finite subset A⊆, and every  probability measure μ∈({0,1}^), we can define a probability measure μ∩ A∈({0,1}^A)⊆^2^A by 

    ∀ B⊆ A   μ∩ A(B)=μ({H⊆| H∩ A=B}).

Then,

    μ_nμ∀ A⊆, |A|<∞   μ_n∩ Aμ∩ A,

where the convergence on the right is the standard one in the Euclidean space ^2^A.

 
The space ({0,1}^ℱ) is compact in the weak* topology.



Since {0,1}^ℱ is compact, the fact can be deduced from the Banach–Alaoglu Theorem (cf. Theorem 3.15 in <cit.>).





 §.§ Pseudo subgroups

For every B⊆ C⊆, the restriction map R_B⊆ C{0,1}^C→{0,1}^B is 
    ∀ A⊆ C   R_B⊆ C(A)=A∩ B.
 When C= we remove it from the notation and write R_B instead of R_B⊆.
For sets K,L⊆ B⊆ we can define

    _B(K,L)={A⊆ B| K⊆ A, A∩ L=∅}.

Note that _(K,L)=(K,L).
 
    Let K,L⊆ B⊆ C⊆. Then R_B⊆ C(_C(K,L))=_B(K,L) and R_B⊆ C^-1(_B(K,L))=_C(K,L).


    Let A⊆ B and A'⊆ C such that A=A'∩ B. As K,L⊆ B, the intersection of A' with K and L depends only on A'∩ B=A. Hence  
    
    A∈_B(K,L)  A'∈_C(K,L).

  Adding this observation to the fact R_B⊆ C is onto proves the desired claim. 


Now, we can pushforward measures along the inverse system of restrictions. Namely, given  B⊆ C⊆ and a probability measure π on {0,1}^C we can define the probability measure R_B⊆ C*π on {0,1}^B by

    ∀ Y ⊆_ Borel{0,1}^B   R_B⊆ C*π(Y)=π(R_B⊆ C^-1(Y)).


 
   Claim <ref> implies that R_B⊆ C is continuous. Therefore,  R_B⊆ C*({0,1}^C)→({0,1}^B) is continuous.





    


This proof is very standard. As oppose to the other basic claims we attach proofs to, this one doesn't seem to contribute to the understanding of the main theorems. So maybe it should be removed. Waiting for your thoughts on it.
 Let f{0,1}^B→ be a continuous function. Define f{0,1}^C→ be the pullback of f along R_B⊆ C, namely 
    
    ∀ A⊆ C   f(A)=f∘ R_B⊆ C(A)=f(A∩ B).

    The pullback is a composition of continuous functions and hence continuous. For every μ∈({0,1}^C),
    
    _H∼μ[f(H)]=_H∼μ[f∘ R_B⊆ C(H)]=_H∼ R_B⊆ C*μ[f(H)].

       So, if  μ_n μ  in ({0,1}^C), then 
       
    _H∼ R_B⊆ C*μ_n[f(H)]=_H∼μ_n[f(H)]_H∼μ[f(H)]=_H∼ R_B⊆ C*μ[f(H)].

       Since this was true for any continuous f, we have that R_B⊆ C*μ_nR_B⊆ C*μ and we are done.






   Let B⊆. A subset  A⊆ B is said to be a pseudo subgroup of B if there is a subgroup H≤ such that R_B(H)=H∩ B=A. Denote by P(B) the collection of pseudo subgroups of B.[Note that this collection is a closed subset of the subsets of B.]

 
    Let A⊆ B⊆. Then, A is a pseudo subgroup of B if and only if ⟨ A⟩∩ B = A, where ⟨ A⟩ is the subgroup of  generated by A. 




    If there is a subgroup H for which  A=R_B(H)=H∩ B⊆ H, then in particular ⟨ A⟩⊆ H. Because A is always contained in ⟨ A⟩∩ B, we can conclude.



    Let A⊆ B⊆ C⊆. If A is not a pseudo subgroup of B, and A'⊆ C restricts to A (i.e., A'∩ B=A), then A' is not a pseudo subgroup of C.


    By Claim <ref>, if A is not a pseudo subgroup of B, then there is a w∈ B such that w∉ A while w∈⟨ A⟩.  Since w∈ B⊆ C, and A'∩ B=A, the same w satisfies that w∉ A' while w∈⟨ A⟩⊆⟨ A'⟩, proving that A' is not a pseudo subgroup of C.





Let B⊆ be a subset.



  * The set 𝒫_B is the collection of all π∈({0,1}^B)  that are concentrated on pseudo subgroups of B, namely π(P(B))=1. Equivalently, π({0,1}^B∖ P(B))=0.

  * The set 𝒬_B is the subset of _B of distributions that are S-invariant when defined. Namely, assuming  K, L ⊆ B are finite subsets such that 
    ∀ s∈ S∪ S^-1    sKs^-1, sLs^-1⊆ B,
 we have π( _B(K,L)) = π(_B( sKs^-1, sLs^-1)).  We call  probability distributions in Q_B pseudo invariant random subgroups over B, or pseudo IRSs in short.

  * Let _B=R_B*^-1(_B) and _B=R_B*^-1(_B).




    Note that _=_=(()) and _=_=().



Let B⊆ C⊆ be subsets.
    
       
        
  * The restriction to B of a distribution over pseudo subgroups of C, is a distribution over pseudo subgroups of B. Namely,
        
    ∀π∈_C   R_B⊆ C*π∈_B.
 
        Similarly, the restriction to B of a pseudo IRS of C is a pseudo IRS of B. Namely, 
        
    ∀π∈_C   R_B⊆ C*π∈_B.

        
  * The (Borel) probability distributions over subgroups of  are the intersection of pull-backs of all distributions over pseudo subgroups in finite subsets. Similarly, the IRSs of  are the intersection of pull-backs of all pseudo IRSs of finite subsets. Namely,
        
    ⋂_B⊆, |B|<∞_B=(()), ⋂_B⊆, |B|<∞_B=().

    




        
      Let π∈_C.  By definition, 
    π({0,1}^C∖ P(C))=0.
 By Corollary <ref>, 
    R_B⊆ C^-1({0,1}^B∖ P(B))⊆{0,1}^C∖ P(C).
 Hence, by the definition of the pushforward, 
      
    R_B⊆ C*π({0,1}^B∖ P(B))   =π(R_B⊆ C^-1({0,1}^B∖ P(B)))
       ≤π({0,1}^C∖ P(C))
       =0,

    and  so R_B⊆ C*π∈_B.  

        Assume further that π∈_C. By Claim <ref>, for every finite K,L⊆ B, 
    R^-1_B⊆ C(_B(K,L))=_C(K,L).
 
        Thus  
        
    R_B⊆ C*π(_B(K,L))   =π(R_B⊆ C^-1(_B(K,L)))
    =π(_C(K,L)).

        If K,L⊆ B such that sKs^-1,sLs^-1⊆ B for every s∈ S∪ S^-1, then the same applies to C since it contains B, and hence
        
    R_B⊆ C*π(_B(K,L))   =π(_C(K,L))
       =π(_C(sKs^-1,sLs^-1))
       =R_B⊆ C*π(_B(sKs^-1,sLs^-1)).

        Therefore R_B⊆ C*π is in _B and (1) is proved.
 

 
  By clause (1), for every B⊆ C⊆ we have _C⊆_B and _C⊆_B. In particular, when C= we have (())⊆_B and ()⊆_B. Hence  
    (())⊆⋂_B⊆, |B|<∞_B,  ()⊆⋂_B⊆, |B|<∞_B.

     We are left to prove the reverse containments. Recall the open cover of the complement of the subgroups described in the proof of Claim <ref>.
     Given π∈({0,1}^) such that π∉(()), we have 
    π((⋃_w,w'∈  C_w,w')∪(⋃_w∈ C_w ) ∪  C_)=π ({0,1}^∖())>0.

    Hence, by the union bound (Boole's inequality), there is either a pair w,w'∈ such that π(C_w,w')>0, or a single element w∈ such that π(C_w)>0, or π(C_)>0. Thus, if B={_,w,w',w· w',w^-1},  then π∉_B. In particular π∉⋂_B, proving the first reverse containment.
     
     Lastly, let π∈⋂_B.  For every finite K,L⊆, there is a larger enough finite set B such that K,L⊆ B and also sKs^-1,sLs^-1⊆ B for every s∈ S∪ S^-1. Since π∈_B,  by Claim <ref> applied to C=, we have for every s∈ S∪ S^-1 that
    
    π((K,L))   =π(R_B^-1(_B(K,L)))
       =R_B*π(_B(K,L))
       =R_B*π(_B(sKs^-1,sLs^-1))
       =π(R_B^-1(_B(sKs^-1,sLs^-1)))
       =π((sKs^-1,sLs^-1)).

    Hence, by Fact <ref>, (2) is deduced.



    

The following claim is never used in the text. It is used as a reference in the introduction, but it seems we do not need it. Should we remove it?

The Aldous–Lyons Conjecture <ref> is true if and only if for every finite subset B⊆ the set  R_B*(_())  is dense in  R_B*(()).


    Let π∈(), and let (B_t)_t=1^∞ be a sequence of subsets of  such that B_t⊆ B_t+1 and ⋃ B_t=. Since  R_B_t*(_())  is dense in  R_B_t*(()) (as subsets of the finite dimensional vector space ℝ^B_t equipped with some norm ‖·‖), we can find π_t∈_() such that ||R_B_t*π-R_B_t*π_t||≤1/t. By Claim <ref>, ({0,1}^) is compact. 
    Moreover, _B is a closed set when B is finite[This is true for any B⊆, but it is straightforward for finite ones, and we only use it in this case.] and R_B⊆ C* is continuous (Corollary <ref>), so by item (2) of Lemma <ref> () is the intersection of closed sets in the weak* topology, 
 and thus it is closed. 
 Hence, π_t has a subsequence with weak* limit π_∞∈() (without loss of generality, the subsequence is π_t itself). For every s≥ t we have 
    
    ||R_B_t*π-R_B_t*π_∞||≤ ||R_B_t*π-R_B_t*π_s||+||R_B_t*π_s-R_B_t*π_∞||.

    Since R_B_t* is continuous, and since π_sπ_∞, there is a large enough s for which 
    
    ||R_B_t*π_s-R_B_t*π_∞||≤1/s.

    We did not choose a specific norm up until now. By the triangle inequality 
    
    ∀σ,σ'∈({0,1}^B_s)   ||R_B_t⊆ B_s*σ-R_B_t⊆ B_s*σ'||_1≤ ||σ-σ'||_1.

    Hence by choosing ||·||=||·||_1, we have 
    
    ||R_B_t*π-R_B_t*π_s||≤ ||R_B_s*π-R_B_s*π_s||≤1/s.

   Therefore for every s≥ t,
    
    ||R_B_t*π-R_B_t*π_∞||≤2/s.

    But the left hand side is independent of s. Thus R_B_t*π=R_B_t*π_∞ for every natural number t. Let K,L⊆ be finite subsets. Since B_t⊆ B_t+1 and ⋃ B_t=, there is a large enough t for which K,L⊆ B_t. By Claim <ref>, 
    
    R_B_t*π(_B_t(K,L))   =π(R_B_t*^-1(_B_t(K,L)))=π((K,L))
    
         R_B_t*π_∞(_B_t(K,L))   =π_∞(R_B_t*^-1(_B_t(K,L)))=π_∞((K,L))

    which implies π((K,L))=π_∞((K,L)).  By Fact <ref>, we deduce π=π_∞. Therefore π_tπ and _() is weak* dense in (), proving Conjecture <ref>. 

    The reverse implication is immediate from the fact the pushforward along restrictions maps R_B*({0,1}^)→({0,1}^B)  are continuous, as was deduced in Corollary <ref>.


 
    Conjecture <ref> is false if and only if there is a finite set K⊆ and a functional φℝ^{0,1}^K→ℝ, such that 
    
    sup{φ(π)|π∈ R_K(_())}<sup{φ(π)|π∈ R_K(())}.



This formulation of Conjecture <ref> should motivate our approach of studying subgroup tests and their values, which are exactly linear functionals of the above format.




The system of finite subsets of   forms a directed system induced by inclusion. The pushforwards along  restriction maps R_B⊆ C* thus form an inverse system of maps between the compact sets (B) for B⊆, |B|<∞. In this language, Lemma <ref> says that 

    (())=_B_B    and    ()=_B _B.





 §.§ Computational aspects of pseudo invariant random subgroups


 
Given A⊆ B⊆ where |B|<∞, there is an algorithm deciding whether A is a pseudo subgroup of B. 





Given a finite set of elements A={ w_1,...,w_k }⊆, and another element w∈, there is an algorithm to decide whether w∈⟨ A⟩. Indeed, one can use the following:


  * Construct Core(A) the core graph of ⟨ A⟩  via the method of Stallings' foldings  <cit.>.

  * Check whether the path that begins at the basepoint of Core(A) and traverses w is closed.

  * If the path is closed, then w∈⟨ A⟩. Otherwise w∉⟨ A⟩.


By Claim <ref>, A is not a pseudo subgroup of B if and only if there exists a w∈ B such that w∉ A but w∈⟨ A ⟩. Hence, a sufficient and necessary condition for A to be a pseudo subgroup of B is to be accepted by the following algorithm:

    
  * Construct  Core(A).
    
  * For every w∈ B satisfying w∉ A, reject if w is a closed path in Core(A).
    
  * Accept if A passed (2) for every w∈ B.




    Recall that a free group is LERF (locally extended residually finite), i.e., for every finite set A={w_1,...,w_k} and every w∈ such that w∉⟨ A⟩, there exists a finite index subgroup H for which 

    A⊆ H and w∉ H.

This was originally proved by M. Hall <cit.>, and can be proven using the method of Stallings' foldings <cit.>. It follows that for  every A a pseudo subgroup of a finite B, there exists a finite index subgroup H for which R_B(H)=A.



    Let B⊆ be a finite set. Then, the set _B of distributions over pseudo subgroups of B and the set _B of pseudo IRSs over B are computable polytopes in ℝ^{0,1}^B. Namely, they are defined by (computable) integral linear equations and inequalities.


         Recall that _B⊆_B⊆({0,1}^B)⊆^{0,1}^B. Namely, all these distributions  are vectors which are indexed by subsets of B.     
     We describe the defining system of equations and inequalities forcing a vector π∈ℝ^{0,1}^B to belong to _B. First, it needs to be a probability measure, hence  
        
    ∀ A⊆ B   π(A)   ≥ 0,
    ∑_A⊆ Bπ(A)   =1.

        Further, for every A⊆ B, we can run the algorithm in Claim <ref> to decide whether A is a pseudo subgroup, and if not to add the constraint π(A)=0. This is the defining system of _B. To get the defining system of _B, we need to add the following:
        For every K,L⊆ B, we can check whether for all s∈ S∪ S^-1, the sets sKs^-1 and sLs^-1 are also contained in B, and if so add the constraints
        
    ∀ s∈ S∪ S^-1   ∑_A∈_B(K,L)π(A)_π(_B(K,L))=∑_A∈_B(sKs^-1,sLs^-1)π(A)_π(_B(sKs^-1,sLs^-1)).

        








 








§ MAIN THEOREM I: APPROXIMATING THE VALUES OF SUBGROUP TESTS
 

Recall the definitions of challenges, tests and values from Section <ref>.
Let  be a test over the generating set S, with challenges (K_i;D_i)_i∈ Q and rational distribution μ over Q. Let K=⋃_i∈ Q K_i, which is a finite subset of (S)=. We can extend the definition of (,·) in the following way: For every  B satisfying K⊆ B⊆, and for every distribution π∈({0,1}^B),  let 

    (,π)=_A∼π_i∼μ[D_i(A∩ K_i)].

The next claim is formalizing the following intuitive idea: Since D_i depends only on the restriction of H to K_i, then replacing π by its push-forward along any restriction R_B will not change its value against .


Under the extended definition of the value, for every B and C such that K⊆ B⊆ C⊆, and for every π∈({0,1}^C), we have

    (,π)=(,R_B⊆ C*π).




    By definition, sampling A∼ R_B⊆ C*π is the same as sampling A'∼π and outputting A=A'∩ B.
    Also, K_i=B∩ K_i since K_i⊆ K⊆ B. Hence,
    
    (,R_B⊆ C*π)   =_A∼ R_B⊆ C*π_i∼μ[D_i(A∩ K_i)]
       =_A'∼π_i∼μ[D_i((A'∩ B)∩ K_i)]
       =_A'∼π_i∼μ[D_i(A'∩ K_i)]
       =(,π).



Recall the first clause of Theorem <ref>: There is a Turing machine M_1 that takes as an input the encoding of a test , and outputs an infinite sequence of non-decreasing numbers (α_t)_t=1^∞⊆ [0,1]  such that lim_t→∞α_t=_().




 
Let σ S→(X) be a finite action. The value of Φ(σ) against  is 
 
    (,Φ(σ))   =_H∼Φ(σ)_i∼μD_i(H∩ K_i)
       =∑_x∈ X∑_i∈ Qμ(i)/|X|· D_i((σ,x)∩ K_i)

 which is a rational combination of  evaluations of D_i. Since the sets K_i, functions D_i and coefficients μ(i) can be calculated in finite time from  the encoding of , and (σ,x)∩ K_i can be calculated from σ for every x∈ X, the value of Φ(σ) against  can be computed in finite time. 

  There is a computable countable enumeration of all (equivalence classes of) finite (S)-actions. To see that, note that a map σ S→(n) is defined by the tuple (n ; {σ(s)}_s∈ S). Since the elements of (n) can be ordered (e.g., lexicographically), as well as the elements of S, the infinite set of tuples (n ; {σ(s)}_s∈ S) inherits a countable enumeration. Denote by σ_t the t^ th action according to this enumeration.
 For t∈ℕ, let 
 
    α_t=max_s≤ t{(,Φ(σ_s))}.

 By our discussion so far, the number α_t can be computed in finite time. Moreover, (α_t)_t=1^∞ is a non-decreasing sequence by definition. 
 Since  Im(Φ)=_(), the sequence (α_t)_t=1^∞ tends to the supremum of (,·) over _(), which is, by (<ref>), _().


We now recall the second clause of Theorem <ref>:  There is a Turing machine M_2 that takes as an input the encoding of a test , and outputs an infinite sequence of non-increasing numbers (β_t)_t=1^∞⊆ [0,1]  such that lim_t→∞β_t=_().







 Let K=⋃ K_i as before, and let B⊆ be a finite subset containing K.  Recall the definition of _B the pseudo invariant random subgroups over B (Definition <ref>), and its pre-image _B. Recall also  Claim <ref> and the discussion before it, where we extended the notion of the value to probability distributions over  subsets of  that contain K. The B-approximation to the ergodic value of  is defined to be

    _B() = sup{(,π)|π∈_B}=sup{(,π)|π∈_B}.


Let {B_t}_t=1^∞ be a sequence of finite subsets of , all containing K, such that B_t⊆ B_t+1 and ⋃_t=1^∞ B_t=. 
We now claim the following:

    
  * _B_t() is computable.
    
  * The sequence _B_t() is non-increasing.
    
  * lim_t→∞_B_t()=_().


By Lemma <ref>, _B is defined by a computable integral system of equations and inequalities. Thus, maximizing a linear functional (,·) over it can be done using linear programming in finite time, proving (1).

By Lemma <ref>, whenever B⊆ C⊆ we have R_B⊆ C*(_C)⊆_B. Hence _B_t+1⊆_B_t, which implies _B_t+1()≤_B_t(), proving (2).

Moreover,  ()=_⊆_B_t  which implies _B_t()≥_().
On the other hand, by Claim <ref> the space ({0,1}^) is compact, and since _B are closed sets they are also compact. Hence there is a distribution π_t∈_B_t satisfying _B_t()=(,π_t). By the compactness of ({0,1}^), there is a weak limit to some subsequence of (π_t)_t=1^∞, which we denote by π_∞. Without loss of generality the subsequence is the original sequence. Since π_s∈_B_t for every s≥ t, and since   _B are closed, we deduce that π_∞∈⋂_B_t=(). By the definition of a weak limit, since (,·) is a continuous functional, we have 

    lim_t→∞_B_t()=lim_t→∞(,π_t)=(,lim_w^*π_t)=(,π_∞)≤_(),

proving (3). Therefore, choosing β_t=_B_t() finishes the proof.




§ EXAMPLES OF TESTS
 




 §.§ Relations verification test
  Given a finite subset R⊆(S), the R-verification test 𝒱_R checks whether R is contained in the sampled subgroup H. It contains a single challenge (R;D) with D= 1_R, namely

    ∀ A⊆ R   D(A)=
        1     A=R,
    
        0     A≠ R.


The sofic value of _R is always 1, e.g., using the Dirac measure concentrated on the group  itself (which associates via Φ with the trivial action of  on a singleton). Moreover, if the conjugation orbit of a finite index subgroup H passes _R with probability 1, then R⊆⋂_w∈ wHw^-1. 




Though the value of the R-verification test does not give us much information about R, we will see  in Section <ref> that the potential robustness of this test is an important property of R.
Moreover, this test has variations where you sample r∈ R according to some distribution and check only that r∈ H and not that all of R is in H. See for example CL_part1,CVY_efficient for more on that.




 §.§ Separation test
  
Given two finite subsets R,L⊆(S), the (R,L)-separation test 𝒱_R||L checks whether R is contained in the sampled subgroup H but L∖ R is not. It contains a single challenge (R∪ L;D) with D= 1_R, namely

    ∀ A⊆ R∪ L   D(A)=
        1     A=R,
    
        0     A≠ R.


 
A group Γ is residually finite if for every finite subset L⊆Γ where _Γ∉ L, there is a finite index normal subgroup N⊴Γ such that L∩ N=∅.




    A finitely presented group Γ=⟨ S|R⟩ is residually finite if and only if for every finite L⊆ such that L∩⟨⟨ R⟩⟩=∅[The subgroup ⟨⟨ R⟩⟩ is the smallest normal subgroup containing R.] there is a finitely described IRS passing _R||L with probability 1. 



    Let Γ=(S)/⟨⟨ R⟩⟩ be residually finite, and L⊆ a finite set disjoint from ⟨⟨ R⟩⟩. By Definition <ref>, there is a finite index normal subgroup N⊴Γ such that L⟨⟨ R⟩⟩∩ N=∅. Using the sequence of epimorphisms
    
    ↠Γ↠Γ/N

    we can define the left action of  on X=Γ/N which passes _R||L with probability 1. 
    
    On the other hand, given a finite subset _Γ∉ L⊆Γ, let  L'⊆ be any set of representatives for L. Since _Γ∉ L, we have L'∩⟨⟨ R⟩⟩=∅. 
    Hence, by our assumption,  there is an  action ρ S→(X) with (_R||L',Φ(ρ))=1. Without loss of generality, ρ is transitive (otherwise, its restriction to any orbit will do).  
    The fact (_R||L',Φ(ρ))=1 implies in particular that R⊆(ρ,x) for any x∈ X. Thus, ρ factors through Γ and defines an action ρ'Γ→(X) by ρ'(w⟨⟨ R⟩⟩)=ρ(w). By the fact (_R||L',Φ(ρ))=1, the permutation ρ(w) has no fixed points for every w∈ L', and thus ρ'(u) has no fixed points for every u∈ L. Hence N=⋂_x∈ X(ρ',x) is a normal subgroup of Γ satisfying N∩ L=∅, as required to deduce Γ is residually finite.



    Let X be a finite set. Given two permutations σ,τ∈(X), we define their normalized Hamming distance to be

    d_H(σ,τ):=|{ x∈ X|σ(x)≠τ(x)}|/|X|=ℙ_x∈ X[σ(x)≠τ(x)].





    A finitely presented group Γ=⟨ S|R⟩ is sofic if for every L⊆(S) such that L∩⟨⟨ R⟩⟩=∅, and for every >0, there is a finite set X and an action ρ S→(X) such that 
    
    ∀ r   ∈ R   d_H(ρ(r),_X)≤,
    ∀ w   ∈ L   d_H(ρ(w),_X)≥ 1-.




It is straightforward to see that soficity is a relaxation of residual finiteness. 



    A finitely presented group Γ=⟨ S|R⟩ is sofic if and only if for every finite L⊆ such that L∩⟨⟨ R⟩⟩=∅ we have _(_R||L)=1.



Assume Γ is sofic, and let  L⊆ be a finite set such that L∩⟨⟨ R⟩⟩=∅. For every >0, there is an action ρ_ S→(X_), such that 
  
    ∀ r   ∈ R   d_H(ρ_(r),_X_)≤,
    ∀ w   ∈ L   d_H(ρ_(w),_X_)≥ 1-.

    Hence,
    
    (_R||L,Φ(ρ_))   =_x∈ X_[∀ r∈ R,∀ w∈ L  ρ_(r).x=x, ρ_(w).x≠ x]
       ≥ 1-∑_r∈ R_x∈ X_[ρ_(r).x≠ x]-∑_w∈ L_x∈ X_[ρ_(w).x=x]
       ≥ 1-∑_r∈ Rd_H(ρ_(r),_X_)-∑_w∈ L(1-d_H(ρ_(w),_X_))
       ≥ 1-|R|-|L|.

    Since |R| and |L| are fixed, and >0 is arbitrarily small, we have _(_R||L)=1.

    On the other hand, if for every L we have _(_R||L)=1, then for every >0 there is an action ρ S→(X) with (_R||L,Φ(ρ))≥ 1-. Now, since 
    
    (_R||L,Φ(ρ))   =_x∈ X[∀ r∈ R,∀ w∈ L  ρ(r)x=x, ρ(w)x≠ x]
       ≤min(min_r∈ R(_x∈ X[ρ(r)x=x]), min_w∈ L(_x∈ X[ρ(w)x≠ x]))
       =min(min_r∈ R(1-d_H(ρ(r),_X)), min_w∈ L(d_H(ρ(w),_X))),

    we deduce that
 
    ∀ r   ∈ R   d_H(ρ(r),_X)≤,
    ∀ w   ∈ L   d_H(ρ(w),_X)≥ 1-.

    and Γ is sofic.



 §.§ Formula satisfaction test

Let φ(y_1,...,y_n) be a boolean formula. The test _φ is defined over the formal set of generators S={x_1,...,x_n}. It has a single challenge (K;D), where K=S. Note that the characteristic function of every subset A⊆ S is already in the form 1_A{x_1,...,x_n}→{0,1}, and thus induces an assignment for the formal variables. The decision predicate D accepts A⊆ S if 1_A induces a satisfying assignment for φ, i.e.,

    D(A)=φ( 1_A(x_1),..., 1_A(x_n)).

It is straightforward that (_φ)=1 if and only if φ is satisfiable. 

There is a more interesting version of this test, when the formula φ is a 3CNF. Recall that a 3CNF is an and of clauses of the form φ_t=y_i^_1 y_j^_2 y_k^_3, where y_i,y_j,y_k are taken from a fixed set of formal generators (as before), _1,_2,_3∈{0,1}, and we interpret y^0=y and y^1=¬ y. Now, checking that φ is satisfied by 1_A is the same as going over all the clauses and checking that they are all satisfied. Thus, we can define a randomized version of this check by associating a challenge (K_t;D_t) with each clause φ_t, by letting K_t={x_i,x_j,x_k} and D_t(A)=φ_t( 1_A(x_i), 1_A(x_j), 1_A(x_k)). Then, by choosing a uniform distribution over challenges,  the associated test _φ checks whether a uniformly chosen clause is satisfied by 1_A. The famous PCP theorem <cit.> says that any 3CNF φ can be (efficiently) transformed into a not much larger 3CNF φ', such that if _(_φ)=1, then _(_φ')=1, and if _(_φ)=0, then  _(_φ')≤1/2.
The scaled up version of this argument shows that any  problem (see <cit.>) can be reduced to approximating the sofic value of a test.[This last remark is meaningless as long as we do not specify the exact way we encode tests. The point  is that there is a natural way to encode tests such that  can be reduced to approximating the sofic value immediately from <cit.>. In any case,  this paper is devoted to proving a much stronger conclusion, which is that approximating the sofic value is as hard as the Halting Problem.]




    





§ ROBUSTNESS
 
 In this section we define  a rigidity property of tests called robustness (see Definition <ref>). Loosely, a test is robust if every almost optimal strategy against it is close to an optimal one. Though this notion is natural, one needs to clarify what is the measure of distance between strategies. 



 §.§ Edit distance

We first define a generalized version of the normalized Hamming distance (<ref>). Let X and Y be two finite sets, and assume X⊆ Y. Given two permutations σ∈(X) and τ∈(Y), we define their normalized Hamming distance with errors to be

    d_H(σ,τ)=1-|{ x∈ X|σ(x)=τ(x)}|/|Y|.

If for every y∉ X we let σ(y)=error ∉ Y,
then we can define equivalently 

    d_H(σ,τ)=ℙ_y∈ Y[σ(y)≠τ(y)],

which is the way we defined the normalized Hamming distance in (<ref>). Since the generalized version is the same as the usual Hamming distance when Y=X, we use the same notation for both and just call them the Hamming distance from now on.

Let S→ℝ_>0 be a function, which we call the significance function. 
Given two actions ρ S→(X) and φ S→(Y), we  define their -weighted distance as follows:

    d^
    (ρ,φ)=∑_s∈ S(s)· d_H(ρ(s),φ(s)).[Though the Hamming distance is always bounded by 1, the weighted distance may be larger than 1.]

For every θ∈(Y), let φ^θ S→(Y) be φ^θ(s)=θφ(s) θ^-1.
Now, the -weighted edit distance[
There is a more graph theoretical way of viewing the edit distance, in the spirit of the  section <ref>. It essentially measures how much one Schreier graph needs to be changed to get to the other graph. For more on that, see Section 4 of <cit.> and the Γ-graph notion in <cit.>.] of ρ and φ is  the following minima: 

    d^_ edit(ρ,φ)=min_θ∈(Y)(d^(ρ,φ^θ)).

Note that the edit distance is independent of the specific embedding of X in Y.
The edit distance induces a metric on _(). For every π,π' ∈_(), let

    d^_ edit(π,π')=inf{ d^_ edit(ρ,φ) |ρ∈Φ^-1(π),φ∈Φ^-1(π')} ,

where recall the definition of Φ in (<ref>).






  §.§.§ A graph theoretic perspective
 


 Let Γ be a  group with S a finite  generating set. Let X be a set and σΓ→(X) an action of Γ on X. The generalized Schreier graph  (σ,S) is a directed graph with vertex set X and edges labeled by S. The edges of (σ,S) are defined as follows: For every x∈ X  and s∈ S, there is a directed edge from  σ(s).x to x labeled by s, namely

    σ(s).x x.

We may say that an edge is labeled by some inverse of s∈ S, such as xy. By that we mean the edge labeled by s and oriented in the other direction, namely yx. This is natural as σ(s^-1)=σ(s)^-1. Moreover, when S generates Γ, it includes words with inverses, and so when one associates words in the generators to paths in a labeled graph, this interpretation is natural (cf. <cit.>). 


    This  generalizes the usual notion of a Schreier coset graph (Γ,H,S) by choosing σΓ→_H\^Γ to be the natural action of Γ on right H-cosets. 
 
The Schreier graph does not provide any new data about σ, but it  provides a graph theoretic perspective for approaching IRSs.
For example, a fact that we repeatedly use is that orbits of the action are in correspondence with the connected components of the Schreier graph. Moreover, the edit distance is better understood as a distance between graphs: Given two labeled graphs, one wants to transform one to the other. Whenever the endpoint of an edge labeled by s is changed, there is an associated cost (s). The same cost is incurred when deleting completely or adding such an s-labeled edge. The goal is to minimize the cost of moving between the graphs (up  to isomorphism). The edit distance is exactly the minimal cost of such a transformation.







  §.§.§ Topology induced by the edit distance

 It turns out that regardless of the significance function , the topology induced by the edit distance is strictly stronger than the weak^* topology:

  We have the following:


  * Given two sequences π_n,π'_n∈_() such that π_n converges  to π_∞∈() in the weak^* topology and lim_n→∞d^_ edit(π_n,π'_n)=0, then π'_n converge to the same π_∞ in the weak^* topology.

  * There are sequences π_n∈_() that converge in the weak^* topology but are not Cauchy sequences in the edit distance topology.




    The above proposition should be contrasted with the amenable case, studied in <cit.>, where the authors showed that the converse of (1) is true when the free group is replaced by an amenable group. See <cit.> for a different proof of the above clause (2).





Let min() be the minimum value of . Since S is finite, min()>0. So d^_ edit≥min()d_ edit where d_ edit is the edit distance with significance function equal to the constant 1. So lim_n→∞d_ edit(π_n,π'_n)=0. This reduces the problem to the ordinary edit distance.

Let K,L be finite subsets of . By Remark <ref>, we have

    lim_n→∞π_n((K,L)) = π_∞((K,L)).

So it suffices to prove

    lim_n→∞ |π_n((K,L))-π'_n((K,L))| = 0.

Let r>0 be a large enough radius so that K∪ L is contained in the ball B(r) ⊂ of radius r with respect to the word metric induced by S.

Because π_n and π'_n are finitely described, there are actions ρ_n S→(X_n) and ρ'_n S→(Y_n) which induce π_n and π'_n respectively. We may assume that for each n, either X_n ⊂ Y_n or Y_n ⊂ X_n. After conjugating if necessary, we may further assume

    d_ edit(ρ_n,ρ'_n) = d(ρ_n,ρ'_n) = ∑_s∈ S d_H(ρ_n(s),ρ'_n(s)).


Given an integer k ≥ 1, let (k) be the set of all v∈ X_n ∪ Y_n such that there exists g in the ball B(k)⊂ with ρ_n(g)^-1v ρ'_n(g)^-1v. The definition of edit distance gives

    |(1)| ≤ d_ edit(ρ_n,ρ'_n) |X_n ∪ Y_n|.

Suppose v ∈(k) ∖(k-1) for some k≥ 2. Then there is a g ∈ B(k) such that ρ_n(g)^-1v ρ'_n(g)^-1v. Let g=s_1⋯ s_k with s_i ∈ S ∪ S^-1. Since v ∉(k-1), if h=s_1⋯ s_k-1, then w:=ρ_n(h)^-1v = ρ'_n(h)^-1v and ρ_n(s_k)^-1w ρ'_n(s_k)^-1w. Hence w ∈(1) and so v ∈ B(k-1)(1). Therefore,

    |(k)| ≤ |B(k-1)| |(1)| ≤ |B(k-1)|d_ edit(ρ_n,ρ'_n) |X_n ∪ Y_n|.

Since |B(k-1)| is independent of n, and d_ edit(ρ_n,ρ'_n) 0, we can deduce that

    lim_n→∞|(k)|/|X_n ∪ Y_n| = 0

for all k. 

Because K ∪ L ⊂ B(r), if v ∉(r), then (ρ_n, v) ∈(K,L) if and only if (ρ'_n, v) ∈(K,L). Hence

    |π_n((K,L))-π'_n((K,L))| ≤|(k)|/|X_n ∪ Y_n|→ 0

as n→∞. This finishes the proof of (1).

To prove item (2), recall that if G=(V,E) is a graph then a subset I ⊂ V is independent if no edge in E has both endpoints in I. If G is finite then the independence ratio of G is α(G)=|I|/|V| where I is an independent set of maximum cardinality. If ρ:S →(n) is an action then we let α(ρ)=α(G) where G is the associated action graph.

For any actions ρ and ρ',  we have d_edit(ρ,ρ') ≥ |α(ρ)-α(ρ')|. So it suffices to show there exist action sequences {ρ_n}_n, {ρ'_n}_n whose corresponding IRSs weak^* converge to the same IRS and  |α(ρ)-α(ρ')|>c for some constant c>0.

By using a first moment argument, it can be shown that if ρ_n:S →(n) is chosen uniformly at random and if G_n is the associated action graph then with high probability α(G_n)<1/2 - c where c>0 is a constant depending only on |S| (which we assume is at least 2). Moreover,  there are sets Ω_n ⊂(n)^S with |Ω_n|/|(n)^S| → 1 as n→∞ such that if ρ_n ∈Ω_n for all n and π_n are the corresponding IRSs then π_n weak^* converges to the trivial subgroup {}. This can be derived by estimating the expected number of short cycles in the graph G_n <cit.>. A more precise estimate on Ω_n is obtained in <cit.>.

Therefore, there exists a sequence {ρ_n} of actions such that if π_n are the corresponding IRSs then π_n weak^* converges to the trivial subgroup {} and α(G_n) < 1/2 - c.

On the other hand, there exist ρ'_n:S →(n) whose action graphs are bi-partite such that if π'_n is the corresponding IRS then π'_n also weak^* converges to the trivial subgroup {}. Since the independence ratio in this case is 1/2, it follows that the sequence π_1,π'_1,π_2,π'_2,… cannot be Cauchy in the edit distance.

















 §.§ Robustness

Given s∈ S, we can define a map v_s→ℕ that given a reduced word w∈ counts the number of appearances of s or s^-1 in w. Namely, if w=∏_j=1^ℓ s_j^_j,  where s_j∈ S and _j∈{± 1}, is a reduced word, then v_s(w)=∑_j=1^ℓ 1_s(s_j).

Let  be 
a test with its usual associated data S, (K_i;D_i)_i∈ Q and μ. We define _ S→ℝ_>0, the significance function associated with , as follows: 

    _(s)=_i∼μ[∑_w∈ K_i v_s(w)].


 
    Let >0, and  be 
a test. Let X and Y be finite sets, and ρ S→(X) and φ S→(Y) be actions. Assume d^__ edit(ρ,φ)≤. Then |(,Φ(ρ))-(,Φ(φ))|≤.



    Assume without loss of generality that X⊆ Y and that d^__ edit(ρ,φ)=d^_(ρ,φ). Let i∈ Q. For every w=∏_j=1^ℓ s_j^_j∈, we have
    
    _y∈ Y[ 1_(ρ,y)(w)≠ 1_(φ,y)(w)]≤_y∈ Y[ρ(w).y≠φ(w).y].

Inequality (<ref>) is a consequence of the following implication: If  ρ(w) acts on y the same way as φ(w), then in particular  w is either in both (ρ,y) and (φ,y) or in none of them. For 1≤ k≤ℓ, let w_k be the suffix of w from position k, namely w_k=∏_j=k^ℓ s_j^_j. Moreover, let w_ℓ+1 be the empty word _. Then 

    _y∈ Y[ρ(w).y≠φ(w).y]≤_y∈ Y[⋁_1≤ k≤ℓρ(w_k). y ≠φ(w_k). y].

Inequality (<ref>) is a consequence of  the following implication: If the endpoint of the path beginning at y and labeled by w^-1 in (ρ,S) is different than the endpoint of  the same path in (φ,S), then the two paths diverged at some point. Now,
   
    _y∈ Y[⋁_1≤ k≤ℓρ(w_k). y ≠φ(w_k) .y]≤∑_k=1^ℓ_y∈ Y[ρ(w_k). y ≠φ(w_k). y ρ(w_k+1). y = φ(w_k+1). y].

Inequality (<ref>) is a consequence of the union bound and the logic tautology ⋁_j=1^m φ_j=⋁_j=1^m(φ_j¬φ_j-1), where φ_0= False. But, since 
   
    ρ(w_k) .y ≠φ(w_k) .y ρ(w_k+1) .y = φ(w_k+1) .y ρ(s_k^_k)∘ρ(w_k+1).y≠φ(s_k^_k) ∘ρ(w_k+1).y,

   and since ρ(w_k+1).y is uniformly distributed given that y is uniformly distributed, we can deduce that 
    
    _y∈ Y[ρ(w_k) .y ≠φ(w_k) .y ρ(w_k+1). y = φ(w_k+1) .y]
        ≤_y∈ Y[ρ(s_k) .y≠φ(s_k) .y ]_=d_H(ρ(s_k),φ(s_k)).

   All in all, 

    _y∈ Y[ 1_(ρ,y)(w)≠ 1_(φ,y)(w)]≤∑_j=1^ℓ d_H(ρ(s_j),φ(s_j))=∑_s∈ Sv_s(w)d_H(ρ(s),φ(s))

Hence, by the union bound and inequality (<ref>), we have

    _i∼μ_y∈ Y[(ρ,y)∩K_i≠(φ,y)∩K_i]   ≤_i∼μ[∑_w∈ K_i∑_s∈ S v_s(w)d_H(ρ(s),φ(s))]
       =∑_s∈ S_i∼μ[∑_w∈ K_iv_s(w)]_(<ref>)d_H(ρ(s),φ(s))
       =∑_s∈ S_(s)d_H(ρ(s),φ(s))
       =d^_(ρ,φ)≤.

Lastly, by combining  

    |(,ρ)-(,φ)|   ≤_i∼μ_y∈ Y[D_i((ρ,y)∩K_i)≠ D_i((φ,y)∩K_i)]
       ≤_i∼μ_y∈ Y[(ρ,y)∩K_i≠(φ,y)∩K_i]

with 
 (<ref>), we deduce the claim.



    For those familiar with  robustness of non-local games, d^__ edit plays a somewhat analogous role to the state-dependent distance (cf. Section 4 of <cit.>, for example).


Robustness is a reverse implication  to the one in Claim <ref>.


Let   be a test and δℝ_≥0→ℝ_≥0 a function satisfying δ()0. We say that  is δ-robust if: 


  * There exists an optimal strategy, namely π∈_() such that (,π)=_().

  * Almost optimal strategies are close to optimal ones, namely, if  π'∈_() satisfies (,π')≥_()-, then there is an optimal strategy π such that d^__ edit(π,π')≤δ().




Let Γ=⟨ S|R⟩ be a finitely presented group. Then, Γ is said to be δ-homomorphism stable, where δ()0, if for every ρ S→(X) where  _r∈ R[d_H(ρ(r),_X)]≤ there is a Γ-action φ S→(Y), where X⊆ Y and such that _s∈ S[d_H(ρ(s),φ(s))]≤δ().


    This notion is more commonly known as flexible pointwise group stability in permutations. See GlebskyRivera,ArzhantsevaPaunescu,BLT,BeckerLubotzky,CL_part1,CVY_efficient for various results in this theory.


    The group Γ=⟨ S|R⟩ is δ-homomorphism stable if and only if the  R-verification test _R is δ'-robust. The exact relation between δ and δ' can be calculated explicitly — they are constant multiples of one another, where the constants depend only on S and R.





§ TAILORED NON-LOCAL GAMES


We commonly use both {0,1} and _2 for the set with 2-modular arithmetic.



 §.§ Non-local games preliminaries

We repeat the definitions which appeared in Section <ref> of the introduction. Since this paper is not focused on non-local games, we treat them somewhat technically. For formal definitions associated with the complexity class ^*, we refer to <cit.>. For the connection between ^* and nonlocal games, a good starting point is <cit.>.


   A (synchronous) non-local game  consists of a finite graph G=(V,E), a length function ℓ V→ℕ, formal sets of generators S_x={^x,i| 1≤ i≤ℓ(x)} for every vertex x∈ V, a distribution μ over the edges E, and decision functions D_xy{0,1}^S_xy→{0,1} for every edge xy∈ E, where S_xy=S_x∪ S_y. We denote by S the set ⋃_x∈ V S_x.


   Anyone familiar with the definition of a non-local game will immediately notice that the standard formalism is different than the one we chose here. Usually, a (synchronous) non-local game is defined as a pair of finite sets X,A, commonly referred to as the question and answer sets respectively, together with a   distribution μ over pairs of questions X× X and a decision predicate D X× X× A× A→{0,1} satisfying D(x,x,a,b)=0 for any x∈ X and a≠ b∈ A. 

   It is quite straightforward to move between the definitions.   A way one can extract the data of Definition <ref> from the above is as follows: Let ℓ be the constant function Λ=⌈log |A|⌉, and fix an embedding of A into {0,1}^Λ. The vertices V of the underlying graph G will be X, and the support of μ will be the edge set E⊆ X× X. The formal generator ^x,i corresponds to the i^ th bit of the answer a when x∈ X is asked as a question. 
   Then, given that x,y were asked, a pair of answers a,b can be encoded as a map γ S_x∪ S_y→{0,1}, where 
   
    γ|_S_x=a   and  γ|_S_y=b .
 
  The  notation (<ref>) will be used repeatedly in the text. 
   Lastly, D_xy(γ)=D(x,y,a,b), where γ is the aforementioned encoding. 

   In the other direction, we can define A to be all bit strings shorter than Λ=max_x∈ Vℓ(x), X to be the vertex set V, and μ stays the same. Finally, D(x,y,a,b)=D_xy(γ) when a,b are of the correct lengths, i.e. a=ℓ(x),b=ℓ(y), and is 0 when either of them is of the incorrect length. Since we consider only synchronous strategies, the condition D(x,x,a,b) is never checked in practice for a≠ b and we can assume it was satisfied beforehand.
Because of this correspondence, we may refer to ℓ as the answer length function.



Non-local games are often dramatized as two-prover interactive proofs with one round <cit.>. This perspective  may help some readers to absorb the upcoming technicalities better. 
Two players, that can share some resources beforehand (random bits in the classical case and entangled qubits in the quantum case), are separated spatially — e.g., they are seated in far away rooms. 
A referee  samples a pair of questions and sends one to each player. 
The players then use their shared resources to come up with answers, and send them back to the referee. 
The referee then decides, using the decision predicate, whether the players won or lost. 
The decision predicate as well as the distribution over possible questions are assumed to be known to the players beforehand.  
We demonstrate this dramatization in Example <ref>.
   


The following is the quantum mechanics' analogue of a probability distribution. Similar to the way probability distributions (over finite sets) are collections of non-negative real numbers that add up to 1, the quantum analogue would be “non-negative” matrices that add up to the identity. 


    A positive operator valued measure (POVM) of dimension n with outcomes in a set A  is a mapping A→ M_n× n(ℂ) such that _a is a positive (semi-definite) matrix for every a∈ A and ∑_a∈ A_a=_n. It is called projective, or a projection valued measure (PVM), if in addition _a is an orthogonal projection for every a∈ A, namely (_a)^2=_a=(_a)^*, where * is the conjugate transpose operation. 
Given that  is a PVM and A={0,1}^Λ,  we define for 1≤ i≤Λ the i^ th marginal of  to be ^i{0,1}→ M_n× n(ℂ)  by 

    ^i_0=∑_aΛ→{0,1}
     a_i=0_a   and  ^i_1=∑_aΛ→{0,1}
     a_i=1_a.

Thus, _a=∏_i=1^Λ^i_a_i. Furthermore, the matrix ^i=^i_0-^i_1 is an order 2  unitary, which we call the i^ th binary observable. All in all, a PVM can be given either as a map {0,1}^Λ→ M_n× n(ℕ), or as a map [Λ]→ U(n) such that the images of  are commuting involutions — i.e., they square to the identity.

As its name suggests, every POVM  defines a probability distribution over its answer set A as follows

    [a  is sampled]=τ(_a),

where τ is the dimension normalized trace on n× n matrices. Such an answer is said to be sampled according to  and we denote it by a∼.


    In the case of a PVM with outcome set A={0,1}^Λ, the observables {^i}_i=1^Λ generate a unitary representation of _2^Λ. The _a's in this case can be read from the unitary representation using the Fourier transform of the representation. Hence, for the case of A={0,1}^Λ, a PVM can be given as a collection of mutually perpendicular ortohgonal projections that sum up to the identity, or as a unitary representation of _2^Λ, and one can move from one perspective to the other without losing information.


     A (synchronous quantum) n-dimensional strategy for  is a map ρ S→ U(n), where the images are involutions, and ρ(S_x) commutes for every fixed x∈ V — by saying that a set of matrices (or later permutations) commutes, we mean that every pair of elements in this set commute.
     Another way of viewing strategies is by saying that they associate with every vertex x∈ V an n-dimensional PVM (in observable form).
     The strategy ρ is said to be commuting along edges if ρ(S_xy) commutes for every edge xy∈ E. 

     
     Such a strategy defines for every edge xy∈ E a probability distribution over functions γ S_xy→{0,1} as follows:
     
    [γ S_xy→{0,1}  is sampled]=τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()),

where τ is again the dimension normalized trace on n× n matrices, ^_0ℂ^n→ℂ^n is the projection on the (+1)-eigenspace of ρ(), and ^_1ℂ^n→ℂ^n is the projection on its (-1)-eigenspace.[Note that ρ()=^_0-^_1, which exactly shows how to move from this observable form of the PVMs induced by ρ back to its projection based form.] We say that γ was sampled according to ρ if it has this distribution, and we denote it by γ∼ρ.

For γ S_xy→{0,1}, let ^x_γ=∏_∈ S_x^_γ(). Then, the map ^xy{0,1}^S_xy→ M_n× n(ℂ) defined by ^xy_γ=^x_γ^y_γ^x_γ  is  a POVM. Thus, γ∼ρ is a special case of sampling according to a POVM (<ref>), which in this case is ^xy. In the spirit of the notation in (<ref>), it may be appropriate to write ^x_γ=^x_a and ^y_γ=^y_b.


    The images of  according to ρ are usually called binary observables, and the value γ() is usually called the measurement outcome.


 
    There is a procedural (i.e., algorithmic) way of  sampling  γ S_xy→{0,1} according to ρ: 
Since ρ(S_x)⊆ U(n) commutes, these matrices have a mutual orthonormal basis of eigenvectors B_x. Similarly, ρ(S_y) have a mutual orthonormal basis of eigenvectors B_y.
 First, sample v⃗∈ B_x uniformly at random. 
 Then, sample u⃗∈ B_y according to the  squared length of its projection on v, namely 
    [u⃗∈ B_y  is sampled|v⃗∈ B_x  was sampled]=|⟨v⃗|u⃗⟩ |^2,

        where ⟨·| ·⟩ is the standard inner product on ℂ^n, i.e., ⟨v⃗|u⃗⟩=∑_i=1^n v_i· u_i.
 Finally,  γ S_xy→{0,1} is determined as follows — for ∈ S_x, γ()=
        0    ρ()v⃗=v⃗,

        1    ρ ()v⃗ =-v⃗,, and for ∈ S_y, γ()=
        0    ρ()u⃗=u⃗,

        1    ρ ()u⃗ =-u⃗..
        All in all, 
        
    [ (v⃗,u⃗)  is sampled]=|⟨v⃗|u⃗⟩|^2/n,

        which is independent of the ordering between x and y. Note also that the probability a specific γ is measured is  independent of the  chosen bases B_x and B_y.



The function γ S_xy→_2 is sampled in the above procedure if for the resulting pair (v⃗,u⃗) we have v⃗∈( ∏_∈ S_x^_γ()) and u⃗∈(∏_∈ S_y^_γ()). Denote by ^x_γ the product ∏_∈ S_x^_γ(), and similarly ^y_γ. So, 

    [γ  is sampled]=∑_v⃗∈(^x_γ)
    u⃗∈(^y_γ)|⟨v⃗|u⃗⟩ |^2/n.

By the fact that B_x is a mutual orthonormal basis of eigenvectors for ρ(S_X), every v⃗∈ B_x is either in the image of ^x_γ or in its kernel. Therefore, the above sum can be written as 

    ∑_v⃗∈(^x_γ)
    u⃗∈(^y_γ)|⟨v⃗|u⃗⟩ |^2/n=∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^x_γv⃗|^y_γu⃗⟩ |^2/n=∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2/n.

Since B_y is an orthonormal basis and ^x_γ,^y_γ are orthogonal projections,  for every v⃗∈ B_x we have 

    ∑_u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2=^y_γ^x_γv⃗^2=⟨^y_γ^x_γv⃗|^y_γ^x_γv⃗⟩ =⟨^x_γ^y_γ^x_γv⃗|v⃗⟩.

Thus,

    ∑_v⃗∈ B_x
    u⃗∈ B_y|⟨^y_γ^x_γv⃗|u⃗⟩ |^2/n=∑_v⃗∈ B_x⟨^x_γ^y_γ^x_γv⃗|v⃗⟩/n=τ(^x_γ^y_γ^x_γ)=τ(^x_γ^y_γ),

where the last equality uses the cyclicity of the trace and the fact ^x_γ is a projection. All in all, 

    [γ  is sampled]=τ(^x_γ^y_γ),

as needed.


 
The formula (<ref>) for the probability a specific γ is sampled is motivated by the Born measurement rule from quantum mechanics. In quantum mechanics, a (projective) measurement is specified by choosing a PVM  A→ B(ℋ) acting on a separable Hilbert space ℋ (Definition <ref> provided the finite dimensional case). 
Given a quantum state (unit vector) ψ∈ℋ, a measurement of ψ returns the outcome a with probability 
    [a   is the  measurement outcome]=ψ^* _a ψ.
 
If there are two systems (e.g. two “players”), then a Hilbert space ℋ_A, ℋ_B is associated to each of them
respectively, such that the joint Hilbert space is ℋ=ℋ_A ⊗ℋ_B. 
The joint distribution on outcomes is then given by probabilities 

    [(a,b)  are the  measurement outcomes]=ψ^* (_a^x ⊗ (^y_b)^†) ψ,

where † is transposition with no complex conjugation, as opposed to *.[It is common to drop the transposition † on the ℋ_B POVM in (<ref>). But, in this case, a transposition will appear in the calculation using traces (<ref>). So, we decided on this equivalent formulation.]
The case where ℋ_A and ℋ_B have the same (finite) dimension n, and ψ= n^-1/2∑_i e_x,i⊗ e_y,i for orthonormal bases {e_x,i} and {e_y,i} of ℋ_A and ℋ_B respectively, termed “maximally entangled,” yields 

    [(a,b)  are the  measurement outcomes]=τ(^x_a ^y_b).
 
 


    The  value of the strategy ρ against the non-local game  is

    (,ρ)   =_xy∼μ_γ∼ρ[D_xy(γ)]
       =∑_xy∈ E∑_γ S_xy→_2μ(xy)D_xy(γ)τ(∏_∈ S_x^_γ()∏_∈ S_y^_γ()).

 The (synchronous) quantum value of , ^*(), is the supremum over all possible strategies against .
 




The main result in <cit.> is a reduction from the Halting Problem to approximating  the quantum value of a game:

There exists a polynomial time algorithm that takes as input (the encoding of) a Turing machine M and outputs (the encoding of) a non-local game _M such that:

    
  * Sampling xy∈ E according to  μ and evaluating  D_xy(·) from the encoding of the  game _M  can be done in time poly(|M|), where |M| is the bit-length of the encoding of M.
    
  * If M halts, then there exists a quantum strategy ρ that commutes along edges, for which  (_M,ρ)=1. In particular, ^*(_M)=1.
    
  * If M never halts, then ^*(_M)≤1/2.


The  goal of the next section is to show that a somewhat stronger variation of Theorem <ref>, which is proved in <cit.>,  provides a negative solution to the Aldous–Lyons Conjecture <ref>.



 §.§ Permutation strategies

Our goal in  Section <ref> is to relate values of subgroup tests to values of non-local games and vice versa. The strategies for subgroup tests are finitely described IRSs, which are (induced by) maps of the form σ S→(n), while quantum strategies for non-local games are maps of the form ρ S→ U(n) adhering to certain conditions. These already seem quite related, as embedding (n) into U(n) as permutation matrices is natural, and indeed transforms every (nice enough) finitely described IRS into a quantum strategy. But, there is a problem with doing this naively: 
Say that we want to construct a quantum strategy, such that  for a certain ∈ S, regardless of anything else,  γ() will always be 1. Then, we can achieve it by choosing ρ()=-. But, - is not a permutation matrix. Therefore, any σ that will be naively transformed into a quantum strategy by embedding (n) into U(n) as permutation matrices cannot ensure a specific  is always evaluated to 1. 
To amend that, we use a tool developed in the study of solution groups of linear constraint system games (cf.  <cit.>): 
We add a special generator  that plays the role of -. 
Since - is a central involution without fixed points, the image of  under σ needs to behave the same. 
This forces σ to act on an even sized set. 
Now, if we want σ()=σ() to translate into γ() being always 1, we will need to restrict ourselves to the (-1)-eigenspace of σ(). This leads us to the following two definitions. 


    A permutation strategy for a non-local game  is a map 
σ S∪{}→(2n)
with the following properties:

    
  * The image of  is a central involution with no fixed points. Namely, 
    
    σ()^2=,  d_H(σ(),)=1,
       ∀∈ S   σ()σ()=σ()σ().

    
  * The rest of the images are involutions as well. Namely,
    
    ∀∈ S   σ()^2=.

    
  * The image of S_x commutes. Namely, for every fixed vertex x∈ V, 
    
    ∀,'∈ S_x   σ()σ(')=σ(')σ().





    Let σ S∪{}→(2n) be a permutation strategy for . Embed (2n) in U(2n) as permutation matrices. Note that in this viewpoint, U(2n) is acting naturally on functions f [2n]→ℂ, and the matrices are represented with respect to the standard basis, which consists of the indicators 
    
    ∀⋆∈ [2n]    1_⋆(♢)=
            1    ♢=⋆,
    
            0    ♢≠⋆.

    Moreover, for a permutation ζ∈(2n) and function f [2n]→ℂ, we have
    
    ∀⋆∈ [2n]   (ζ.f)(⋆)=f(ζ^-1.⋆).

    By (1) of Definition <ref>, σ() is a fixed point free involution, and thus its (+1)-eigenspace W^+ and (-1)-eigenspace W^- are of the same dimension n. Note that W^+ consists of all functions f [2n]→ℂ that are constant along -labeled edges in the Schreier graph[Generalized Schreier graphs were defined in  Section <ref>.] (σ,S∪{}), while W^- consists of all functions that flip their signs along -labeled edges. Let =^_1ℂ^2n→ W^- be the orthogonal  projection on W^-. Note that ^*=^_1 and ^*=_W^-.
    
    We can choose bases for W^+ and W^- as follows. 
    If we denote 
    
    ⋆'=σ().⋆

    for every ⋆∈ [2n], then ⋆⋆' is the perfect matching induced by -labeled edges in  (σ,S∪{}). 
    So, an orthogonal basis of W^+ may be taken to be  B^+={ 1_⋆+ 1_⋆'}_⋆∈ [2n], while for W^- one takes  B^-={ 1_⋆- 1_⋆'}_⋆∈ [2n] —  B^- contains for each vector also its opposite, so this is a basis when choosing a representative for each such pair. The union of these bases is an orthogonal basis for ℂ^2n. Furthermore, we later refer to B^- as the standard basis of W^-≅ℂ^n.



    Given that ρ S∪{}→(2n) is a permutation strategy, there are n orbits of σ(), and σ acts on them in a well-defined manner. If we arbitrarily denote each orbit as {+⋆,-⋆}, instead of {⋆,⋆'} a la (<ref>), then σ acts naturally on the set Ω_±={±}×Ω in a way that commutes with the sign flip. Hence, a permutation strategy can be defined as a map into signed permutations, where  is mapped to the sign flip, and such that the variables at each vertex are mapped to commuting involutions. This point of view may be helpful for understanding some of the next claims and remarks. We decided not to use the language of signed permutations in this paper, though in the companion paper <cit.> it is more pronounced. 



 Let σ S∪{}→(2n) be a permutation strategy for . Recall from Remark <ref> that W^- is the (-1)-eigenspace of σ(), and that ℂ^2n→ W^- is the orthogonal projection on W^-. Then,
the quantum strategy ρ S→End(W^-) induced by σ is  the mapping to the W^--corners
    
    ρ()=σ()^*.





    Let  σ be a permutation strategy. Then, the quantum strategy ρ induced by σ, as in Definition <ref>, is indeed a quantum strategy for , as in Definition <ref>.


    The criterion for being a quantum strategy as in Definition <ref> is to be a map into unitaries such that the image of the generators S_x associated with any specific vertex commute, and all the images are involutions.
First,
 
    ρ()^*=(σ()^*)^*=σ()^* ^*=σ()^-1 ^*.

    Now,  the fact that σ() commutes with  σ() for any ∈ S implies that ^*σ()=σ()^*. Therefore, 
    
    ρ()^*ρ()   =σ()^-1^*σ()^*
       =σ()^-1σ()__2n^*^*__W^-
       =^*=_W^-,

   and the image of ρ is contained in U(W^-). Item (2) of Definition <ref> states that  σ()^-1=σ() for all ∈ S,  and combined with (<ref>) gives
   
    ρ()^*=σ()^-1^*=σ()^*=ρ(),

   i.e., the images of ρ are involutions. Finally, since σ(S_x) commutes for every fixed x∈ V, we have
   
    ∀,∈ S_x   ρ()ρ()   =σ()^*σ()^*
       =^*σ()σ()^*
       =^*σ()σ()^*
       =σ()^*σ()^*
       =ρ()ρ(),

   which finishes the proof.



[Classical permutation strategies]
Let (2)={,ζ} be the group of permutations acting on two elements. For every fixed f S→{0,1}, we can define an action σ S∪{}→(2) as follows

    σ()=ζ  ;  σ()=    f()=0,
    ζ    f()=1.

It is straightforward to see that the quantum strategy associated with σ satisfies ρ()=(-1)^f(), and thus γ S_xy→{0,1} that is sampled according to ρ is deterministically f|_S_xy.
Strategies  of this form are usually called deterministic. By taking direct sums of such actions (for potentially different f's), we can get any (rational) distribution over deterministic strategies. These strategies are usually called classical. So, every (rational) classical strategy can be obtained as a permutation strategy. 





    Let σ S∪{}→(2n) be a permutation strategy, and ρ the associated quantum strategy. Then, if for a word w∈(S∪{}) we have ρ(w)=, then σ(w)=.


    Recall from Remark <ref> that the images of ρ act naturally, and in the same way as σ,  on W^-, the (-1)-eigenspace of σ(), which consist of functions f [2n]→ℂ that flip along -edges. Recall also the notation ⋆'=σ().⋆ from (<ref>).
    So, the fact that ρ(w)= means that for any  function f∈ W^-, σ(w).f=f as well.  
    This is true in particular for B^-={ 1_⋆- 1_⋆'}_⋆∈ [2n], the basis of W^- described in the same remark. Note that for every ζ∈(2n) we have ζ. 1_⋆= 1_ζ.⋆.
    Hence, for every ⋆∈[2n], we have

    1_⋆- 1_⋆'=σ(w).( 1_⋆- 1_⋆')= 1_σ(w).⋆- 1_σ(w).⋆'.

By evaluating the functions on both sides of the above equation on the point ⋆∈ [2n], we get

    1= 1_σ(w).⋆(⋆)- 1_σ(w).⋆'(⋆),

which means σ(w).⋆=⋆. Since ⋆ was any point in [2n], σ(w)= and we are done.





     A non-local game  is said to have a perfect permutation strategy if there exists a permutation strategy σ such that the  quantum strategy ρ induced by it, as in Definition <ref>, satisfies (,ρ)=1.
 


In the rest of this subsection we further analyze permutation strategies, and specifically discuss a procedural sampling — similar to  Claim <ref>  — of a γ S_xy→{0,1} according to ρ which is induced by a permutation strategy σ. 



 §.§ Interlude — Fourier bases of _2^m actions



    Let σ_2^k→(_2^k) be the standard action of _2^k on itself, namely σ(w).v=v+w for every v,w∈_2^k. The Fourier basis of σ is the following collection of functions, denoted by _2^k: For every u∈_2^k, let u_2^k→ℂ be 

    ∀ w∈_2^k   u(w)=(-1)^⟨ u,w⟩/√(2^k),

where ⟨ u,w⟩=∑_i=1^k u_iw_i is the standard bilinear form on _2^k.[We try to keep ⟨·,·⟩ for the bilinear form over a finite vector space, and ⟨· |·⟩ for the complex inner product.]



    The collection _2^k is an orthonormal basis of eigenvectors for the action  σ. 


    The action σ acts as permutations on the standard basis of ℂ^_2^k, and thus extends to act on the whole vector space of functions from _2^k to ℂ. Using the standard inner product ⟨·|·⟩ on  ℂ^_2^k as before, we can check that 
    
    ∀ u∈_2^k   ⟨u|u⟩ =∑_w∈_2^ku(w)·u(w)=∑_w∈_2^k1/2^k=1.

    Furthermore, 
    
    ∀ u≠ v∈_2^k   ⟨v|u⟩   =∑_w∈_2^kv(w)·u(w) 
       =1/2^k∑_w∈_2^k(-1)^⟨ v+u,w⟩
       =0,

    where the last equality is derived from the fact v+u≠0⃗. Lastly, 
    
    ∀ u,v,w∈_2^k   (σ(w).v)(u)   =v(u+w)
       =(-1)^⟨ u+w,v⟩/√(2^k)
       =(-1)^⟨ w,v⟩·(-1)^⟨ u,v⟩/√(2^k)
       =(-1)^⟨ w,v⟩v(u),

    and _2^k is indeed a collection of mutual eigenvectors for σ.


Recall basic group actions definitions. 
Let G be a group, and σ G→(X),τ G→(Y) be two homomorphisms, which induce actions of G on X and Y respectively. A map 
f X→ Y  is a factor (of G-actions)  — also known as a morphism of G actions or a  (σ,τ)-equivariance — if
for all w∈ G and x∈ X, we have f(σ(w).x)=τ(w).f(x).
Note that for every set S⊆ G, such a map induces a morphism of oriented edge-labeled graphs between (σ,S) and (τ,S).
The actions σ and τ are said to be equivalent if there exists a bijective factor between them, which in the graph case would be a (edge-labeled oriented) graph isomorphism between (σ,S) and (τ,S).
Given two actions, we can define their sum  σ⊕τ G→(X⊔ Y), where  ⊔  is the  disjoint union, to be

    ∀ w∈ G, z∈ X⊔ Y   
     (σ⊕τ(w)).z=σ(w). z     z∈ X,
    τ(w). z     z∈ Y.


and their  product σ⊗τ S→(X× Y) to be

    ∀ s∈ S, x∈ X,y∈  Y   (σ⊗τ(s)).(x,y)=(σ(s).x ,τ(s).y).


If σ is a transitive action and  x∈ X, then it is equivalent to the  action of G on G/(σ,x), where (σ,x) is defined (as in Section <ref>) to be
(σ,x)={w∈ G|σ(w).x=x}.
The bijective factor in this case is associating with σ(w).x the coset w·(σ,x), and recalling that by transitivity every y∈ X is of the form σ(w).x for some w∈ G. So,  every action σ G→(X) is equivalent to the sum of actions of G on quotients of it G/H, for some collection of subgroups H≤ G. 

Assume now that G is _2^m for some positive integer m. Let σ G→(X) be a transitive action. Since every subgroup H of  G is normal, this action factors through  some _2^k for k≤ m. Namely, we can find a bijective factor between X and the action of _2^k on itself: 
 Fix some ⋆∈ X; (σ,⋆) is a subspace of _2^m, and we implicitly assumed it is isomorphic to _2^m-k; for every ♢∈ X, there is an element g of _2^m for which σ(g).⋆=♢, and thus σ(g').⋆=♢ for every g'∈ g+(σ,⋆); so we can associate g+(σ,⋆) with ♢ and _2^m/(σ,⋆) with _2^k.  
For every g∈_2^m, let g̃∈_2^k be its image in the bijection between the quotient space _2^m/(σ,⋆) and _2^k. Then,  

    ∀ v∈_2^k   σ(g).v=(-1)^⟨ v,g̃⟩v.

In particular, for s∈ S⊆_2^m, the action of s on v can be read from what happens along edges labeled by s in (σ,S) — if v(⋆)=v(σ(s).⋆), then v is in the (+1)-eigenspace of σ (s), and otherwise it is in its (-1)-eigenspace.

All in all, given a not necessarily transitive action σ of _2^m on X, its orbits have Fourier bases as in Definition <ref> (extended to being zero on all vertices outside of the orbit), and  the disjoint union of these will be called  the Fourier basis of σ. 
Note that since the bijections between the orbits and _2^k can be chosen in many ways, the Fourier basis of σ is determined only up to the sign of each basis vector. 





 §.§ Procedural sampling according to permutation strategies

Let σ S∪{}→(2n) be a permutation strategy for , and recall Remark <ref> and the notation therein. 
By Definition <ref>, σ(S_x∪{}) consists of commuting involutions. 
Therefore, the group generated by σ(S_x∪{}) is a quotient of _2^S_x∪{} — this can be acquired by linearly extending the map 1_↦σ() for every ∈ S_x∪{}, where 1_S_x∪{}→_2 is the indicator of . 
This means that 

    σ_x=σ|_S_x∪{}
 
induces an action of  _2^S_x∪{} on [2n]. 
Choose B_x ⊂ℂ^2n to be the Fourier basis of σ_x, as was defined in the Interlude Section <ref>.
Let B_x^-= B_x∩ W^-⊂ B_x be the collection of n eigenvectors in B_x on which σ() acts as (-1), namely all v⃗∈ B_x such that  σ().v⃗=-v⃗. These are the Fourier basis elements that flip along edges labeled by  in (σ_x,S_x∪{}). Then, B_x^- is an orthonormal basis of eigenvectors for ρ(S_x).


    Let  σ S∪{}→(2n) be a permutation strategy for , and ρ S→ U(W^-) the quantum strategy associated to σ. Then, the sampling procedure of γ S_xy→{0,1} according to ρ can be described  as follows:
    
        
  * Sample ⋆∈ [2n] uniformly at random. Then, it is part of an orbit O_x⊆ [2n] of σ_x (defined in (<ref>)) and O_y⊆ [2n] of σ_y.
        
  * Choose a pair v⃗∈ B^-_x and u⃗∈ B^-_y such that v⃗ is supported on O_x and u⃗ is supported on O_y, with probability 
        
    2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|.

        
  * For ∈ S_x, let γ()=0 if ρ().v⃗=v⃗ and 1 otherwise. Similarly, for ∈ S_y, let γ()=0 if ρ().u⃗=u⃗ and 1 otherwise.
    
    We may use the notation γ∼σ instead of γ∼ρ to refer to sampling according to a permutation strategy.



The probability of sampling ⋆∈ [2n] which is in O_x∩ O_y is |O_x∩ O_y|/2n. Thus, the probability of sampling a specific pair v⃗∈ B_x^-,u⃗∈ B_y^- whose intersection of supports is O_x∩ O_y is
    
    |O_x∩ O_y|/2n·2⟨v⃗|u⃗⟩^2/|O_x∩ O_y|=⟨v⃗|u⃗⟩ ^2/n.

    This shows that the above sampling procedure provides the same distribution as the one in Claim <ref>.

    IS THIS PROOF REDUNDANT.
    We mainly need to show that the above is genuinely a sampling scheme, which boils down to the second step defining a probability distribution, i.e.,
    
    ∑_v⃗∈ B_x^-Supp(v⃗)=O_x 
    u⃗∈ B_y^-Supp(u⃗)=O_y⟨v⃗|u⃗⟩ ^2=|O_x∩ O_y|/2.

    This is because, once this is true, 
    the probability of getting a specific pair is still ⟨v⃗|u⃗⟩ ^2/n. To see that, 
    note that  the probability of sampling ⋆∈ [2n] which is in O_x∩ O_y is |O_x∩ O_y|/2n, and thus the probability of sampling a specific pair v⃗,u⃗ whose intersection of  supports is O_x∩ O_y is
    
    |O_x∩ O_y|/2n·2⟨v⃗|u⃗⟩^2/|O_x∩ O_y|=⟨v⃗|u⃗⟩ ^2/n.

    Note that if v⃗ and u⃗ do not have a common support, then their inner product is zero and the calculation remains valid.

Let 1_O_x be the indicator of O_x (it is actually a scalar multiple of a basis element in B_x, though not in B_x^-). Let u⃗· 1_O_x be the restriction of u⃗ to O_x, namely the function which agrees with u⃗ on O_x, and outputs 0 everywhere else. 
Since u⃗ is a Fourier basis vector supported on O_y, its non-zero entries are all of absolute value 1/√(|O_y|). Thus, ⟨u⃗· 1_O_x|u⃗· 1_O_x⟩=|O_x∩ O_y|/|O_y|. 
On the other hand, u⃗· 1_O_x is supported on O_x. 
Moreover, since  preserves O_x and O_y (by their definition), whenever ⋆∈ O_x∩ O_y, also σ().⋆∈ O_x∩ O_y. 
Thus, u⃗· 1_O_x is a function that flips along  labeled edges in (σ,S∪{}). All in all, we have 

    u⃗· 1_O_x=∑ _v⃗∈ B_x^-Supp(v⃗)=O_x ⟨v⃗|u⃗· 1_O_x⟩_⟨v⃗|u⃗⟩v⃗,

    and in particular 
    
    u⃗· 1_O_x^2=∑ _v⃗∈ B_x^-Supp(v⃗)=O_x ⟨v⃗|u⃗⟩^2.

    Summing up over the |O_y|/2 many u⃗∈ B^-_y which are supported on O_y, we deduce the claim.











 §.§ Tailored games

 As permutation strategies allow us to translate finitely described strategies of subgroup tests to quantum strategies of non-local games, the soon to be defined Z-aligned permutation strategies of tailored non-local games will allow us to move in the other direction. It is harder to motivate our choices beforehand in this case, but we hope that Section <ref>  clarifies what may seem cryptic in this section. We begin by restricting the family of non-local games that we focus on — for the reader familiar with linear constraint system games, we note that this family is a generalization of them.
 

Colloquially,  a tailored non-local game is one where D_xy reads part of γ, and decides according to this partial view which parity checks to apply on the whole of γ.[We considered calling such games controlled linear, since it is more informative. But, since conditionally linear is a term used in MIPRE,Tailored_MIPRE, and terms containing linear are generally overused, we decided to use a less informative notion.]

Formally,
   a tailored non-local game  is equipped with an extra structure, described shortly, and its decision functions D_xy behave canonically with respect to this extra data.
    Instead of a single length function ℓ,  has two length functions ℓ^ V→ℕ and ℓ^ V→ℕ, and ℓ=ℓ^+ℓ^. Before, the length function described the size of the formal generating set at each vertex. 
    Now, the formal set of generators S_x at x∈ V will be a disjoint union of the sets S_x^ and S_x^, where S_x^={^x,,i| 1≤ i≤ℓ^(x)} and S_x^={^x,,i| 1≤ i≤ℓ^(x)}. The elements of  S_x^ are  called readable variables at x∈ V and the elements of S_x^ the linear or unreadable variables at x. 
    Let S→{0,1} be the readability function, namely, the indicator of whether ∈ S is  readable or not.
      In addition,  is equipped with  a collection of controlled linear constraints functions L_xy that take as input a function γ^ S_x^∪ S_y^→_2, and outputs a collection of subsets of S_xy∪{}. Namely,
    
    L_xy_2^S_x^∪ S_y^→_2^_2^S_xy∪{}.

    The image of L_xy is interpreted as a collection of linear constraints that will be verified by the decision function.   
    Finally, the decision function D_xy(γ) behaves as follows: It restricts γ to the readable variables, namely looks at γ^=γ|_S_x^∪ S_y^ S_x^∪ S_y^→_2, and calculates L_xy(γ)=L_xy(γ^). Then, it extends γ such that γ()=1. Finally, for every  α∈ L_xy(γ), we have α S_xy∪{}→_2,  and D_xy verifies that 
    
    ⟨α,γ⟩=∑_∈ S_xy∪{}α() ·γ()=0.
 
    Namely, L_xy(γ) consists of linear constraints that γ needs to satisfy. 



    Though being tailored  seems to be quite a restrictive form for a non-local game, every game can be tailored in a trivial manner. First, all variables are declared readable, namely ℓ^=ℓ and ℓ^=0. Then, if the decision function D_xy decided to accept γ according to the original game, then it lets L_xy(γ) be empty (and thus all linear conditions will be satisfied regardless of what γ is). And, if D_xy decided to reject γ according to the original game, 
    then it chooses L_xy(γ) to contain the singleton {} as the single subset appearing in  L_xy. Note that {} represents the linear equation 1·γ()=0, which is 1=0, and thus cannot be satisfied. 


The last remark raises the question: What have we gained by defining tailored non-local games, if any game can be tailored in a straightforward manner? 


   A permutation strategy σ for a tailored non-local game  is said to be Z-aligned if the assignments  σ() for readable variables act on each point in [2n] either like the identity  or like σ(). This is equivalent to ρ() being a diagonal matrix when presented according to  the standard basis B^- of W^-.[ The standard basis is usually called the Z-basis in quantum information theory, and thus the name Z-aligned strategies.]  



    The classical permutation strategies described in Example <ref> are Z-aligned. But, one can construct permutation strategies that induce a classical strategy in the usual sense (i.e., whose images are all commuting) without being Z-aligned. 

It is clearer now why the way one tailors a non-local game matters: The existence of a perfect Z-aligned permutation strategy for the game  depends on it. Let us demonstrate this with binary linear constraint system (LCS) games, and specifically with the magic square game. For  a full description of  the magic square game we refer to <cit.> or <cit.>, and for a general  introduction to LCS games see <cit.>. 

[Linear constraint system games]
   Let A be an m× n matrix with _2 coefficients, and let b⃗ be a column vector in _2^m. Classically, such a pair defines a system of linear equations Ax⃗=b⃗ over _2.  It also defines a certain non-local game (A,b⃗) which is the quantum counterpart of this classical system of equations. Let us provide a dramatization of this game, in the spirit of Remark <ref>. In  (A,b⃗), the referee chooses a random linear constraint in Ax⃗=b⃗ (i.e., a row) and sends it to one of the players, and chooses  a random variable that appears  in the chosen constraint (i.e., a column whose intersection with the chosen row is non-zero) to the other player. It then expects the row player to respond with an assignment to the variables appearing in this constraint, and from the column player to respond with an assignment to its single variable. The referee declares this to be a winning condition if the row player's assignment satisfied the constraint, and both players agree on the value of the variable given to the column player. 
   The column player is thought to hold a global assignment and answer according to it, while the row player is assumed to have a satisfying assignment to each constraint — intuitively, by  checking consistency between the players, the referee verifies that the global assignment of the column player indeed satisfies all the constraints. 
   
   Let us define this game rigorously. 
   The vertices in the underlying graph of (A,b⃗) will be indexed by the rows and columns of the matrix A, namely {r_i| i∈ [m]} and {c_j| j∈ [n]}. There is an edge between a row r_i and a column c_j if and only if A_ij=1. The length of every column vertex is 1, and we denote by _j the variable associated with the j^ th column c_j. The length of each row vertex is the number of 1's in the row, and we associate variables S_r_i={_ij'| A_ij'=1} to every row r_i. The decision function D_r_ic_j gets as input an assignment γ to _j and {_ij'| A_ij'=1}, and accepts if and only if 

    ∑_j' A_ij'=1γ(_ij')=b_i   and  γ(_j)=γ(_ij).

Though for our discussion the distribution μ over edges in this game is not important, one can consider the following standard sampling scheme: 1) Choose a row uniformly at random. 2) Choose a uniform column out of the support of the chosen row.

Let us describe a non-trivial tailoring of  (A,b⃗). First, all variables are unreadable, namely ℓ^=0 and ℓ^=ℓ. Given that the edge r_ic_j was sampled, the system L_r_ic_j will consist of two checks, which are derived from 
(<ref>):[note that since there are no readable variables, L_r_ic_J is constant.] 

    α_ consistency()   =
        0    ≠_j,_ij
    
        1    otherwise
    α_ linear()   =
        0    =_j
    
        1    =_ij'∈ S_r_i
    
        b_i    =

Then, α_ consistency will force D_r_ic_j to check that γ(_j)=γ(_ij), and α_ linear will force it to check that ∑_j' A_ij'=1γ(_ij')=b_i, and thus it acts the same way as before. 


   The difference between the above tailored form of (A,b⃗) and the one suggested in Remark <ref> may seem technical. But, here all the variables are unreadable, and in the version of Remark <ref> version all variables are readable. 
   If all variables of a tailored game are readable, for it to have a perfect  Z-aligned permutation strategy is the same as having a perfect classical strategy — which in turn is the same as for the linear system Ax⃗=b⃗ to have a solution. 
   But, when all the variables are unreadable, there could be a perfect Z-aligned permutation strategy without Ax⃗=b⃗ having a solution. 
   For example, the perfect strategy for the Mermin–Peres magic square game (see for example <cit.>) can be derived from a permutation strategy (see our companion paper <cit.>), and since there are no readable variables in this case,  it is automatically Z-aligned.
   Furthermore, the magic square game has no perfect classical strategy. This shows that if we care about perfect Z-aligned permutation strategies, the way we tailor a game matters.







§ MAIN THEOREM II: TESTS ASSOCIATED WITH TAILORED GAMES


The goal of this section is to define a mapping from tailored non-local games to subgroup tests such that: 

    
  * Completeness: Perfect Z-aligned permutation strategies that commute along edges for the tailored game are translated to perfect finitely described strategies of the subgroup test.
    
  * Soundness: Almost perfect finitely described strategies for the subgroup test are close to almost perfect Z-aligned permutation strategies for the tailored non-local game. 
 

Throughout this section,  is a tailored non-local game with underlying graph G=(V,E),  length functions ℓ^,ℓ^ V→ with ℓ=ℓ^+ℓ^ and Λ =max_x∈ V(ℓ(x)), formal sets of generators S_x=S_x^∪ S_x^ for every vertex x∈ V, where S_x^={^x,,i| 1≤ i≤ℓ^(x)} and S_x^={^x,,i| 1≤ i≤ℓ^(x)}, a readability function S→{0,1} where S=⋃_x∈ V S_x, a distribution μ over  edges E, and decision functions D_xy_2^S_xy→_2 for every edge xy∈ E with a controlled linear constraints map L_xy_2^S_x^∪ S_y^→_2^_2^S_xy∪{}. 




    The synchronous subgroup test =() associated with the tailored non-local game  is defined as follows: The set of generators S̃ in the test  is S∪{}. The set  Q̃ which parametrizes the challenges of  is the edge set E of , and the distribution μ̃ over challenges is μ of . The decision function D̃_xy works as follows (and the subsets K̃_xy can be derived from this description): Given a subgroup H of (S̃), 
    
        
  *  It verifies that
    
    ∉ H  ;  ^2∈ H  ;  ∀∈ S_xy   [,]∈ H.

    
  *  It verifies that
    
    ∀∈ S_xy      ^2∈ H,
    ∀,'∈ S_x      [,']∈ H,
    ∀,'∈ S_y      [,']∈ H.

        
  *  
    It verifies that 
    
    ∀∈ S_x^∪ S_y^   ∈ H  or  ∈ H.

        
  *  If H passed all the previous checks, then the following function γ̃^ S_x^∪ S_y^→_2 can be calculated: γ̃^()=0 if ∈ H and γ̃^()=1 if ∈ H. After recovering this γ̃^, the subset 
        
    L̃_xy(H)=L_xy(γ̃^)

        can  be calculated.
        So, the decision function verifies that 
        
    ∀α∈L̃_xy(H)   ^α()·∏_∈ S_x^α()·∏_∈ S_y^α()∈ H,

          where the product is according to some pre-fixed ordering of S_x and S_y.
    
  
    If any of the above checks did not go through, D̃_xy rejects H, and otherwise it accepts it.



    Let us motivate the checks that the decision function D̃_xy is applying on H. Let σS̃→(X) be a finitely described strategy. If σ always passes clause:Check_1_in_associated_test, it means that  acts as a fixed point free central involution on X, which is part of the requirement of a permutation strategy  for  (see Definition <ref>). If σ always passes clause:Check_2_in_associated_test, then the images of all other variables in S̃ are involutions, and for any fixed x∈ V, σ(S_x) commutes, which are again requirements for σ to be a permutation strategy of . If σ always passes clause:Check_3_in_associated_test, then the readable variables always act as either  or , which implies σ is a Z-aligned permutation strategy for  (see Definition <ref>). So, the goal of the first three Checks is to force any finitely described strategy of  to be a permutation Z-aligned strategy for . Lastly, clause:Check_4_in_associated_test verifies “the same”  linear relations as , which means that the winning probability of σ  against   and  are similar.


We are ready to formulate our main theorem:

    Let  be a tailored non-local game with max_x∈ V (ℓ(x))=Λ>0, and let =() be its associated synchronous subgroup test, as in Definition <ref>. 
    
        
  *  Completeness: If  has a perfect Z-aligned permutation strategy that commutes along edges, then  has a perfect finitely described strategy.
        
  *  Soundness: If  has a finitely described strategy σ with (,σ)≥ 1-, then there exists a quantum strategy ρ such that (,ρ)≥ 1-CΛ ^4· 2^6Λ, where C>0 is a universal constant.
    

The next section is devoted to the proof of Theorem <ref>.  


    There exists a polynomial time algorithm that takes as input (the encoding of) a Turing machine M and outputs (the encoding of) a tailored non-local game _M such that:

    
  *  Sampling xy∈ E according to  μ and evaluating  D_xy(·) from the encoding of the  game _M  can be done in time poly(|M|), where |M| is the bit-length of the encoding of M.
    
  *  If M halts, then _M has a perfect  Z-aligned permutation strategy that commutes along edges.
    
  *  If M never halts, then ^*(_M)≤1/2.


Theorem <ref> is proved in <cit.>.

   The Aldous–Lyons Conjecture <ref> has a negative solution.


The idea is as follows. Given a Turing machine M, we can define  the tailored non-local game  _M calculated in Theorem <ref>, and then calculate the associated subgroup test (_M)  from Definition  <ref>. In the next paragraph, the following two facts are proved:

    
  * If M halts, then (_M) has sofic value 1.
    
  * On the other hand, if M does not halt, then the sofic value of (_M) is smaller than 1-λ(M), where λ(M)=2^-c|M|^c for some universal constant c>0 independent of M. 
 
 Recall that by Theorem <ref>, there is a computable sequence approaching _((_M)) from above and another computable sequence approaching _((_M)) from below. If the Aldous–Lyons Conjecture <ref> has a positive solution, then these values are the same, and as Corollary <ref> states, one can approximate _((_M)) up to any predetermined additive error θ. As λ(M) can be computed directly from |M|, one can choose θ=λ(M)/2, and thus deduce whether _((_M))<1 or _((_M))>1-λ(M). This in turn allows one to decide whether M halts or not. Since the Halting Problem is undecidable, this is a contradiction, which implies that Conjecture  <ref> must have a negative solution.



Let us prove the two bullets above. Since the running time of the verification procedure of _M bounds Λ=max_x∈ V(ℓ(x)), we can use clause:1_tailored_MIP* of Theorem <ref> to deduce that Λ=poly(|M|). 
    If M halts, then by clause:2_tailored_MIP* of Theorem <ref> and clause:1_in_values_associated_test of Theorem <ref>, we have _((_M))=1.
    If M does not halt, then by clause:3_tailored_MIP* of Theorem <ref>, we have ^*(_M)≤1/2. Hence, by clause:2_in_values_associated_test of Theorem <ref>,
    we have
    
    1/2≥ 1-CΛ^4· 2^6Λ(1-_((_M))).

    Rearranging this inequality, we get 
    
    _((_M))≤ 1-1/2CΛ^4· 2^6Λ.

    The quantity 1/2CΛ^4· 2^6Λ can be bounded from below by λ(M)=2^-c|M|^c, where c>0 is a universal constant that depends on the constants implicit in the notation  Λ=poly(|M|) (guaranteed by Theorem <ref>), as well as the constant C>0 appearing in Theorem <ref>. This finishes the proof.







§ PROVING THEOREM <REF>






 §.§ Proving the completeness in Theorem <ref>


This is the easy direction, we have done most of the work towards it, and here we use the notation that we have established before. Recall that S̃=S∪{}. Let σ S∪{}→(2n) be a perfect Z-aligned permutation strategy that commutes along edges for the tailored non-local game . Further recall that the finitely described IRS Φ(σ) samples a subgroup H of  as follows — it takes a uniform ⋆∈ [2n] and outputs H=(σ,⋆). By Definition <ref>,

    d_H(σ(),)=1   ;   σ()^2=  ;  ∀∈ S   σ()σ()=σ()σ().

Therefore, for every ⋆∈ [2n], 

    ∉(σ,⋆)   ;  ^2∈(σ,⋆)  ;  ∀∈ S   [,]∈(σ,⋆),

and when Φ(σ) runs against  it  always passes clause:Check_1_in_associated_test. Also by  Definition <ref>,

    ∀∈ S σ()^2=  ;  ∀ x∈ V, ∀,'∈ S_x   σ()σ(')=σ(')σ().

Therefore, for every ⋆∈ [2n],

    ∀∈ S ^2∈(σ,⋆)  ;  ∀ x∈ V, ∀,'∈ S_x   [,']∈(σ,⋆),

and when Φ(σ) runs against  it always passes clause:Check_2_in_associated_test.
By Definition <ref>, for every ⋆∈ [2n] we have 

    ∀∈^-1(1)   σ().⋆=⋆  or  σ().⋆=σ().⋆.

Namely, for every ⋆∈ [2n],

    ∀∈^-1(1)   ∈(σ,⋆)  or  ∈(σ,⋆),

and when Φ(σ) runs against  it always passes clause:Check_3_in_associated_test.

We are left to show that Φ(σ) always passes clause:Check_4_in_associated_test.
To that end, we need a couple of claims.
    The first is a well known and commonly used observation in quantum information theory. For the convenience of the reader, we provide a proof.

    Let S be a finite set, ρ S→ U(n) such that ρ(S) commutes[This claim can be generalized so that a weaker assumption is used. See Section 3 of <cit.>.] and all the images are involutions.  Then, given a fixed α S→_2, we have ∑_∈ Sα()γ()=0  with probability 1 when  γ∼ρ if and only if ∏_∈ Sρ()^α()=. Namely,

    _γ∼ρ[⟨α,γ⟩=0]=1 ∏_∈ Sρ()^α()=.



Assume without loss of generality that α is fully supported. Otherwise, we can focus on ρ|_Supp(α) and proceed in the same manner. 
    As before, we can write ρ()=_0^-_1^, where ^_0 is the projection on the (+1)-eigenspace of ρ(), and _1^ is the projection on its (-1)-eigenspace.  By Definition <ref>, 

    _γ∼ρ[∑_∈ Sγ()=0]   =∑_γ S→_2
    ∑γ()=0τ(∏_∈ S_γ()^).

Note that ^_0+^_1=, and thus 
∑_γ S→_2τ(∏_∈ S_γ()^) is always 1. 
Therefore, 

    τ(∏_∈ Sρ())   =τ(∏_∈ S(^_0-^_1))
       =∑_γ S→_2
    ∑γ()=0τ(∏_γ()^) -∑_γ S→_2
    ∑γ()=1τ(∏_γ()^)
       =2·∑_γ S→_2
    ∑γ()=0τ(∏_γ()^)-1
       =2·_γ∼ρ[∑_∈ Sγ()=0]-1.

Thus,  ∏ρ()= if and only if τ(∏ρ())=1 if and only if  _γ∼ρ[∑_∈ Sγ()=0]=1, which finishes the proof.



    Let σ S∪{}→(2n) be a Z-aligned permutation strategy for a tailored game , and let ρ be the quantum strategy associated with σ (as in Definition <ref>). 
    Then, the restriction of  γ∼ρ to the readable variables in S_xy (which we denoted by γ^) depends only on the vertex ⋆∈ [2n] sampled in the beginning of the procedure outlined in Claim <ref>.
    Furthermore, if we denote the sampled L_xy(γ) by L_xy(⋆), emphasizing its dependence on the sampled vertex, then L̃_xy((σ,⋆))=L_xy(⋆), where L̃_xy was defined in (<ref>).


    Let ⋆∈ [2n] be some fixed vertex, O_x be the σ_x orbit of ⋆, and ∈ S_x be a readable variable. Since σ is Z-aligned, either σ().⋆=⋆ or σ().⋆=σ().⋆. Now, as discussed in Section <ref>, σ acting on  O_x is equivalent to an action of _2^k on itself. Furthermore, checking whether  v⃗∈ B^-_x that is supported on O_x is in the (+1) or (-1) eigenspace of σ() depends only on whether v⃗ flips along an -labeled edge in (σ,S) for some vertex in O_x. Since v⃗∈ B^-_x, it must flip along -labeled edges. So, if σ().⋆=⋆ then σ()v⃗=v⃗ and γ()=0, and if σ().⋆=σ().⋆ then σ()v⃗=-v⃗ and γ()=1. 
    Since this was independent of the specific v⃗ that we considered, the first part of the claim is deduced.  
   This proves that γ̃^()=γ^(), where γ̃^() was defined in clause:Check_4_in_associated_test of Definition <ref>, and therefore L̃_xy((σ,⋆))=L_xy(γ̃^)=L_xy(γ^)=L_xy(⋆) which is the second part of the claim.


We can now come back to our proof. Let us, in the spirit of previous notations, denote  σ_xy=σ|_S_xy∪{}. Let ⋆∈ [2n], and let O_xy be the orbit of ⋆ under σ_xy. Since σ commutes along edges (see Definition <ref>), σ_xy induces an action of _2^S_xy∪{}. Since (σ_xy) is commuting, the action of readable variables in O_xy is constant, namely if a readable variable acts as  on some ⋆∈ O_xy, it will act that way on all ⋆∈ O_xy, and similarly for acting as . Furthermore, by Claim <ref>, the linear constraints sampled in the associated game L̃_xy at a specific vertex ⋆∈ O_xy are the same as the ones sampled by the tailored non-local game.
Joining these observations, there is a set of constraints L such that for every ⋆∈ O_xy we have  L̃_xy((σ,⋆))=L_xy(⋆)=L.  So, we can focus on the orbit O_xy, and study the corner of ρ, which was the quantum strategy induced by σ, with respect to the projection ^O_xyℂ^2n→ℂ^O_xy. 
Since ρ is a perfect strategy, for every γ that was sampled (according to Claim <ref>) by first choosing ⋆∈ O_xy, and for every α∈ L, we have ∑α()γ()=0. 
By Claim <ref>,  this means that ∏ρ()^α() is the identity when restricted to functions supported on O_xy that flip along -labeled edges. By Claim <ref>, we can deduce that ∏σ()^α() acts as the identity on O_xy. Since this was true for every α in L, and L=L̃_xy((σ,⋆)) for every ⋆∈ O_xy, we can conclude that σ always passes clause:Check_4_in_associated_test when run against . All in all, σ is a perfect strategy for , proving the completeness clause of Theorem <ref>.



 §.§ Proving the soundness in Theorem <ref>


The plan is as follows:

    
  * Recall that S̃= S∪{}.  Given a finitely described strategy σS̃→(n) for , with (,σ)≥ 1-, we  perturb it bit by bit until it passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test from the definition of  (while potentially worsening the probability of passing  clause:Check_4_in_associated_test). 
    
  * Then, we can bound from below the value of this perturbed strategy using the mechanisms of Section <ref>. 
    
  * Since the resulting perturbed strategy satisfies the first three checks of , it is a Z-aligned permutation strategy for . For such strategies, we prove that their value against  induces a lower-bound on their value against . This will conclude the proof.
   

The following three propositions are the manifestation of the above plan. We abuse notations and denote by μ the marginal distribution on vertices of 𝒢, namely

    ∀ x∈ V   μ(x)=1/2∑_xy∈ Eμ(xy).

Also, recall the notion of the significance function _  associated with a test , defined in (<ref>), and the notion of the edit distance d^_ edit (<ref>) between finitely described IRSs. Finally, recall that  Λ=max_x∈ V(ℓ(x)), and assume Λ>0.

   Let σS̃→(n) be a finitely described strategy  for , with (,σ)≥ 1-. Denote by _x the losing probability in  given that an edge containing x∈ V was sampled as the challenge — note that _x∼μ[_x]=. 
   Then, there is a strategy σ'S̃→(2m), where m= ⌈n/2⌉, that passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test in the definition of the associated test , and which is close to σ in the following sense:
   
    d_H(σ(),σ'())≤  C_0
    ∀ x∈ V, ∈ S_x      d_H(σ(),σ'())≤  C_0· 2^2Λ·Λ^3(+_x).

    for some universal constant C_0>0.



     Every  finitely described strategy 
 σS̃→(2n)   for  that passes with probability 1 clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test in the definition of the associated test  is a Z-aligned permutation strategy for  and vice versa. Moreover, there is a universal constant C_1>0 such that  if  (,σ)≥ 1-, then (,ρ)≥ 1-C_1· 2^2Λ where ρ is the quantum strategy associated with σ (Definition <ref>). 


    Let  be the test associated with the tailored non-local game .  Then, there is a universal constant C_2>0, such that the significance _τ̃ associated with the test   satisfies
    
    _()   ≤ C_2· 2^2Λ,
    ∀ x∈ V, ∈ S_x   _()   ≤μ(x)· C_2· 2^2Λ.



    We did not try to optimise the parameters in the three preceding propositions. Actually, we decided to choose a simpler version of the associated synchronous test (Definition <ref>) that provides an exp(Λ) deterioration in the value, as seen in Proposition <ref>. We have variations of tailored games and the associated synchronous test which allow for only a poly(Λ) deterioration in the value, but we could not find any reduction that is independent of Λ.


Now, we can deduce the soundness clause of Theorem <ref> as a corollary of the preceding propositions.


    Let σS̃→(n) be the given finitely described strategy with value ≥ 1-. By Proposition <ref> and Proposition  <ref>, there is a strategy σ'S̃→(2⌈n/2⌉) passing clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test with probability 1, and such that the significance weighted distance between σ and σ' is
    
    d^_(σ,σ')   =∑_s∈S̃_(s)· d_H(σ(s),σ'(s))
       =_()d_H(σ(),σ'())+∑_x∈ V∑_∈ S_x_()d_H(σ(),σ'())
       ≤ C_2· 2^2Λd_H(σ(),σ'())+∑_x∈ V∑_∈ S_xμ(x)· C_2· 2^2Λ d_H(σ(),σ'())
       ≤ C_0C_2·2^2Λ+∑_x∈ V|S_x|·μ(x)· C_2· 2^2Λ· C_0· 2^2Λ·Λ^3(+_x)
       ≤ C_0C_2·2^2Λ+ C_0C_2Λ ^4 · 2^4Λ(∑_x∈ Vμ(x)(+_x))__x∼μ[+_x]=2
       ≤ 3C_0C_2Λ ^4· 2^4Λ.

    Therefore, by Claim <ref>, 
    (,σ')≥ 1--3C_0C_2Λ ^4· 2^4Λ≥ 1-4C_0C_2Λ ^4· 2^4Λ.
 By applying Proposition <ref> on σ',  its associated quantum strategy ρ' satisfies  
    
    (,ρ')≥ 1-4C_0C_1C_2Λ ^4· 2^6Λ.

By    choosing C=4C_0C_1C_2, we deduce the claim.


We are left to prove Propositions <ref>, <ref> and 
 <ref>. This is done in a reverse manner.




   Note that  participates in most of  the checks in  regardless of the specific challenge. 
   In clause:Check_1_in_associated_test, it appears 3+2|S_xy| times, and since |S_xy|≤ 2Λ, this is bounded by 3+4Λ. 
   In clause:Check_2_in_associated_test it is not used. 
   In clause:Check_3_in_associated_test it is used at most |S_xy| times, which is bounded from above  by 2Λ. 
   Lastly, the number of times   appears in L_xy(γ) is bounded by the number of subsets of S_xy∪{} that contain it, which is 2^|S_xy|≤ 2^2Λ.   All in all, the significance of  is at most (3+4Λ)+2Λ+2^2Λ≤ 4· 2^2Λ. 
    
    Now, a variable  in S_x appears only if we sampled an edge in E with x as one of its endpoints. If this happened, it will appear twice in clause:Check_1_in_associated_test, 2+2|S_x| times in clause:Check_2_in_associated_test, at most twice in clause:Check_3_in_associated_test, and at most 2^|S_xy∪{}|-1 times in clause:Check_4_in_associated_test. Again, |S_x|≤Λ and |S_xy|≤ 2Λ, and thus  appears at most 2+2+2Λ+2+2^2Λ≤ 3· 2^2Λ times when an edge containing x is sampled. This means that the significance of  is at most μ(x)· 3· 2^2Λ. Choosing C_2=4 completes the proof.




Let σS̃→(2n) be a finitely described strategy that passes clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test of  with probability 1. Then, 
from perfectly passing clause:Check_1_in_associated_test and  clause:Check_2_in_associated_test, σ is a permutation strategy, and by perfectly passing clause:Check_3_in_associated_test  it is Z-aligned. The other direction was proved in the completeness analysis in Section <ref>. We are going to prove that 

    (,ρ)≥ 1- 2· 2^2Λ(1-(,σ)),
 and thus C_1=2 satisfies the claim.

We start by giving expressions for the success probability of σ in , and of the associated ρ in 𝒢 respectively. 
Fix an edge xy∈ E, and denote by σ_x=σ|_S_x∪{} and σ_y=σ|_S_y∪{} the appropriate restrictions of the permutation strategy σ.  The intersection of orbits O_x∩ O_y, where O_x is an orbit of σ_x and O_y is an orbit of σ_y, partitions [2n] into disjoint sets. Fixing a pair of orbits O_x,O_y such that O_x∩ O_y≠∅,  by Claim <ref>, there is a set L_xy^O_x,O_y  of subsets of  S_xy∪{} such that 

    ∀⋆∈ O_x∩ O_y   L^O_x,O_y_xy=L_xy(⋆)=L̃_xy((σ,⋆)).
 
For α∈𝔽_2^S_xy∪{}, define

    ^α=∏_∈ S_x∪{}^α()   and    ^α=∏_∈ S_y^α().

Let Υ̃^O_x,O_y(α) be the set of all vertices ⋆∈ O_x∩ O_y such that σ(^α).⋆≠σ(^α).⋆, namely all vertices in the intersection of orbits that do not satisfy the subgroup test constraint induced by α.  Let 

    Υ̃^O_x,O_y_xy = ⋃_α∈ L^O_x,O_y_xyΥ̃^O_x,O_y(α),

and let 

    Υ̃_xy=⋃_O_x,O_yΥ̃^O_x,O_y_xy,

where the union is over all orbits of σ_x and σ_y that have a non-empty intersection.
Then Υ̃_xy is the set of all  ⋆∈ [2n] such that D̃_xy( (σ,⋆))=0, hence

    (,σ)=1- _xy∼μ|Υ̃_xy|/2n =1-_xy∼μ_⋆∈ [2n]_xy(⋆)   ,

 where
 
    _xy(⋆)=|Υ̃_xy^O_x,O_y|/|O_x∩ O_y|

 and O_x and O_y are the unique σ_x and σ_y orbits for which ⋆∈ O_x∩ O_y.
 
Recall that according to the procedural sampling described in  Claim <ref>, any pair v⃗∈ B_x^-, u⃗∈ B^-_y of Fourier basis elements supported on σ_x-orbit O_x and σ_y-orbit O_y respectively, is sampled with probability 2⟨v⃗|u⃗⟩^2/|O_x∩ O_y| (conditional on the sampled ⋆ being in O_x∩ O_y), and it induces an “assignment” γ S_xy∪{}→_2 for which γ()=1.
 Let Υ^O_x,O_y(α) be the set of all such (v⃗,u⃗) for which 
  γ does not satisfy a specific linear constraint α∈_2^S_xy∪{}, namely ∑_∈ S_xy∪{}γ()α()≠ 0. 
  Let  
 
    Υ^O_x,O_y_xy = ⋃_α∈ L^O_x,O_y_xyΥ^O_x,O_y(α)

 be the collection of Fourier basis pairs (v⃗,u⃗)∈ B_x^-× B_y^- supported on O_x,O_y, such that the assignment γ induced by the pair is rejected by D_xy. 
 Finally let 
 
    Υ_xy=⋃_O_x,O_yΥ^O_x,O_y_xy⊆ B^-_x× B^-_y,

 which is exactly the collection of possible sampled pairs by the quantum strategy ρ induced by σ, given that the edge xy was asked,  that produce an assignment γ satisfying D_xy(γ)=0. 
 Hence,

    (,ρ)=1-_xy∼μ∑_(v⃗,u⃗)∈Υ_xy⟨v⃗|u⃗⟩^2/n= 1-_xy∼μ_⋆∈ [2n]δ_xy(⋆) ,

where 

    δ_xy(⋆) =∑_(v⃗,u⃗)∈Υ^O_x,O_y_xy2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|

and O_x,O_y are the unique orbits for which ⋆∈ O_x∩ O_y.
Observe the following claim.

    For every α∈𝔽_2^S_xy∪{},  σ_x-orbit O_x and σ_y-orbit O_y such that O_x∩ O_y≠∅, we have

    ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗|u⃗⟩ ^2 ≤ 1/2|Υ̃^O_x,O_y(α)|  .


Let us show how to deduce our conclusion assuming the  validity of this claim, and then we will address the claim itself.
Given ⋆∈ O_x∩ O_y, we have

    δ_xy(⋆)     (<ref>)=∑_(v⃗,u⃗)∈Υ^O_x,O_y_xy2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|
          ≤ ∑_α∈ L_xy^O_x,O_y ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)2⟨v⃗|u⃗⟩ ^2/|O_x∩ O_y|
        Claim <ref>≤∑_α∈ L_xy^O_x,O_y |Υ̃^O_x,O_y(α)|/|O_x∩ O_y| .

Now, for every α∈ L_xy^O_x,O_y,  we have by (<ref>) that Υ̃^O_x,O_y(α)⊆Υ̃^O_x,O_y_xy, which using the above analysis means 

    δ_xy(⋆)≤∑_α∈ L_xy^O_x,O_y |Υ̃^O_x,O_y_xy|/|O_x∩ O_y|≤ 2^2Λ+1·_xy(⋆),

where the last inequality uses (<ref>) and  the fact |L^O_x,O_y_xy|≤|_2^S_xy∪{}|≤  2^2Λ+1. 
Therefore,

    1-(𝒢,ρ)(<ref>)=_xy∼μ_⋆∈ [2n][δ_xy(⋆)]≤ 2· 2^2Λ_xy∼μ_⋆∈ [2n][_xy(⋆)](<ref>)=2· 2^2Λ(1-( 𝒯̃,σ)),

proving (<ref>) and thus the proposition. 




Let Γ̃^O_x,O_y(α) be the complement of Υ^O_x,O_y(α) in O_x∩ O_y, namely 

    Γ̃^O_x,O_y(α)= (O_x∩ O_y) ∖Υ̃^O_x,O_y(α).

Recall the notation ^α and ^α from (<ref>).
For ⋆∈ O_x∩ O_y, we have σ(^α).⋆∈ O_x and σ(^α).⋆∈ O_y by the fact O_x is a σ_x-orbit and O_y is a σ_y-orbit. If ⋆∈Γ̃^O_x,O_y(α), then by its definition we have ♢:=σ(^α).⋆=σ(^α).⋆, and ♢∈ O_x∩ O_y.  As σ(^α) and σ(^α) are involutions, we have σ(^α).♢=⋆=σ(^α).♢, which implies ♢∈Γ̃^O_x,O_y(α).  All in all, Γ̃^O_x,O_y(α) is invariant under the actions of ^α and ^α — graphically, if we add ^α and ^α labels to the Schrier graph of the action σ on [2n], then they agree on every ⋆∈Γ̃^O_x,O_y(α) and are contained in it.

For (v⃗,u⃗) ∈Υ^O_x,O_y(α)  we know that  σ(^α)v⃗ =±v⃗ while σ(^α)u⃗=∓u⃗. 
This means that for every ^α-labeled edge between vertices in Γ̃^O_x,O_y(α), v⃗ and u⃗ agree on one of its endpoints and evaluate to opposites on the other endpoint. 
This implies

    ⟨v⃗|u⃗⟩   =∑_⋆∈ O_x∩ O_yv⃗(⋆)u⃗(⋆)
       =∑_⋆∈Υ̃^O_x,O_y (α)v⃗(⋆)u⃗(⋆)+∑_⋆∈Γ̃^O_x,O_y(α)v⃗(⋆)u⃗(⋆)
       =∑_⋆∈Υ̃^O_x,O_y (α)v⃗(⋆)u⃗(⋆)+1/2∑_⋆∈Γ̃^O_x,O_y(α)v⃗(⋆)u⃗(⋆)+v⃗(σ(^α).⋆)u⃗(σ(^α).⋆)_=0
       =⟨v⃗· 1_Υ̃^O_x,O_y (α)|u⃗⟩,

where  1_Υ̃^O_x,O_y is the indicator function of the set Υ̃^O_x,O_y, and · is the pointwise product of functions.
Thus,

    ∑_(v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗|u⃗⟩^2   =∑_v⃗Supp(v⃗)=O_x(∑_u⃗
      (v⃗,u⃗)∈Υ^O_x,O_y(α)⟨v⃗· 1_Υ̃^O_x,O_y (α) |u⃗⟩^2)
       ≤∑_v⃗Supp(v⃗)=O_x(∑_u⃗Supp(u⃗)=O_y⟨v⃗· 1_Υ̃^O_x,O_y (α) |u⃗⟩^2)
       =(*),

and since 1_Υ̃^O_x,O_y(α) is supported on O_y, we have 

    ∑_u⃗Supp(u⃗)=O_y⟨v⃗· 1_Υ̃^O_x,O_y(α) |u⃗⟩^2=v⃗· 1_Υ̃^O_x,O_y (α)^2=|Υ̃^O_x,O_y (α)|/|O_x| .

Since there are |O_x|/2 many v⃗∈ B^-_x which are supported on O_x, we deduce that
(*)≤|Υ̃^O_x,O_y(α)|/2,
as desired.









































































































A short detour into (quite elementary) group stability results is needed for proving  Proposition <ref>.


 §.§ Interlude — Group stability


 [Almost involutions are close to involutions]
 Let ζ∈(X). Then,
there is τ∈(X) such that τ^2= and
d_H(ζ,τ)= d_H(ζ^2,). 


Let W={ x∈ X |ζ^2 .x=x}.
Note that the restriction ζ|_W is an involution W→ W.
Define 

    τ(x)=ζ(x)     x∈ W,
    
    x     x∉ W.

Then τ^2= and d_H(ζ,τ)=1-|W|/|X|=d_H(ζ^2,).








[Almost fixed point free are close to fixed point free]
        Let ζ∈(X) be an involution. Assume d_H(ζ,)≥ 1-. Then, there is an involution with no fixed points τ∈(Y), where |Y|=2·⌈|X|/2⌉, such that d_H(ζ,τ)≤ 2.


    Let W={x∈ X|ζ.x=x}. Since d_H(ζ,)=1-|W|/|X|, we can deduce that |W|≤|X|. If W is odd, add a vertex to it and make it even. Now that W is even sized, we can choose any perfect matching on it and define τ to be the involution induced by this perfect matching. Extend τ to the rest of X by letting it act as ζ on the vertices out of W. Then the resulting τ is an involution with no fixed points, and 
    
    d_H(ζ,τ)≤|W|+1/|X|≤ 2.



[Almost commuting involutions are close to commuting involutions]
    Given two involutions ζ,τ∈(X), there is an involution τ'∈(X) such that
    
    [τ',ζ]=   and   d_H(τ,τ')≤ d_H([τ,ζ],).

    
    
   Let δ=d_H([τ,ζ],), and let W={x∈ X |ζτ .x=τζ .x}. Then |W|= (1-δ)|X|. Moreover, for every x
    ∈ W, 
    
    ζτ.(ζ. x)   =ζ .(τζ .x)=ζ^2 τ .x=τ .x=τζ .(ζ .x)ζ .x∈ W;
    τζ. (τ .x)   =τ .(ζτ .x)= τ^2 ζ .x=ζ.x=ζτ .(τ .x)τ .x∈ W.

    Hence, W is invariant under the actions of ζ and τ. Thus, the following is well defined
    
    τ'(x)=τ(x)     x∈ W,
    
                x     x∉ W.

    We therefore can conclude that d_H(τ,τ')≤ 1-|W|/|X|=δ, and 
    
    ∀ x∈ W   τ'ζ.x_∈ W   =τζ. x=ζτ .x=ζτ'. x;
    ∀ x∉ W   τ'ζ. x_∉ W   = ζ .x=ζτ'.x,

    which implies [τ',ζ]= as required.
 
 
     Claim <ref> can be proved even without assuming that the permutations are involutions (while allowing both permutations to change and not only τ). This was originally proved in <cit.> and was later reproved with effective bounds in <cit.>.
 


The following is a special case of item 1 of Theorem 2 in <cit.>.
[Almost actions of finite groups are close to actual actions]
    Let G be a finite group, and let f G→(X) be a function. Assume that for every g,h∈ G, d_H(f(gh),f(g)f(h))≤. Then, there is a homomorphism φ G→(X) such that 
    
    ∀ g∈ G   d_H(f(g),φ(g))≤ |G|^2.



    Let x∈ X be a vertex such that 
    
    ∀ g,h∈ G   f(gh).x=f(g)f(h).x.

    Then, if O⊆ X is the orbit of x with respect to the action of (the group generated by) (f), then for every y∈ O we have f(gh).y=f(g)f(h).y. To prove that, it is enough to prove it for a single step, namely for  y=f(k).x for some k∈ G (the rest is by induction). So, for every g,h∈ G, we have
    
    f(gh).y=f(gh)f(k).x=f(ghk).x=f(g)f(hk).x=f(g)f(h)f(k).x=f(g)f(h).y,

    where all equalities are either f(k).x=y or (<ref>). This means that the restriction of f to its action on O induces a homomorphism from G to (O).
    
    We assumed for every g,h∈ G, that d_H(f(gh),f(g)f(h))≤. Therefore, there are at most |G|^2 many vertices z in X for which there is some g,h∈ G such that f(gh).z≠ f(g)f(h).z. So, the union of orbits as O consists of at least (1-|G|^2)|X| of the vertices in X. So, if we define φ(g).x=f(g).x whenever x satisfies (<ref>), and φ(g).x=x otherwise, then φ is an action of G it satisfies for every g∈ G, d_H(f(g),φ(g))≤ |G|^2.

 






 §.§ Proof of Proposition <ref>

Throughout this proof, we use the term triangle inequality for iterative applications of the Hamming distance's triangle inequality,  as was used  in the proof of Claim <ref>.


   Let σS̃→(n) be a finitely described strategy  for , with (,σ)≥ 1-. By clause:Check_1_in_associated_test, d_H(σ()^2,)≤. By Claim <ref>, there is a function σ_1S̃→(n) that agrees with σ on all ∈ S, but σ_1() is an involution and d_H(σ(),σ_1())≤. Again by clause:Check_1_in_associated_test, d_H(σ(),)≥ 1-, and by the triangle inequality d_H(σ_1(),)≥ 1-2. Let m=⌈n/2⌉. Then, by Claim <ref>, there is a function σ_2S̃→(2m) such that for every ⋆∈ [n]⊆ [2m], we have σ_2().⋆=σ_1().⋆=σ().⋆, and σ_2() is a fixed point free involution with d_H(σ_1(),σ_2())≤ 4. All in all, for every ∈ S∪{} we have d_H(σ_2(),σ())≤ 5.

   Now, let _x be the losing probability when an edge containing x is sampled in . Then, _x∼μ[_x]=. Furthermore, by clause:Check_2_in_associated_test, for every ∈ S_x we have d_H(σ()^2,)≤_x. Therefore, by the triangle inequality, d_H(σ_2()^2,)≤ d_H(σ_2()^2,σ()^2)+d_H(σ()^2,)≤ 10+_x. By Claim <ref>, there is a function σ_3S̃→(2m)  such that  σ_3()=σ_2(), and for every x∈ V and ∈ S_x both σ_3()^2= and d_H(σ_3(),σ_2())≤ 10+_x. Now, by clause:Check_1_in_associated_test, for every ∈ S_x we have d_H(σ()σ(),σ()σ())≤_x. Thus, by the triangle inequality, d_H(σ_3()σ_3(),σ_3()σ_3())≤ 30+2_x. Since σ_3() and σ_3() are involutions, we may apply Claim <ref> and find a function σ_4S̃→(2m) such that σ_4()=σ_3(), σ_4()σ_4()=σ_4()σ_4() and d_H(σ_4(),σ_3())≤ 30+2_x. Again by the triangle inequality, we deduce that d_H(σ(),σ_4())≤ 5, and that for ∈ S_x we have d_H(σ(),σ_4())≤ 40 +3_x.

   Recall that S_x={^x,i| i≤ℓ(x)} (ignoring readability). We extend σ_4 to _2^S_x in the following way: For ξ S_x→_2, let
   
    σ_4(ξ)=∏_i=1^ℓ(x)σ_4()^ξ(),

   where the product is ordered according to the index i. By clause:Check_2_in_associated_test, for every ,'∈ S_x we have 
    d_H(σ()σ('),σ(')σ())≤_x.

   Thus, by the triangle inequality, d_H(σ_4()σ_4('),σ_4(')σ_4())≤ 160+ 13_x. Note also that the images of σ_4 are involutions. Therefore, for every ξ_1,ξ_2 S_x→_2 we have 
   
    d_H(σ_4(ξ_1+ξ_2),σ_4(ξ_1)σ_4(ξ_2))≤ℓ(x)^2(160+ 13_x)≤Λ^2(160+ 13_x).

   By Claim <ref> applied to each _2^S_x individually, we have a function σ_5S̃→(2m) such that σ_5(S_x) commutes and consists only of involutions, and 
    d_H(σ_4(),σ_5())≤ |_2^S_x|^2 ·Λ^2(160+13_x)≤ 2^2Λ·Λ^2(160+13_x).
 Furthermore, because of the way σ_5 is constructed in the proof of Claim <ref>, by letting σ_5()=σ_4(), the image of  is still a central involution with no fixed points. So, σ_5 always passes clause:Check_1_in_associated_test and clause:Check_2_in_associated_test. 

   Since σ passes clause:Check_3_in_associated_test with probability _x when an edge with x∈ V  is sampled, we deduce that every readable variable ∈ S_x^ satisfies

    _⋆∈ [n][σ().⋆=⋆ or σ().⋆=σ().⋆]≥ 1-_x.

Hence, by the triangle inequality, 
   
    _⋆∈ [2m][σ_5().⋆=⋆ or σ_5().⋆=σ_5().⋆]   ≥ 1-_x-d_H(σ_5(),σ())-d_H(σ_5(),σ())
       ≥ 1-_x-5-2^2Λ·Λ^2(160+13_x)
       ≥ 1-2^2Λ·Λ^2(165+14_x).

   By the fact that σ_5(S_x) commutes for every x∈ V,  the property σ_5().⋆=⋆ implies σ_5().♢=♢ for every ♢∈ O_x, where O_x is the orbit of ⋆ induced by σ_5(S_x) (which we usually denoted as (σ_5)_x beforehand). Since σ_5() commutes with all images, the same is true for the property σ_5().⋆=σ_5().⋆.  
   Hence, in each orbit of σ_5(S_x), either all vertices ⋆ satisfy σ_5().⋆=⋆ or σ_5().⋆=σ_5().⋆ or none of them. 
   We can thus define a function σ_6 that agrees with σ_5 on the orbits in which all readable variables ∈ S_x^ act appropriately, and on the other orbits we just define for every ∈ S_x that σ_6().⋆=⋆ (note that we keep  acting the same no matter what). So, we have 
   
    ∀ x∈ V, ∈ S_x    d_H(σ_6(),σ_5())   ≤ |S_x^|· 2^2Λ·Λ^2(165+14_x)
       ≤ 2^2Λ·Λ^3(165+14_x),

and now σ_6 always passes clause:Check_1_in_associated_test, clause:Check_2_in_associated_test and clause:Check_3_in_associated_test. By the triangle inequality, d_H(σ_6(),σ())≤ 5 and for every ∈ S_x we have

    d_H(σ_6(),σ())   ≤ 2^2Λ·Λ^3(165+14_x)+2^2Λ·Λ^2(160+13_x)
       +30+2_x+10+_x+5
       ≤ 2^2ΛΛ ^3(370+30_x).

Choosing σ'=σ_6 and C_0=370, we deduce the claim.



    How better can the parameters in Propositions <ref>, <ref> and <ref> be? We know how to  make the parameters in Proposition <ref>  polynomial in Λ by implementing a Hadmamard code in every vertex x∈ V and using <cit.> or <cit.> in the analysis instead of <cit.> — since we did not see any theoretical gain by doing that, and it was a lengthier proof, we decided not to implement this parameter improvement. We are also not sure whether the dependence in Proposition <ref> is needed or not. Finally, the significance analysis performed in Section <ref> is very crude. Can a better analysis allow to remove the dependence on Λ altogether, at least for some rich enough family of games?




    



§ SUBGROUP STRATEGIES

We need to decide if and where to keep this

    A subgroup strategy for a non-local game  consists of an IRS κ∈((  S∪{})) with the following properties for κ-a.e. subgroup H.

    
  * The image of  is a central involution with no fixed points. Namely, we have 
    
    ^2 ∈ H, ∉ H,
       ∀∈ S   [,]∈ H.

    
  * The rest of the images are involutions as well. Namely,
    
    ∀∈ S   ^2 ∈ H.

    
  * The image of S_x commutes. Namely, for every fixed vertex x∈ V, and 
    
    ∀,'∈ S_x   [,'] ∈ H.





 Let κ∈((  S∪{})) be a subgroup strategy for .
The quantum strategy ρ induced by κ is defined as follows. Let =(S∪{}) and sub() be the space of pairs (H,Hg) with H ∈sub() and Hg ∈ H \. Let κ̃ be the measure on this space which disintegrates as 

    dκ̃(H,Hg) = dκ(H).

In other words, we are putting counting measure on the fibers of the map sub() →sub(). So if F is a non-negative Borel function on sub() then

    ∫ F(H,Hg) dκ̃(H,Hg) = ∫∑_Hg F(H,Hg)  dκ(H).

Let L^2(κ̃) denote the associated L^2-space. 

For f ∈, define an operator σ̃(f) on L^2(κ̃) by

    (σ̃(f)·ϕ)(H,Hg) = ϕ(f^-1Hf, f^-1Hg) =ϕ (f^-1Hf, f^-1Hf · f^-1g).

It is an exercise to verify that σ̃ defines a homomorphism from  into the unitary group of L^2(κ̃). 

Let L^2_1(,κ̃) ⊂ L^2(κ̃) be the subspace of functions ϕ such that σ̃()·ϕ = -ϕ. Let 

    ^_1: L^2(κ̃) → L^2_1(,κ̃)

be the orthogonal projection map onto L^2_1(,κ̃). Let

    ^_1: L^2(κ̃) → L^2(κ̃)

be the composition of ^_1 with the inclusion map L^2_1(,κ̃) → L^2(κ̃).

Note that (^_1)^*^_1=^_1 and ^_1(^_1)^*=_n.

Given a Hilbert space ℋ, let B(ℋ) denote the algebra of bounded operators on ℋ. Define ρ S→ B( L^2_1(,κ̃)) by
      
    ρ()=^_1 σ̃()(^_1)^*.








    Let  κ be a subgroup strategy. Then, the quantum strategy ρ induced by κ, as in Definition <ref>, is indeed a quantum strategy for , as in Definition <ref>.


The proof is the exact same as the proof of Claim <ref>. 








plain





 Paper Title: Asymptotics of the overlap distribution of branching Brownian motion at high temperature.



Quantum Simulation via Stochastic Combination of Unitaries
    Prineha Narang
    August 5, 2024
==========================================================



§ ABSTRACT 
    At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity.
    We investigate the precise decay rate of the probability to obtain an overlap greater than a, for some a>0, in the whole subcritical phase of inverse temperatures β∈ [0,β_c).
    Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally.
    Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases.



	 
	



§ INTRODUCTION


The (binary) branching Brownian motion (BBM) is a continuous-time branching Markov process constructed as follows.
At time t = 0, a single particle starts a standard Brownian motion in  from the origin.
After a random time following an exponential distribution with parameter 1, it splits in two, or equivalently, it dies while giving birth to two children.
These new particles then repeats the same process, independently of each other.
More precisely, the children perform independent Brownian motions from the position of their parent at its death and, after independent exponential times, they in turn split in two.
For a formal construction, see <cit.>.

Branching Brownian motion has been widely studied over the last fifty years, initially for its link with reaction-diffusion equations <cit.> which motivated a precise study of its extremes <cit.>.
Other important motivations for this model come from physics.
In spin glass theory, the study of disordered systems with a hierarchical structure of correlation for the energies dates back to the generalized random energy model introduced by Derrida and Gardner <cit.> and includes the BBM (see e.g. Bovier's book <cit.>).
The BBM can also be seen as an infinite dimensional limit of directed polymers <cit.> and is also related to diffractive scattering in high-energy particle physics <cit.>.
See Section <ref> for more details.
In all these frameworks coming from physics, understanding the overlap distribution of particles chosen according to the Gibbs measure plays an important role, and this paper is set in this context, by investigating the probability to have an atypical overlap in the high temperature phase.




 §.§ Definitions and previous results


From a statistical physics perspective, for some fixed time t ≥ 0, one can identify the positions of particles alive at time t with the energy levels of a physical system.
Each particle can then be thought of as a configuration of the system.
Let (t) be the set of particles alive at time t and X_u(t) the position of particle u at time t.
The Gibbs measure of the BBM at time t and at inverse temperature β≥ 0 is the (random) probability measure _β, t on (t) that assigns to each particle u a weight proportional to ^β X_u(t).
By rescaling the associated partition function, we obtain the so-called additive martingale of the BBM at inverse temperature β, introduced by McKean <cit.>,

    W_t(β) = ^-ψ(β)t∑_u ∈(t)^β X_u(t),

where ψ(β) = 1 + β^2/2.
Since it is a positive martingale, it converges to a random variable W_∞(β).
This limit plays a key role in the study of the long-term behavior of the BBM, as illustrated in Theorem <ref> and Theorem <ref> below.
A phase transition occurs: by <cit.>, if β≥√(2) then W_∞(β) = 0 , if β < √(2) then W_∞(β) > 0 a.s.
In the latter case, the additive martingale is uniformly integrable, or equivalently, it converges in L^1.
In the the critical case β = √(2), it is more suitable to consider the derivative martingale, introduced by Lalley and Sellke <cit.>,

    Z_t = ∑_u ∈(t) (√(2)t - X_u(t)) ^√(2)X_u(t) - 2t.

The authors showed that it converges to a random variable Z_∞ and that Z_∞ > 0 The critical additive martingale is related to the derivative martingale through the following convergence

    √(t) W_t(√(2)) √(2/π) Z_∞,   in probability,

proved for the branching random walk by Aïdékon and Shi <cit.>.
For the BBM, this is a direct consequence of <cit.>.


In this paper, we are interested in the behavior of the overlap between particles chosen according to the Gibbs measure.
The overlap between two particles u, v ∈(t), denoted q_t(u, v) ∈ [0,1], is defined as 
the last time where u and v have had a common ancestor alive, rescaled by a factor of t.
In other words, it is the proportion of time where u and v have shared a common trajectory.
Then, the overlap distribution at time t ≥ 0 and inverse temperature β is the following random probability measure on [0,1]:

    ν_β, t = _β, t^⊗ 2(q_t(u, v) ∈·) = 1/W_t(β)^2∑_u, v ∈(t)^β(X_u(t) + X_v(t)) - 2 ψ(β) tδ_q_t(u, v),

so that, for a ∈ [0,1], ν_β, t([a,1]) is the conditional probability, given the BBM, that two particles chosen independently according to _β,t have an overlap at least a.
This quantity was introduced by Derrida and Spohn <cit.> by analogy with the overlap between two configurations of spin glasses.
We are also interested in the mean overlap distribution [ν_β, t], which is a deterministic probability measure on [0,1] such that [ν_β, t([a,1])] is now the (non-conditional) probability that two particles chosen independently according to _β,t have an overlap at least a.


The overlap distribution satisfies the following phase transition:

    ν_β, tδ_0,  
    		   if β≤√(2), 
    
    		(1 - π_β) δ_0 + π_βδ_1, 
    		   if β > √(2),

for the weak topology, where π_β is a random variable with values in (0,1), mean 1-√(2)/β and a law which depends explicitely on β.
This result, conjectured in <cit.>, has been proved for the branching random walk by Mallein <cit.> and for the BBM by Bonnefont <cit.>. Also note that this result was proved earlier for the mean overlap distribution in <cit.>.
In this paper, we focus on the subcritical phase β < √(2) and obtain the precise decay rate of the overlap distribution on (0,1], as well as its limit, once rescaled by this rate.
We investigate separately the typical overlap distribution in Theorem <ref> and the mean overlap distribution in Theorem <ref>.
Each theorem reveals two main regimes, though the threshold differs between them.



Theorem <ref> involves a new probability measure _β, introduced by Chauvin and Rouault <cit.>.
If we denote by (_t)_t ≥ 0 the natural filtration of the BBM, we can define it through the Radon–Nikodym derivatives

    _̣β̣|̣_̣_̣ṭ/|̣_̣_̣ṭ = W_t(β),

for t ≥ 0.
Under _β, one of the particles, called spine, performs a Brownian motion with drift β and splits at rate 2 while the behavior of the other particles remains unchanged, see <ref> for details.
We denote by __β the associated expectation.

Throughout this paper, the letters t and s denote some points in time [0, ∞).
The letters C and c denote constants, deterministic and time-invariant quantities, which may vary from one line to another.
We set = {0,1,2,…} and ^* = ∖{0}.
We use the Bachmann–Landau notations o, O, ∼, and Hardy's notation ≍ with their usual meaning.
More precisely, for f _+ → and g _+ →_+^*, we say 
that f(t) = o(g(t)) as t →∞ if lim_t→∞ f(t)/g(t) = 0, 
that f(t) = O(g(t)) as t →∞ if lim sup_t→∞f(t)/g(t) < ∞, 
that f(t) ∼ g(t) as t →∞ if lim_t→∞ f(t)/g(t) = 1, 
and that f(t) ≍ g(t) as t →∞ if 0 < lim inf_t→∞ f(t)/g(t) ≤lim sup_t→∞ f(t)/g(t) < ∞.



 §.§ Main results


In the following theorems, we describe the asymptotic behavior at high temperature of the overlap distribution and its expectation.


    Let 0 ≤β < √(2) and 0 < a < 1.
    
        
  *  If 0 ≤β < √(2)/2, then
        
    ^(1 - β^2)atν_β, t([a, 1]) W_∞(2β)/W_∞(β)^2W_∞(β)^2,

        
  *  If β = √(2)/2, then
        
    √(at)^at/2ν_β, t([a, 1]) √(2/π)Z_∞/W_∞(β)^2W_∞(β)^2,   in probability.

        
  *  If √(2)/2 < β < √(2), then
        
    (at)^3 β/√(2)^(√(2) - β)^2 atν_β, t([a, 1]) C_β Z_∞^√(2)β/W_∞(β)^2 S_2β,   in distribution,

        where S_2β is a non-degenerate (√(2)/2β)-stable random variable independent of the BBM and C_β > 0 is a constant that depends only on β.
    



    Let 0 ≤β < √(2) and 0 < a < 1.
    As t →∞,
    
        
  *  if β = 0, then
        
    ν_β, t([a, 1])∼ 2at ^-at,

        
  *  if 0 < β < √(2/3), then W_t(β) converges -and _2β-to some positive and finite variable W_∞(β), and
        
    ν_β, t([a, 1])∼__2β[ 1/W_∞(β)^2] W_∞(β)^2^-(1 - β^2)at,

        
  *  if β = √(2/3), then
        
    [ ν_β, t([a, 1]) ] ≍ t^-1/2^-at/3,

        
  *  if √(2/3) < β < √(2), then
        
    [ ν_β, t([a, 1]) ] ≍ t^-3/2^-(2 - β^2)^2at/8β^2.

    


The exponents appearing in the exponential decay of the overlap distribution in Theorems <ref> and <ref> are represented in Figure <ref>.



 
    If 0 ≤β < √(2)/2, the additive martingale W_t(2β) is uniformly integrable and therefore the measure _2β is uniformly continuous w.r.t.  with Radon–Nykodim derivative W_∞(2β).    
    In particular, if 0 < β < √(2)/2, we can rewrite Theorem <ref>.<ref> as
    
    ^(1 - β^2)atν_β, t([a, 1])W_∞(2β)/W_∞(β)^2W_∞(β)^2.

    Note that this limit is the expectation of the limit in Theorem <ref>.<ref>.
    This no longer holds if √(2)/2 ≤β < √(2/3) since the right-hand side of (<ref>) becomes zero.
    The probability measure _2β then appears as the appropriate framework to extend this formula.





 §.§ Comments, related literature and heuristics





  
Motivations.
As mentioned earlier, our motivation for studying the overlap comes from the physics literature.
In mean-field spin glass models such as the Sherrington–Kirkpatrick model, the overlap between two configurations is defined as an explicit function of the correlation between the energies of these configurations, and the overlap distribution plays a key role in the interpretation of Parisi's formula for the free energy <cit.> and of the ultrametric structure of the Gibbs measure in the infinite-volume limit <cit.>.
To understand the free energy and the overlap distribution on more tractable examples, Derrida and Gardner <cit.> introduced the Generalized Random Energy Model (GREM), where the ultrametric structure is directly implemented in the definition of the Hamiltonian at finite volume: more precisely, the energies have a hierarchical structure of correlations with a fixed number of levels, whereas the BBM corresponds to a number of levels growing like the number of particles.
As indicated above, the overlap distribution for the BBM has been first studied in physics by Derrida and Spohn <cit.>, motivated by both spin glass and polymers theories. 
Indeed, the BBM can also be seen as an infinite limit of the directed polymer model and, in this framework, the overlap distribution describes the proportion of time where two polymers chosen independently according to the Gibbs measure are overlapping.

More recently, Derrida and Mottishaw <cit.> investigated the finite-size corrections for the mean overlap distribution in the critical and supercritical phases β≥√(2): to be more precise than the known convergence [ν_β,t] →√(2)/βδ_0 + (1-√(2)/β) δ_1 as t →∞, they describe how the mass of [ν_β,t] between 0 and 1 vanishes. 
More precisely, they conjecture that, for any 0<a<b<1,

    [ν_β,t([a,b])] 
        ∼1/β√(2 π t)∫_a^b x̣/x^3/2 (1-x)^(1+2·1_β>√(2))/2.

This prediction is based both on numerical simulations and on an approximation by the GREM, for which Derrida and Mottishaw <cit.> calculated the finite-size corrections of the overlap as well.
In the case β>√(2), the asymptotic (<ref>) also has an interpretation in high-energy particle physics, uncovered by Mueller and Munier <cit.>: using a link between the BBM and the scattering of high-energy hadrons dating back to <cit.>, they show a link between the statistics of rapidity gaps in diffractive electron-nucleus scattering and statistics of the extremal particles of the BBM, including their overlap distribution, which is well-predicted by (<ref>) because supercritical Gibbs measures are supported by extremal particles.


In this paper, we study this question in the subcritical phase β < √(2) instead, and investigate moreover the behavior of the typical overlap distribution.
Observe that the exponent appearing in the exponential decay of the mean overlap distribution is vanishing as β approaches √(2) (see Figure <ref>), which is consistent with the fact that the decay is expected to be polynomial only for β≥√(2).
This is also the case for the typical overlap distribution, so we also expect a polynomial decay for β≥√(2) in that case.
However, we expect the typical overlap distribution to decay faster than the mean overlap distribution when β≥√(2), as it is the case for β∈ [√(2)/2,√(2)), which means that [ν_β,t([a,b])] is governed by a rare event (called “fluctuation” in <cit.>).





  
Typical overlap distribution.
Most heuristics can be deduced from the following rewriting of the overlap distribution (see (<ref>) for details)

    ν_β, t([a, 1]) = ^(β^2 - 1) at/W_t(β)^2∑_w ∈(at)^2β X_w(at) - ψ(2β)at W_t - at^(w, at)(β)^2,

where (W_s^(w, at)(β))_s ≥ 0 is the additive martingale of the BBM starting from particle w at time at.
Note that, in the sum of (<ref>), these additive martingales are tilted by Gibbs weights at inverse temperature 2β and time at.
Consequently, the particles u,v∈(t) that mainly contribute to the overlap distribution (see the expression in (<ref>)) are those contributing to W_t-at^(w, at)(β), with w itself being among the particles contributing to the Gibbs measure _2β, at, or equivalently, to the additive martingale W_at(2β).
The description of these particles' behaviors follows from this.

There is a first regime β∈ [0, √(2)/2) for which the Gibbs measure _2β, at is subcritical, in the sense that it is supported by an exponential number of particles.
These ones perform a Brownian motion with drift 2β (see <cit.> for a weaker version of this property and <cit.> for the case of the branching random walk).
So do the particles that contribute to the overlap distribution, up to time at, after which their trajectories relax to a drift β (see Figure <ref>).
The large amount of such particles induces a law of large numbers that allows to approximate (<ref>) with

    ^(β^2 - 1) at/W_t(β)^2∑_w ∈(at)^2β X_w(at) - ψ(2β)atW_∞(β)^2 = ^(β^2 - 1) at/W_t(β)^2 W_at(2β) W_∞(β)^2.

Theorem <ref>.<ref> then follows from the convergence of the additive martingales.



At β = √(2)/2, the Gibbs measure _2β, at is critical.
It is supported by a number of order ^C√(t) of particles, having a drift √(2) altered by an upper barrier of slope √(2) (see <cit.> for the branching random walk).
The same holds up to time at for the overlap distribution.
Theorem <ref>.<ref> can be seen as a consequence of the approximation (<ref>) and the convergence (<ref>).

In the remaining regime β∈ (√(2)/2, √(2)), the Gibbs measure _2β, at is supported by the extremal process, particles at a bounded distance from the top of the BBM.
Roughly, their trajectories are Brownian bridges conditioned to stay under a barrier of slope √(2) (see <cit.> for the branching random walk).
This again induces the behavior of the particles that contribute to the overlap distribution (see Figure <ref>).
The α-stable limit with α = √(2)/2β∈ (1/2, 1) is reminiscent of the limit appearing for the renormalized supercritical additive martingale, obtained by Barral, Rhodes and Vargas <cit.> for the branching random walk (see <cit.> for the BBM case): for β > β_c,

    t^3β/2√(2)^(β/√(2)-1)^2t W_t(β)
         Z_∞^β/√(2) S_β,   in distribution,

where S_β is a non-degenerate (√(2)/β)-stable random variable independent of Z_∞.
Here, it is not exactly W_at(2β) which appears in (<ref>) because of the additional factors W_t - at^(u, at)(β)^2, but we show that they do not affect the stable nature of the limit.


The transition at β = √(2)/2 described above is similar to the one appearing in the fluctuations of the additive martingales, obtained by Iksanov, Kolesko and Meiners <cit.> for the branching random walk (see <cit.> for the BBM case).
This fluctuations are obtained using the following decomposition

    W_∞(β) - W_t(β) 
        = ∑_u∈(t)^β X_u(t) - ψ(β) t (W_∞^(u)(β) - 1),

where the main difference with (<ref>) is that this is a sum of centered variables, whereas the terms in (<ref>) are positive.
However, similarly to the case of the overlap, for β∈ (0,√(2)/2), the sum in (<ref>) is dominated by particles contributing to _2β,t (particles u∈(t) such that X_u(t) = 2β t + O(√(t))) and the limit, after rescaling, is normally distributed with variance c W_∞(2β).
Furthermore, for β∈ (√(2)/2,√(2)), the sum is dominated by extremal particles and, as noted in <cit.>, the limit is α-stable with α = √(2)/β∈ (1, 2).
In the later regime, we apply methods similar to those used in <cit.>.



  
Mean overlap distribution.
It is surprising that, in terms of the mean overlap distribution, the threshold at β = √(2)/2 disappears, and this new threshold at β = √(2/3) appears. We do not know other phenomena for the BBM where this threshold plays a role.
Moreover, let us emphasize here that the proof of Theorem <ref> is more involved than the one of Theorem <ref>, even in the case β∈ (0,√(2)/2), partly due to the fact that 1/W_∞(β) is not integrable for any β∈ [0,√(2)), so the denominator in (<ref>) has to be handled carefully.

We now give some crude heuristics to explain the appearance of the threshold at β = √(2/3). 
In the rewriting (<ref>) of ν_β, t([a, 1]), we see that a particle w ∈(at) contributes to the numerator proportionally to ^2β X_w(at).
By Girsanov theorem, the event dominating in [^2β X_w(at)] is the one where (X_w(s))_s∈[0,at] behaves like a Brownian motion with drift 2β. There are typically such particles w in the BBM if and only if β < √(2)/2 and this is the main reason why the behavior of ν_β, t([a, 1]) in the case β∈ (0,√(2)/2) is simply obtained by taking the expectation of the typical behavior (see Remark <ref>).
As soon as β≥√(2)/2, ν_β, t([a, 1]) starts being dominated by a rare event where some particle w∈(at) goes to a position much higher than the typical maximum of the BBM.
However, this position is not necessarily 2β at + O(√(t)) as predicted by Girsanov theorem. 
Indeed, a particle w ∈(at) also contributes to the denominator in (<ref>): its contribution to W_t(β) equals ^β X_w(at) - ψ(β)at W_t - at^(w, at)(β).
If X_w(at) ≪ψ(β)at/β, then ^β X_w(at) - ψ(β)at≪ 1 and the contribution of w in W_t(β) is negligible (W_t(β) is typically of order 1), so only the numerator dictates the optimal behavior of the particle w. 
On the other hand, if X_w(at) ≫ψ(β)at/β, then W_t(β) is dominated by the contribution of particle w and then the numerator and the denominator compensate each other: forcing w to reach a higher position does not increase significantly ν_β, t([a, 1]) because it saturates at 1.
Therefore, particles w∈(at) contributing mostly to ν_β, t([a, 1]) are at a position roughly v(β) at, where v(β) is given by

    v(β) = 2β∧ψ(β)/β 
        = 
            2 β   if β∈ [0,√(2/3)], 
    1/β + β/2   if β∈ [√(2/3), √(2)).

See Figure <ref> for the graph of function v.
Hence, the threshold at β = √(2/3) corresponds to the inverse temperature above which, in mean, particles contributing mostly to the numerator in (<ref>) also contributes significantly to the denominator.
Note that we have cheated a little in the above heuristic: we took for granted that, in order to make ^2β X_w(at) W_t - at^(w, at)(β)^2 large, one has to make ^2β X_w(at) large, but the best strategy could involve making W_t - at^(w, at)(β)^2 large as well.
However, using the tail (W_t - at^(w, at)(β) ≥ x) ≤ C x^-2/β^2 (see Lemma <ref>), one can check that the best strategy to obtain ^2β X_w(at) W_t - at^(w, at)(β)^2 ≃^2β v(β) at is indeed to make only X_w(at) large.




As a consequence of this discussion, one can describe the behavior of particles contributing mostly to the mean overlap distribution.
If β∈ (0, √(2/3)), then contributing particles have a drift v(β) = 2β until time at and then a drift β (see Figure <ref>).
If β∈ [√(2/3),√(2)), we explained that contributing particles have a drift v(β) = 1/β + β/2 until time at, but one can be more precise than that. 
If the contributing particle has, at some time s ∈ [0,at], an ancestor at v(β) s + x, then one can check that the descendants at time t of this ancestor have a contribution of order ^β x in W_t(β), so any such event with a large x results in a small value of ν_β, t([a, 1]).
Hence, contributing particles are roughly constrained below the straight barrier s ↦ v(β)s.
Therefore, if β = √(2/3), contributing particles move like a Brownian motion with drift v(β) conditioned to stay below the barrier until time at, and then like a Brownian motion with drift β.
If β∈ (√(2/3), √(2)), the factor ^2β X_w(at) in the numerator of (<ref>) additionally forces particles to end up close to the barrier at time at, hence contributing particles move like a Brownian motion with drift v(β) conditioned to stay below the barrier until time at while reaching a position v(β) at + O(1) at time at, and then like a Brownian motion with drift β (see Figure <ref>).
Note that these constraints explain the polynomial corrections t^-1/2 and t^-3/2 appearing in Theorem <ref>.<ref> and <ref>.






 §.§ Organization of the paper


Section <ref> contains a formal definition of the model, which is then used to introduce properly some change of measures, and results concerning the additive martingales, the extremal process of the BBM and ballot theorems for Brownian motions.
In Section <ref>, we prove Theorem <ref>, with one subsection dedicated to each part of the statement (in the same order).
Finally, Section <ref> contains the proof of Theorem <ref>, again with one subsection dedicated to each part of the statement (but this time in the order <ref>, <ref>, <ref> and <ref>).



§ PRELIMINARY RESULTS





 §.§ The model


Following the formalism of Chauvin and Rouault <cit.>, we consider the BBM to be a binary tree endowed with random lifetimes and trajectory data.
In addition, it will be useful to distinguish one line of descent, called the spine.
Let us specify.

We consider the set of Ulam–Harris–Neveu labels = ⋃_i ≥ 0{1, 2}^i, with the convention {1, 2}^0 = {∅}.
For a label u ∈, we denote by |u| its length, with the convention |∅| = 0.
For two labels u, v ∈, we denote by uv their concatenation, with the convention u ∅ = ∅ u = u.
We say that u is an ancestor of v and that v is a descendant of u if there exists w ∈ such that v = uw.
In this case, we write u ≤ v.
If in addition u ≠ v, then we write u < v.
We denote by u ∧ v the most recent common ancestor of two particles u and v, u ∧ v = u_1 … u_i_0 with i_0 = sup{i ≥ 1 : u_1 … u_i = v_1 … v_i} if u_1 = v_1, and u ∧ v = ∅ otherwise.

The probability space on which we work is Ω the set of tuples ((σ_u)_u ∈, (Y_u)_u ∈, ξ), where, for each u ∈, σ_u ≥ 0 and Y_u : [0, ∞) → is a continuous function, and ξ = {∅, ξ_1, ξ_1ξ_2, ξ_1ξ_2ξ_3, …}, with (ξ_k)_k≥ 1∈{1,2}^^*, is an infinite line of descent of .
We denote by b_u = ∑_v < uσ_v the birthtime of a particle u, and d_u = ∑_v ≤ uσ_v its deathtime.
For t ≥ 0, we define

    (t) = {u ∈ : b_u ≤ t < d_u}  and  ([0, t]) = {u ∈ : b_u ≤ t}.

We denote by ξ(t) the only particle in (t) ∩ξ, called spine.
We define inductively the position at time t of a particle u ∈(t)

    X_u(t) = ∑_v < u X_v(d_v) + Y_u(t - b_u).

We extend the notion of position for a particle u ∈(t) to the whole interval [0, t]: for every s ∈ [0, t], we set X_u(s) = X_v(s), where v is the ancestor of u alive at time s.
To avoid redundancy, we denote the spinal position X_ξ(t) = X_ξ(t)(t).

For t ≥ 0, let _t be the σ-algebra containing all the information until time t except the spine,

    _t = σ( (u, d_u ∧ t, X_u|_[0, d_u ∧ t]) : u ∈([0, t]) )   and  _∞ = σ( ⋃_t ≥ 0_t ),

where s ∧ t denotes the minimum between s and t.
Let us define a finer filtration by adding the information about the spine,

    _t = σ( _t, ξ(t) )   and  _∞ = σ( ⋃_t ≥ 0_t ).

We define the BBM law as the unique probability measure  on (Ω, _∞) for which

    
  * the trajectories Y_u are standard Brownian motions,
    
  * the lifetimes σ_u are exponential variables with rate 1,
    
  * the spinal coordinates ξ_1, ξ_2, … are with (ξ_1 = 1) = (ξ_1 = 2) = 1/2,
    
  * (Y_u)_u ∈, (σ_u)_u ∈, ξ are independent.

We denote by  the associated expectation.


    For all t ≥ 0 and u ∈(t), ( ξ(t) = u | _t ) = 2^-|u|.


In order to state a branching property, we define, for t ≥ 0, a shift operator Θ_ξ, t : Ω→Ω by Θ_ξ, t(σ, Y, ξ) = (σ', Y', ξ'), where, if u = ξ(t),

    
  *  σ'_v = 
        d_u - t    if  v = ∅, 
σ_uv   otherwise,
    
  *  Y'_v(s) = 
        X_u(t + s) - X_u(t)    if  v = ∅, 

        Y_uv(s)    otherwise,
    
  * ξ' = {∅, ξ_|u| + 1, ξ_|u| + 1ξ_|u| + 2, …}.

We also define, for u ∈ and t ≥ 0, Ω_u, t = {(σ, Y, ξ) ∈Ω : u ∈(t)} and Θ_u, t : Ω_u, t→Ω by Θ_u, t(σ, Y, ξ) = (σ', Y', ξ') satisfying <ref>, <ref> and ξ' = {∅, 1, 11, …}.
A consequence of <cit.> is the following branching property.
If Y and Y_u, u ∈, are non-negative variables on Ω such that Y is _∞-measurable and Y_u, u ∈, are _∞-measurable, then for every t ≥ 0,

    [ Y ∘Θ_ξ, t×∏_u ∈(t) 
     u ≠ξ(t) Y_u ∘Θ_u, t| _t ] = [Y] ∏_u ∈(t) 
     u ≠ξ(t)[Y_u].

This means that, at time t, all the alive particles start independent BBMs shifted in time, space, label, and that the information about the spine is kept in the BBM started from ξ(t).



 §.§ Change of measure



In this section, we introduce some changes of measure that allow a useful interpretation of the process in term of spinal decomposition.
This approach dates back to Kahane and Peyrière <cit.> for the branching random walk and to Chauvin and Rouault <cit.> for the BBM, and it has been widely used since, see <cit.> and <cit.> for applications to the study of the additive martingales. 
The framework of these techniques has been extended in <cit.> and <cit.> to more general branching processes and the presentation below follows roughly the lines of <cit.>.

Let β≥ 0 and ψ(β) = 1 + β^2/2.
The process (2^|ξ(t)|^β X_ξ(t) - ψ(β)t)_t ≥ 0 is a positive (_t)_t ≥ 0-martingale with mean 1.
By Kolmogorov's extension theorem, we can define a probability measure _β on _∞ by

    _̣β̣|̣_̣_̣ṭ/|̣_̣_̣ṭ = 2^|ξ(t)|^β X_ξ(t) - ψ(β)t.



    The additive martingale defined in (<ref>) is the projection of the above martingale onto _t,
    
    W_t(β) = [ 2^|ξ(t)|^β X_ξ(t) - ψ(β)t| _t ].

    Therefore, the change of measure (<ref>) is consistent with (<ref>).


In <cit.>, Chauvin and Rouault describe the effect of this change of measure on the BBM.
Under _β,

    
  * the spine's motion is a Brownian motion with drift β,
    
  * the spine particle splits at rate 2,
    
  * at each splitting event of the spine particle, the label of the new spine particle is chosen uniformly among the two children,
    
  * at this splitting event, the other child starts an independent standard BBM without spine.

By <cit.>, a remarkable and useful property is that

    _β( ξ(t) = u | _t ) = 1_u ∈(t)^β X_u(t) - ψ(β)t/W_t(β).

As mentioned in Remark <ref>, if β < β_c, then the uniform integrability of (W_t(β))_t≥ 0 implies that the measure _β|__∞ is uniformly continuous w.r.t. |__∞ with Radon–Nykodim derivative W_∞(β) (note however that _β and  on the full σ-field _∞ are singular).
On the other hand, if β≥β_c, then the convergence of W_t(β) towards zero implies that the measures _β|__∞ and __∞ are singular.

In this paper, we will also use a slightly different change of measure, that turns out to be more suited to the study of the overlap distribution.
For β≥ 0 and t ≥ 0, we define _β, t on _∞ by

    _̣β̣,̣ ̣ṭ/ = 2^|ξ(t)|^β X_ξ(t) - ψ(β) t.



    Let Y_t, Y_ξ and Y_u, u ∈, be non-negative variables on Ω such that Y_t is _t-measurable, Y_ξ is _∞-measurable, and Y_u, u ∈, are _∞-measurable.
    Then,
    
    __β, t[ Y_t × Y_ξ∘Θ_ξ, t×∏_u ∈(t) 
     u ≠ξ(t) Y_u ∘Θ_u, t]    = [ 2^|ξ(t)|^β X_ξ(t) - ψ(β)t× Y_t ×[ Y_ξ∘Θ_ξ, t×∏_u ∈(t) 
     u ≠ξ(t) Y_u ∘Θ_u, t| _t ] ] 
       = __β[ Y_t ×[ Y_ξ] ×∏_u ∈(t) 
     u ≠ξ(t)[ Y_u ] ],

    by the branching property (<ref>) and the definition of _β.
    This means that, under _β, t,
    
        
  * between 0 and t, the BBM has the same behavior as under _β,
        
  * at time t, the spine particle starts a standard BBM, shifted in time, space and label,
        
  * at time t, the non-spine particles start standard BBMs without spine, shifted in time, space and label, independent of each other and of the spine-BBM.
    


Similarly to (<ref>), we have the following property.


    For all u ∈ and s, t ≥ 0,
    
    _β, t(ξ(t) = u | _t + s) = 1_u ∈(t)^β X_u(t) - ψ(β)t/W_t(β).




    Let A ∈_t+s.
    By definition (<ref>),
    
    __β, t[ 1_A 1_ξ(t) = u] = [ 1_A 1_ξ(t) = u 2^|ξ(t)|^β X_ξ(t) - ψ(β) t] = [ 1_A 1_ξ(t) = u 2^|u|^β X_u(t) - ψ(β) t].

    But u is the spinal particle at time t if and only if u is alive at time t and admits the spinal particle among its descendants at time t + s.
    This consideration together with Remark <ref> allow us to rewrite (<ref>) as
    
    [ 1_A 1_u ∈(t)(∑_v ∈(t+s) 
     v ≥ u1_ξ(t+s) = v) 2^|u|^β X_u(t) - ψ(β) t] = [ 1_A 1_u ∈(t)( ∑_v ∈(t+s) 
     v ≥ u 2^-|v|) 2^|u|^β X_u(t) - ψ(β) t].

    Note that {v ∈(t+s) : v ≥ u} is a binary tree and that, for its particles v, the number |v| - |u| counts the fission times from u to v.
    Therefore,
    
    ∑_v ∈(t+s) 
     v ≥ u 2^-(|v| - |u|) = 1.

    We can then rewrite (<ref>) as
    
    [ 1_A 1_u ∈(t)^β X_u(t) - ψ(β) t] = __β, t[ 1_A 1_u ∈(t)^β X_u(t) - ψ(β) t/W_t(β)],

    using here that _β,t|__∞ has density W_t(β) w.r.t. |__∞.
    This concludes the proof.




 §.§ Moments and tails of the additive martingale


In what follows, it will be convenient to use the following rewriting of the additive martingale (<ref>).
For t ≥ s ≥ 0,

    W_t(β) = ∑_u ∈(s)^β X_u(s) - ψ(β)s W_t - s^(u, s)(β),

where, for r ≥ 0,

    W_r^(u, s)(β) = W_r ∘Θ_u, s = ∑_v ∈(s + r) 
     v ≥ u^β (X_v(s + r) - X_u(s)) - ψ(β)r.

The above process is the additive martingale of a BBM shifted in time, space and label.
It converges to some limit W_∞^(u, s)(β) and hence we can extend (<ref>) to the case t = ∞.
By the branching property (<ref>), conditionally on _s, the processes (W_r^(u, s))_r ∈[0, ∞] for u ∈(s) are copies of (W_r)_r ∈ [0, ∞].


    Let β∈ [0, √(2)).
    For any p ∈ (1, 2/β^2), the additive martingale at inverse temperature β is bounded in L^p and hence converges in L^p.
    Moreover, if p ∈ (1, 2/β^2) ∩ (1, 2], then there exists C>0 such that, for any 0 ≤ s ≤ t ≤∞,
    
    | W_t(β) - W_s(β) |^p ≤ C ^-(p - 1)(1 - pβ^2/2)s.



The proof of this lemma relies partially on the following inequality, which is due to Von Bahr and Esseen <cit.> and is used repetitively throughout the paper. The application of this type of inequalities to branching processes dates back to Neveu <cit.> and Biggins <cit.>.


Let p ∈ [1,2] and X_1,…,X_n be centered independent real random variables in L^p. Then,

    ∑_k=1^n X_k^p≤ 2 ∑_k=1^n X_k^p.




    Let p ∈ (1, 2/β^2).
    Equation (<ref>) holds for t = ∞ and s = 1.
    Therefore, by <cit.>, W_∞(β) is in L^p if and only if
    
    1 > [ #(1) ] [ ( ^β X_ξ(1) - ψ(β))^p ] = ^1 + p^2 β^2/2 - pψ(β).

    This inequality holds since ψ(β) = 1 + β^2/2 and p ∈ (1, 2/β^2).
    Hence W_∞(β) ∈ L^p.
    But, recalling that (W_t(β))_t≥ 0 is uniformly integrable for β < √(2), we have W_t(β) = [ W_∞(β) | _t ].
    Jensen's inequality for conditional expectations implies that the additive martingale is bounded in L^p.

    Now let p ∈ (1, 2/β^2) ∩ [1, 2] and 0 ≤ s < t ≤∞.
    We rewrite
    
    W_t(β) - W_s(β) = ∑_u ∈(s)^β X_u(s) - ψ(β)s (W_t - s^(u, s) - 1).

    Conditionally on _s, the random variables (W_t - s^(u, s) - 1) for u ∈(s) are centered and independent.
    Therefore, by Lemma <ref> and the branching property (<ref>),
    
    |W_t(β) - W_s(β)|^p_s≤∑_u ∈(s)^p β X_u(s) - p ψ(β)s|W_t-s(β) - 1|^p = W_s(pβ) ^-(pψ(β) - ψ(pβ))s|W_t-s(β) - 1|^p.

    Note that
    
    pψ(β) - ψ(pβ) = (p - 1)(1 - pβ^2/2),

    which is positive, by choice of p.
    This concludes since W_s(pβ) has mean 1 and the additive martingale at β is bounded in L^p.



    Let β∈ [0, √(2)) and fix a number b such that
    
    0 < b < √(ψ(β)^2 + 2β^2) - ψ(β)/β^2.

    There exists K > 0 such that, for every t ∈ [0, ∞] and every x ≥ 0,
    
    W_t(β) ≤ x≤ K x^b.




    Equation (<ref>) holds for t = ∞ and s = 1.
    Therefore, by <cit.>, W_∞(β) ≤ x = O(x^-b) as x → 0 for any b>0 such that
    
    [ ( ^β X_ξ(1) - ψ(β))^-b] < ∞  and  [ ( ^β X_ξ(1) - ψ(β))^-b1_#(1) = 1] = ^(b^2 β^2/2 + b ψ(β) - 1) < 1,

    which is equivalent to (<ref>).
    As a consequence, W_∞(β)^-b < ∞ for any b satisfying (<ref>).
    Then, for any t ∈ [0, ∞], by Markov's inequality and Jensen's inequality (using that W_t(β) = [ W_∞(β) | _t ]),    
    
    W_t(β) ≤ x≤ x^b W_t(β)^-b≤ x^b W_∞(β)^-b,

    which concludes the proof.




 §.§ The extremal process


We first recall some important results concerning the extremes of the BBM. For t>0, let

    m(t) = √(2) t - 3/2 √(2)log t.

The convergence of the maximal position has been proved by Bramson <cit.> and the limit has been identified by Lalley and Sellke <cit.>, who introduced the derivative martingale (<ref>) and showed that

    max_u ∈(t) X_u(t) - m(t) 1/√(2) (G + log(CZ_∞)),   in distribution,

where G is a standard Gumbel variable.
As a consequence, for any δ > 0, the probability of the following event converges to 1 as t →∞,

    Λ_δ, t = {max_u ∈(t) X_u(t) ≤ m(t) + δlog t }.


Later, Arguin, Bovier and Kistler
<cit.> and Aïdékon, Berestycki, Brunet and Shi <cit.> independently established the convergence in distribution
of the extremal process,

    _t = ∑_u ∈(t)δ_X_u(t) - m(t).

This convergence holds in the space of Radon measures on  endowed with the vague topology.
They obtained the following limit,

    _∞ = ∑_i, j ≥ 1δ_p_i + Δ_ij + log(C Z_∞)/√(2),

where

    
  * = ∑_iδ_p_i is a Poisson point process on  with intensity measure √(2)^-√(2)xx̣, independent of Z_∞,
    
  * _i = ∑_j δ_Δ_ij are copies of a point process = ∑_j δ_Δ_j such that max = 0 and are independent of  and Z_∞.

In other words, the extremal process converges in distribution to a randomly shifted decorated Poisson point process, a Poisson point process  with a shift depending on Z_∞, where each point is replaced by an independent copy of a decoration .
One can deduce (see <cit.> for details) that for any continuous function with compact support f : → [0, ∞),

    lim_s →∞lim_t →∞^-_t(f)_s = ^-_∞(f)_∞,   ,

where we emphasize that in _∞, only Z_∞ is _∞-measurable, whereas  and (_i)_i≥1 are independent of _∞.
Cortines, Hartung and Louidor <cit.> studied the level sets of the decoration: in their Proposition 1.5, they obtained the following equivalent as x →∞,

    ([-x, 0]) ∼ C ^√(2) x.

In what follows, we write X d= Y to indicate that X and Y have same distribution.


    If β > √(2) and (ξ_k)_k≥1 denotes the points of _∞, then ∑_j ^βΔ_j < ∞, ∑_k ^βξ_k < ∞ and
    
    ∑_k ≥ 1^βξ_kd= C (Z_∞)^β/√(2) S_β,

    where S_β = ∑_i ^β p_i is a non-degenerate √(2)/β-stable random variable independent of Z_∞.



    As done in <cit.>, we rewrite
    
    ∑_j ≥ 1^βΔ_j = ∫_-∞^0 ^β x(x̣) = ∫_0^∞β^-β y([-y, 0]) ỵ.

    Applying the Fubini-Tonelli theorem and (<ref>), we see that the above quantity is finite.
    Now,
    
    ∑_k ≥ 1^βξ_k = (C Z_∞)^β/√(2)∑_i ≥ 1^β(p_i + X_i),

    where X_i = 1/βlog∑_j ^βΔ_ij.
    The random variables X_i for i ≥ 1 are , independent of  and satisfy
    
    ^√(2) X_1 = (∑_j ≥ 1^βΔ_j)^√(2)/β≤∑_j ≥ 1^βΔ_j < ∞.

    By <cit.>,
    
    ∑_i ≥ 1δ_p_i + X_id=∑_i ≥ 1δ_p_i + 1/√(2)log^√(2) X_1.

    By (<ref>) and (<ref>), we have ∑_k ^βξ_k < ∞ if and only if ∑_i ^β p_i < ∞ Computing the expectations, we see that, with probability 1, we have ([0, ∞)) < ∞ and ∑_i ^β p_i1_p_i < 0 < ∞, so ∑_i ^β p_i < ∞.
    Besides, since Z_∞ is independent of  and (_i)_i ≥ 1, (<ref>) and (<ref>) yield
    
    ∑_k ≥ 1^βξ_kd=(C Z_∞^√(2) X_1)^β/√(2)∑_i ≥ 1^β p_i.

    It remains to show that ∑_i ^β p_i is stable.
    Let λ > 0.
    Applying successively the exponential formula for Poisson point processes (see <cit.>) and the change of variable y = ^β x, we obtain
    
    exp(-λ∑_i ≥ 1^β p_i) = exp(-∫_ (1 - ^-λ^β x) √(2)^-√(2) xx̣) = exp(-√(2)/β∫_0^∞ (1 - ^-λ y) y^-1-√(2)/βỵ).

    We recognize the Laplace transform of a non-degenerate √(2)/β-stable law (see <cit.>).


We now define two point processes on ^2 by

    _t^* = ∑_u ∈(t)δ_(X_u(t) - m(t), W_∞^(u, t)(β))  and  _∞^* = ∑_k ≥ 1δ_(ξ_k, W_k),

where W_k for k ≥ 1 are copies of W_∞(β), independent of _∞ = ∑_kδ_ξ_k and of the BBM.
Let f : ^2 → be a bounded continuous function.
Assume that there exists a ∈ such that f(x, y) = 0 as soon as x ≤ a.
Under these conditions and for the branching random walk, Iksanov, Kolesko and Meiners showed in <cit.> that _t^*(f) converges in distribution to _∞^*(f) as t →∞.
Their result easily extends to BBM (see <cit.> for details).
Furthermore, thanks to (<ref>), one can adapt their proof to obtain the following joint convergence (see <cit.> for details), for any β≥ 0,

    (_t^*(f), W_t(β))  (_∞^*(f), W_∞(β)),   in distribution.




 §.§ Brownian estimates


Let (B_t)_t ≥ 0 be a standard Brownian motion on , α∈ (0, 1/2), and define, f_t(s) = s^α∧ (t - s)^α for t ≥ s ≥ 0.
By <cit.>, there exist C, c > 0 such that, for all t ≥ 0, x, y ∈ [1, √(t)],

    c x/(t + 1)^1/2≤sup_s ∈ [0, t] (B_s + f_t(s)) ≤ x≤C x/(t + 1)^1/2,

and

    c x y/(t + 1)^3/2≤sup_s ∈ [0, t] (B_s + f_t(s)) ≤ x, B_t ∈ [x - y, x - y + 1]≤C x y/(t + 1)^3/2.

Besides, the following bound will be useful.


    Fix δ > 0.
    There exists C_δ > 0 such that, for all x, y ≥ 0 and t ≥ 1,
    
    sup_s ∈ [0, t] B_s ≤ x, B_t ∈ [x - y, x - y + δ]≤C_δ x y/t^3/2( ^-(x - y)^2/2t + ^-(x - y + δ)^2/2t).




    By <cit.>, sup_s ∈ [0, t] B_s ≤ xB_t≤ 2 x (x - B_t)^+/t, where we write a^+ = max(a, 0).
    It follows that the left-hand side of (<ref>) is bounded by
    
    2 x (x - B_t)^+/t1_{B_t ∈ [x - y, x - y + δ]}≤2 x y/tB_t ∈ [x - y, x - y + δ].

    Next, we can use the bounds
    
    B_t ∈ [x - y, x - y + δ]≤
                C δ^-(x - y)^2/2t/√(t)   if  x - y ≥ 0, 
    
                C δ^-(x - y + δ)^2/2t/√(t)   if  x - y + δ≤ 0, 
    
                C/√(t)   otherwise.

    Each of these terms is bounded by the right-hand side of (<ref>) since t ≥ 1.




§ PROOF OF THEOREM <REF>




Let β∈ [0, √(2)) and a ∈ (0, 1).
Recall that ψ(β) = 1 + β^2/2 and that the additive martingale W_t(β) is defined in (<ref>).
In what follows, it will be convenient to use the following rewriting of the overlap distribution defined in (<ref>),

    ν_β, t([a, 1]) = 1/W_t(β)^2∑_u, v ∈(t) 
     u ∧ v ≥ at^β(X_u(t) + X_v(t)) - 2 ψ(β) t = ^(β^2 - 1) at/W_t(β)^2∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_t - at^(u, at)(β)^2,

where we recall that

    W_t^(u, s)(β) = W_t(β) ∘Θ_u, s = ∑_v ∈(s+t) 
     v ≥ u^β (X_v(s + t) - X_u(s)) - ψ(β)t.




 §.§ Upper temperature


The aim of this section is to show Theorem <ref>.<ref> which covers the case β∈ [0,√(2)/2).
Define

    F_t = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_t - at^(u, at)(β)^2,

so that ^(1 - β^2)atν_β, t([a, 1]) = F_t W_t(β)^-2, by (<ref>).
It suffices to show that

    F_t  F_∞ = W_∞(2β) W_∞(β)^2,




  §.§.§ Replacement along a partition of time


Let us fix the following partition of the time [0, ∞),

    T = ⋃_n ≥ 0(1/n+1) ∩ [n, n+1].

Here, we justify that, along this partition, we can replace F_t with

    F̅_t = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_∞^(u, at)(β)^2.

We use first branching property at time at, and then factorize the square and apply Cauchy–Schwarz inequality:

    |F_t - F̅_t|_at   ≤ W_at(2β) |W_t - at(β)^2 - W_∞(β)^2| 
       ≤ W_at(2β) 
        ( W_t - at(β)^2 + W_∞(β)^2)^1/2(W_t - at(β) - W_∞(β))^2^1/2
       ≤ C W_at(2β) ^-(2ψ(β) - ψ(2β))s

using Lemma <ref> in the last inequality.
By (<ref>), it follows that |F_t - F̅_t| ≤^-(1 - β^2)(t-at).
Hence, by the Borel–Cantelli lemma,

    |F_t - F̅_t|  0,




  §.§.§ Almost sure convergence along a partition of time


Here, we show that F_t converges to F_∞ along the partition T.
In view of the previous section, it suffices to show that the following process converges to 0 along the partition T,

    Δ_t = F̅_t - W_at(2β) W_∞(β)^2 = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at(W_∞^(u, at)(β)^2 - W_∞(β)^2).

Let us fix p ∈ [1, 2] such that p < 1/2β^2.
By Lemma <ref>,

    |Δ_t|^p_at≤ W_at(p2β) |W_∞(β)^2 - W_∞(β)^2|^p ^-(pψ(2β) - ψ(p2β))at.

The variable W_∞(β)^2 is in L^p because 2p<2/β^2, hence so is W_∞(β)^2 - W_∞(β)^2.
Hence,

    |Δ_t|^p ≤ C ^-(pψ(2β) - ψ(p2β))at.

By (<ref>) and by the Borel–Cantelli lemma,

    Δ_t  0,




  §.§.§ Extension of the almost sure convergence to continuum


Let us denote (t_n)_n ≥ 0 the sequence of the elements of T in increasing order.
Here, we show that

    sup_t ∈ [t_n, t_n+1] |F_t - F_t_n|  0,

We have

    sup_t ∈ [t_n, t_n+1] |F_t - F_t_n| ≤∑_u ∈(at_n)^2β X_u(at_n) - ψ(2β)at_n S_n^(u),

where

    S_n^(u) = sup_t ∈ [t_n, t_n+1]|∑_v ∈(at)
    v ≥ u^2β(X_v(at) - X_u(at_n)) - ψ(2β)(at - at_n) W_t - at^(v, at)(β)^2 - W_t_n - at_n^(u, at_n)(β)^2|.

Note that, by the branching property, for any u ∈(at_n),

    S_n^(u)_at_n = S_n,

where

    S_n = sup_s ∈ [0, t_n+1 - t_n]|∑_v ∈(as)^2β X_v(as) - ψ(2β)as W_s + t_n - a(s + t_n)^(v, as)(β)^2 - W_t_n - at_n(β)^2|.



    For any p ∈ [1, 1/β^2), the process S_n converges in L^p to 0.



    We first prove that, on the event Ω' where W_t(β) converges as t →∞, the process S_n converges to 0.
    Let us fix a realization ω∈Ω' and denote by τ(ω) the first time of branching of the BBM.
    There exists some rank n_0(ω) such that, for any n ≥ n_0(ω), we have t_n+1 - t_n < τ(ω) and then
    
    S_n(ω)    = sup_s ∈ [0, t_n+1 - t_n]|^2β X_∅(as)(ω) - ψ(2β)as W_s + t_n - a(s + t_n)^(∅, as)(β)^2(ω) - W_t_n - at_n(β)^2(ω)| 
       = sup_s ∈ [0, t_n+1 - t_n]|^(2ψ(β) - ψ(2β))as W_s + t_n - at_n(β)^2(ω) - W_t_n - at_n(β)^2(ω)|.

    Since W_t(β)(ω) converges as t →∞, the sequence S_n(ω) converges to 0. 
    Since the event Ω' has probability 1, we deduce that the process S_n converges to 0.
    
    In order to obtain a domination, we bound
    
    S_n ≤sup_s ∈ [0, 1]∑_v ∈(as)^2β X_v(as) - ψ(2β)as W_s + t_n - a(s + t_n)^(v, as)(β)^2 + W_t_n - at_n(β)^2.

    Note that
    
    ∑_v ∈(as)^2β X_v(as) - ψ(2β)as W_s + t_n - a(s + t_n)^(v, as)(β)^2 ≤^(2ψ(β) - ψ(2β))as W_s + t_n - at_n(β)^2.

    Hence,
    
    S_n ≤(sup_s ∈ [0, 1]^(2ψ(β) - ψ(2β))as + 1) sup_s ∈ [0, 1] W_s + t_n - at_n(β)^2.

    By Lemma <ref>, the process W_t(β) is bounded in L^2p for any p ∈ (1, 1/β^2).
    By Doob's maximal inequality, the domination in (<ref>) is in L^p.
    Thus, by dominated convergence, the process S_n converges in L^p to 0.


Let us come back to (<ref>).
By (<ref>), we have

    ∑_u ∈(at_n)^2β X_u(at_n) - ψ(2β)at_nS_n^(u)_at_n = W_at_n(2β) S_n.

By Lemma <ref> and since the additive martingale converges , the above process converges to 0 as n →∞.
To obtain (<ref>), it is then sufficient to show that

    V_n = ∑_u ∈(at_n)^2β X_u(at_n) - ψ(2β)at_n(S_n^(u) - S_n^(u)_at_n)  0,

Let us fix p ∈ (1, 2] such that p < 1/2β^2.
By Lemma <ref>,

    V_n^p_at_n≤ W_at_n(p2β) |S_n - S_n|^p ^-(pψ(2β) - ψ(p2β))at_n.

By Lemma <ref>, the process S_n - S_n converges in L^p to 0.
In particular,

    V_n^p≤ C ^-(pψ(2β) - ψ(p2β))at_n.

By (<ref>) and by the Borel-Cantelli lemma, the convergence (<ref>) holds, which concludes the proof of Theorem <ref>.<ref>.



 §.§ Intermediate temperature


Let us fix β = √(2)/2 and show Theorem <ref>.<ref>.
By (<ref>), we can rewrite

    √(at)^at/2ν_β, t([a, 1]) = √(at) (G_t + R_t) W_t(β)^-2,

where

    G_t = ∑_u ∈(at)^√(2) X_u(at) - 2at W_∞^(u, at)(β)^2   and   R_t = ∑_u ∈(at)^√(2) X_u(at) - 2at(W_t - at^(u, at)(β)^2 - W_∞^(u, at)(β)^2).

Since the additive martingale converges , it suffices to obtain the convergence in probability of √(at)(G_t + R_t) toward √(2/π) Z_∞W_∞(β)^2.
Let us justify that we can ignore the term R_t.
By the triangle inequality and the branching property,

    √(at)|R_t|_at≤√(at) W_at(√(2)) |W_t - at(β)^2 - W_∞(β)^2|.

By (<ref>) and Lemma <ref>, the above process converges in probability to 0 as t →∞, so does √(at) R_t.

It remains to study G_t.
Let us fix θ∈.
We use the conditional characteristic function

    exp( θ√(at) G_t )_at = ∏_u ∈(at)φ(λ^(u, at)),

where

    λ^(u, at) = θ√(at)^√(2) X_u(at) - 2at,   φ(λ) = ^λ W_∞(β)^2 = 1 + λW_∞(β)^2 + r_1(λ),

and r_1(λ) = o(λ) as λ→ 0.
In order to use the above expansion, we need to restrict to an event with high probability where max_u ∈(at)λ^(u, at) converges to 0 as t →∞.
We can choose the event Λ = Λ_δ, at defined in (<ref>), with an arbitrary δ∈ (0,1/√(2)).
On this event, for t large enough and for all u ∈(at), we have φ(λ^(u, at)) ∈∖ (-∞, 0].
This allows us to use the principal complex logarithm  and to write

    exp( θ√(at) G_t )_at1_Λ   = exp( ∑_u ∈(at)(1 + λ^(u, at)W_∞(β)^2 + r_1(λ^(u, at))) ) 1_Λ
       = exp( θ√(at) W_at(√(2)) W_∞(β)^2 + ∑_u ∈(at) r_2(λ^(u, at)) ) 1_Λ,

where r_2(λ) = o(λ) as λ→ 0.
By choice of Λ, with probability 1, we have

    ∑_u ∈(at) r_2(λ^(u, at)) 1_Λ = o ( ∑_u ∈(at)λ^(u, at)) 1_Λ = o ( √(at) W_at(√(2)) ) 1_Λ.

By (<ref>) and (<ref>), with probability 1, we have

    exp( θ( √(at) G_t - √(at) W_at(√(2)) W_∞(β)^2) ) _at1_Λ = exp( o ( √(at) W_at(√(2)) ) ) 1_Λ.

By (<ref>), the above process converges in probability to 1.
This being true for any θ∈, we deduce that √(at) G_t - √(2/π) Z_∞W_∞(β)^2 converges in probability to 0 as t →∞, which concludes the proof of Theorem <ref>.<ref>.




 §.§ Lower temperature


Let β∈ (√(2)/2, √(2)).
Here we follow the techniques from <cit.>, where Iksanov, Kolesko and Meiners computed the fluctuations of the additive martingales of the branching random walk, using the convergence of the extremal process _t.
The main differences are that we have to deal with this convergence jointly with the additive martingale and that we identify a stable distribution at the limit.
Recall that m(t) = √(2)t - 3 · 2^-3/2log t.
We rewrite

    (at)^3 β/√(2)^(√(2) - β)^2 atν_β,t([a, 1]) = H_t W_t(β)^-2,

where

    H_t = ∑_u ∈(at)^2 β (X_u(at) - m(at)) W_t - at^(u, at)(β)^2.

Let ℓ∈, which has here to be thought as a large negative number.
We will use the following continuous approximation of 1_[ℓ, ∞),

    χ_ℓ^+(x) = 
            0    if  x ≤ℓ, 
    
            x - ℓ   if ℓ≤ x ≤ℓ+1, 
    
            1    if  x ≥ℓ+1,

as well as χ_ℓ^- = 1 - χ_ℓ^+.
We decompose

    H_t    = ∑_u ∈(at)^2 β (X_u(at) - m(at))(W_t - at^(u, at)(β)^2 - W_∞^(u, at)(β)^2) 
          + ∑_u ∈(at)^2 β (X_u(at) - m(at)) W_∞^(u, at)(β)^2 χ_ℓ^+(X_u(at) - m(at)) 
          + ∑_u ∈(at)^2 β (X_u(at) - m(at)) W_∞^(u, at)(β)^2 χ_ℓ^-(X_u(at) - m(at)) 
       = R_t, ℓ^1 + H_t, ℓ + R_t, ℓ^2.

Below, Lemma <ref> and Lemma <ref> state that R_t, ℓ^1 and R_t, ℓ^2 are negligible as t →∞ and ℓ→ -∞.
Lemma <ref> gives a first characterization of the limit distribution.


    For all ℓ∈, R_t, ℓ^1 converges in probability to 0 as t →∞.



    We consider p ∈ (0,1] such that √(2)/2β < p < 1/β^2 which is possible because β∈ (√(2)/2, √(2)).
    Using the subadditivity of x ↦ x^p on [0, ∞), we have
    
    |R_t, ℓ^1|^p_at≤|W_t - at(β)^2 - W_∞(β)^2|^p ∑_u ∈(at)^p2β(X_u(at) - m(at)).

    Since p <1/β^2, the expectation on the right-hand side tends to 0 (see Lemma <ref>).
    By (<ref>), the sum on the right-hand side converges in distribution to a finite limit because p 2β > √(2).
    Therefore, by Slutsky's theorem, the above quantity converges in distribution to 0 as t →∞.
    Then, R_t, ℓ^1 converges in probability to 0 as t →∞.



    For any δ > 0, lim_ℓ→ -∞lim sup_t →∞(R_t,ℓ^2 > δ) = 0.



    Note that if a process (X_t, ℓ)_t ≥ 0, ℓ∈ takes its values in [0, ∞), then the following two statements are equivalent
    
        
  *  for any δ > 0, lim_ℓ→ -∞lim sup_t →∞X_t, ℓ≥δ = 0,
        
  *  lim_ℓ→ -∞lim sup_t →∞X_t, ℓ∧ 1 = 0.
    
    We consider p ∈ (0,1] such that √(2)/2β < p < 1/β^2.
    By Jensen's inequality,
    
    (R_t, ℓ^2)^p ∧ 1≤(R_t, ℓ^2)^p_at∧ 1.

    But, using the subadditivity of x ↦ x^p on [0, ∞),
    
    (R_t, ℓ^2)^p_at   ≤W_∞(β)^2p∑_u ∈(at)^p2 β(X_u(at) - m(at))χ_ℓ^-(X_u(at) - m(at))^p 
            = C _at(^p2 β xχ_ℓ^-(x)^p),

    using that W_∞(β) ∈ L^2p because p < 1/β^2 (see Lemma <ref>).
    Since <ref> and <ref> are equivalent, it suffices to obtain that, for any δ > 0,
    
    lim_ℓ→ -∞lim sup_t →∞_at(^p2 β xχ_ℓ^-(x)^p) ≥δ = 0.

    This is proved by Bonnefont in <cit.> (this requires p2β > √(2), which is ensured by our choice of p).



    We have
    
    (at)^3 β/√(2)^(√(2) - β)^2 atν_β, t([a, 1])  X_over = 1/W_∞(β)^2∑_k ≥ 1^2 βξ_k W_k^2,   in distribution,

    where W_k, k ≥ 1, are copies of W_∞(β), independent of W_∞(β) and _∞.



    We first check that X_over is well-defined by showing
    
    ∑_k ≥ 0^2 βξ_k W_k^2 < ∞, 
              almost surely.

    To see this, fix p ∈ (0,1] such that √(2)/2β < p < 1/β^2. 
    Then, by subadditivity,
    
    (∑_k ≥ 1^2 βξ_k W_k^2)^p_∞ 
            = ∑_k ≥ 1^p2 βξ_kW_∞(β)^2p,

    which is finite, by Lemma <ref> and Lemma <ref>, proving (<ref>).

    Now, in order to prove the lemma, by Lemma <ref>, Lemma <ref> and <cit.>, it is sufficient to show that, for all ℓ∈,
    
    H_t, ℓ/W_t(β)^21/W_∞(β)^2∑_k ≥ 1^2 βξ_kχ_ℓ^+(ξ_k) W_k^2,   in distribution,

    and that
    
    1/W_∞(β)^2∑_k ≥ 1^2 βξ_kχ_ℓ^+(ξ_k) W_k^2  X_over,   in distribution.

    Note that (<ref>) is a driect consequence of (<ref>).
    To prove (<ref>), we introduce the test function f(x,y) = ^2β xχ_ℓ^+(x) y^2.
    By (<ref>)
    [Note that f is not bounded as required in (<ref>). However, one can first apply (<ref>) to f ∧ M for some M> 0, and then let M →∞, using that ^*(f) < ∞ by (<ref>).], 
    (_at^*(f), W_at(β)) converges in distribution to (_∞^*(f), W_∞(β)).
    By Slutsky's theorem, the same is true for (_at^*(f), W_t(β)) and hence we get (<ref>).


We are now able to prove Theorem <ref>.<ref>.
By definition of _∞, we can rewrite

    X_over = (C Z_∞)^2β/√(2)/W_∞(β)^2∑_i≥ 1^2β(p_i + X_i),

where X_i = 1/2βlog∑_j ^2 βΔ_ij W_ij^2 and where W_ij are copies of W_∞(β), independent of _∞, , (_i)_i ≥ 1.
Now, since the variables X_i for i ≥ 1 are and independent of , it follows from <cit.> that

    ∑_i ≥ 1δ_p_i + X_id=∑_i ≥ 1δ_p_i + 1/√(2)log^√(2) X_1,

under the condition that [^√(2) X_1] < ∞, which we check as follows: taking p ∈ (√(2)/β, 2/β^2) ∩ [1, 2], using first Jensen's inequality and then subadditivity of x ↦ x^p/2 on [0, ∞), we obtain

    ^√(2) X_1^β p /√(2)≤^β p X_1
        = (∑_j ≥ 1^2 βΔ_1j W_1j^2)^p/2≤∑_j ≥ 1^p βΔ_1jW_∞(β)^p
        < ∞,

where the last quantity is finite by Lemma <ref> and Lemma <ref>.
By (<ref>) and (<ref>),

    X_overd=(C Z_∞^√(2) X_1)^2β/√(2)/W_∞(β)^2∑_i≥ 1^2β p_i.

Finally, by Lemma <ref>, the sum S_2β = ∑_i ^2β p_i is a non-degenerate (√(2)/2β)-stable random variable.
This remark together with Lemma <ref> and (<ref>) conclude the proof of Theorem <ref>.<ref>.




§ PROOF OF THEOREM <REF>



In this section, the key tools will be the changes of measure (<ref>) and (<ref>) at inverse temperature 2β, as well as the associated spinal decomposition.
We denote by Π the point process whose points s_i for i ≥ 1 are the ordered spinal branching times.
Recall that Π is a homogeneous Poisson point process on [0,∞) with intensity 1 under  and 2 under _2β.
We also write s_0 = 0.
In what follows, it will be convenient to decompose the additive martingale along the spine,

    W_t(β) = ∑_i ≥ 1^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) 1_s_i ≤ t + ^β X_ξ(t) - ψ(β)t,

where, if d_u ∧ v denotes the deathtime of the last common ancestor of u and v,

    W_s^(i)(β) = ∑_u ∈(s_i + s) 
     d_u ∧ξ(s_i + s) = s_i^β (X_u(s_i + s) - X_u(s_i)) - ψ(β)s.

Under both probability measures  and _2β, given Π, the processes (W_s^(i)(β))_s ≥ 0 for i ≥ 1 are the additive martingales of independent BBMs without spine.
Therefore, they converge -and _2β-to copies of W_∞(β), that we denote by W_∞^(i)(β).
This allows us to define

    W(β) = ∑_i ≥ 1^β X_ξ(s_i) - ψ(β)s_i W_∞^(i)(β),

which is the limit of W_t(β) under _2β as showed in the following lemma. Note that we distinguish W(β) and W_∞(β) for now, but, once this lemma is proved, we can afterwards say that they are the same quantity, see Remark <ref>.
We also emphasize here that the convergence of W_t(β) is classically studied under _β (see e.g. <cit.>), but we need it here under _2β.


    If β∈ [0, √(2/3)), the random variable W(β) is _2β-positive and finite.
    If β∈ [√(2/3), √(2)), it is _2β-infinite.
    In both cases, it is the _2β-limit of the process W_t(β) as t →∞.



    Let β∈ [0,√(2)).
    Under _2β,
    
    W(β) = ∑_i ≥ 1^β B_s_i + (3β^2/2 - 1) s_i W_∞^(i)(β),

    where B_s = X_ξ(s) - 2 β s is a standard Brownian motion.
    Note that 3β^2/2 - 1 ≥ 0 if and only if β≥√(2/3).
    In particular, if β∈ [√(2/3), √(2)), the (non-negative) terms of the series defining W(β) do not converge to 0 and then W(β) = ∞, _2β-Besides, by Fatou's lemma,
    
    lim inf_t →∞ W_t(β) ≥∑_i ≥ 1lim inf_t →∞^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) 1_s_i ≤ t,

    and this lower bound is _2β-equal to W(β) = ∞.

    Now let β∈ [0, √(2/3)).
    We have
    
    __2β[ W(β) | Π, X_ξ] = ∑_i ≥ 1^β B_s_i - (1 - 3β^2/2) s_i,

    and, _2β-, this series converges since
    
    √(^β B_s_n - (1 - 3β^2/2) s_n)^-(1 - 3β^2/2)/2 < 1.

    Thus, W(β) is _2β-finite.
    Furthermore, it is _2β-positive since the terms of the series (<ref>) are _2β-positive.
    
    It remains to control W(β) - W_t(β).
    We rewrite it
    
    ∑_i ≥ 1 
     s_i ≤ t^β X_ξ(s_i) - ψ(β)s_i (W_∞^(i)(β) - W_t - s_i^(i)(β)) + ∑_i ≥ 1 
     s_i > t^β X_ξ(s_i) - ψ(β)s_i W_∞^(i)(β) - ^β X_ξ(t) - ψ(β)t = T_1(t) + T_2(t) + T_3(t).

    The term T_3(t) converges _2β-to 0 since the exponent β X_ξ(s) - ψ(β)s is a Brownian motion with drift -1 + 3β^2/2 < 0 (and variance β^2).
    The term T_2(t) converges _2β-to 0 since it is the remainder of a series which converges _2β-To control the term T_1(t), we first show that it converges along the sequence of spinal branching times.
    By Lemma <ref>, for any p ∈ [1, 2] and n ≥ 1,
    
    __2β[ |T_1(s_n)|^p  | Π, X_ξ] ≤∑_i = 1^n ^p β X_ξ(s_i) - p ψ(β) s_i f(s_n - s_i),

    where, for s ≥ 0, f(s) = [|W_∞(β) - W_s(β)|^p].
    By Lemma <ref>, we have f(s) ≤ C ^-(p - 1)(1 - pβ^2/2)s. Hence, by (<ref>), the above quantity is bounded by
    
    C ^-(p - 1)(1 - pβ^2/2)s_n∑_i ≥ 1^p β X_ξ(s_i) - ψ(p β) s_i,

    which is _2β-summable in n as soon as p ∈ (1, 2/β^2) to ensure that the exponential prefactor vanishes, and β^2 (2p-p^2/2) < 1 to ensure that the sum is _2β-finite by proceeding as in (<ref>) (such a choice of p exists because β < √(2/3)).
    By the Borel–Cantelli lemma, T_1(s_n) converges _2β-to 0 as n →∞.
    Now let us control
    
    sup_t ∈ (s_n, s_n+1) |T_1(t) - T_1(s_n)| ≤∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_isup_t ∈ (s_n, s_n+1)| W_t - s_i^(i)(β) - W_s_n - s_i^(i)(β) |_= S_n, i.

    Similarly to (<ref>), we proceed by showing that
    
    ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i__2β[ S_n, i| Π]  0,   _2β-,

    and
    
    ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i( S_n, i - __2β[ S_n, i| Π] )  0,   _2β-

    By applying the Cauchy-Schwarz inequality and Doob's maximal inequality, we obtain, for any p ∈ (1,2],
    
    __2β[ S_n, i| Π] 
            ≤__2β[ (S_n, i)^p | Π]^1/p≤p/p-1 f(s_n - s_i, s_n+1 - s_i)^1/p,

    where, for s ≤ t,
    
    f(s, t) = [ | W_t(β) - W_s(β) |^p ] ≤ C ^-(p - 1)(1 - pβ^2/2)s,

    by Lemma <ref>.
    In particular, since p ∈ (1, 2/β^2), each term of the sum in (<ref>) converges _2β-to 0 as n →∞.
    Then, by dominated convergence with domination
    
    ^β X_ξ(s_i) - ψ(β)s_i__2β[ S_n, i| Π] ≤ C ^β X_ξ(s_i) - ψ(β)s_i,

    we obtain (<ref>).
    As for the convergence (<ref>), it can be deduced from Lemma <ref> and (<ref>), by proceeding as in the proof that T_1(s_n) converges _2β-to 0 above.
    Finally, the bound in (<ref>) converges _2β-to 0 as n →∞ and the term T_1(t) converges _2β-to 0 as t →∞.


 
    In the case 0 ≤β < √(2)/2, a consequence of Lemma <ref> is that W(β) = W_∞(β) -and _2β-since the measures  and _2β are equivalent.
    On the other hand, if β∈ [√(2)/2,√(2)), then  and _2β are mutually singular so we can assume that W_∞(β) = W(β) _2β-. 
    Moreover, one can check that W(β) = W_∞(β) -stays true for β∈ [√(2)/2,√(2)) by following an argument similar to the proof of Lemma <ref> in the case β∈ (0,√(2/3)).
    This ensures that there is no notation conflict if we write W_∞(β) instead of W(β) under  or _2β, which we will do from now on.




 §.§ Upper temperature


Let us fix β∈ (0, √(2/3)) and show Theorem <ref>.<ref>. 
Note that the claim concerning the convergence of W_t(β) has already been proved in Lemma <ref> (see also Remark <ref>), so we focus on the convergence of the rescaled overlap distribution.
We define

    F_t = ∑_u ∈(at)^2β X_u(at) - ψ(2β)at W_t - at^(u, at)(β)^2,

so that ^(1 - β^2)atν_β, t([a, 1]) = F_t W_t(β)^-2, by (<ref>).
We have

    ^(1 - β^2)atν_β, t([a, 1])   = [ F_t W_t(β)^-2 - F_t W_at(β)^-2] + [ F_t W_at(β)^-2] 
       = [ F_t W_t(β)^-2 - F_t W_at(β)^-2] + __2β[ W_at(β)^-2] [ W_t-at(β)^2 ],

using the branching property and the definition of _2β in the second term.
Then, by Lemma <ref> and Lemma <ref> below, together with Lemma <ref>,

    ^(1 - β^2)atν_β, t([a, 1])__2β[ W_∞^-2] W_∞(β)^2,

which is Theorem <ref>.<ref>.


    For every β∈ (0, √(2/3)),
    
    __2β[W_t(β)^-2] __2β[W_∞(β)^-2] ∈ (0, ∞).



 
    The convergence in this lemma stays true for any β∈ [0, √(2)), but for β∉ (0, √(2/3)) the limit is not in (0,∞) anymore:
    
        
  * If β = 0, the fixed point equation (<ref>) with t=∞ and s=1, together with <cit.> implies that __0[W_∞(0)^-2] =  [W_∞(0)^-1] = ∞.
        Then, by Fatou's lemma, __0[W_t(0)^-2] →∞ as t →∞.
        
  * If β∈ [√(2/3), √(2)), Lemma <ref> states that W_t(β) →∞, _2β-, and Lemma <ref> below implies that __2β[W_t(β)^-2] → 0 as t →∞.
    



    Let β∈ (0, √(2/3)) and define F_t as in (<ref>).
    Then,
    
    [ F_t W_t(β)^-2 - F_t W_at(β)^-2]  0.



The remainder of this section is dedicated to the proofs of Lemma <ref> and Lemma <ref>


    Let X_i for i ≥ 1 be independent random variables and a_i, i ≥ 1, be positive coefficients.
    Assume that there exist b, K > 0 such that, for each i ≥ 1 and every x > 0,
    
    X_i ≤ x≤ K x^b.

    Then, for every p ∈ [1, ∞) such that p/b ∉, there exists C = C(b,K,p) >0 such that, letting n = ⌈ p/b ⌉,
    
    ( ∑_i = 1^n a_i X_i )^-p≤ C a_1^-b⋯ a_n - 1^-b a_n^-p + b(n - 1).




    Note that b(n - 1) < p < bn because p/b ∉.
    We can assume that a_1 ≥ a_2 ≥⋯≥ a_n since the general case follows from
    
    a_i^-b a_n^-p + b(n - 1)≤ a_n^-b a_i^-p + b(n - 1).

    For each j = 1, …, n,
    
    ∑_i = 1^n a_i X_i ≤ x≤a_1 X_1 ≤ x⋯a_j X_j ≤ x≤ K^j a_1^-b⋯ a_j^-b x^bj.

    In particular, we can use the following bound
    
    ∑_i = 1^n a_i X_i ≤ x≤
                1    if  x ≥ a_1, 
    
                K^n - 1 a_1^-b⋯ a_n-1^-b x^b(n-1)   if  a_n ≤ x < a_1, 
    
                K^n a_1^-b⋯ a_n^-b x^bn   if  x < a_n.

    This yields
    
    ( ∑_i = 1^n a_i X_i )^-p   = ∫_0^∞∑_i = 1^n a_i X_i ≤ x^-1/px̣
       ≤∫_0^a_1^-px̣ + ∫_a_1^-p^a_n^-p K^n - 1 x^-b(n-1)/p a_1^-b⋯ a_n-1^-bx̣ + ∫_a_n^-p^∞ K^n x^-bn/p a_1^-b⋯ a_n^-bx̣
       ≤ a_1^-p + C a_1^-b⋯ a_n - 1^-b a_n^-p + b(n-1),

    using b(n-1)/p<1 in the second integral and bn/p >1 in the third one.
    To conclude, it suffices to note that a_1^-p≤ a_1^-b… a_n-1^-b a_n^-p + b(n-1).




    Let β∈ [0, √(2)) and p ∈ [1, ∞).
    There exist n_0≥ 1 and C > 0 such that, for any n ≥ n_0, any I = {i_1, …, i_n}⊂{1, 2, …} with i_1 < … < i_n and any t ∈ [0, ∞], on the event where s_i_n≤ t, we have
    
    __2β[ ( ∑_i ∈ I^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-p| Π] ≤ C ^(p^2β^2/2 + p(1 - 3β^2/2))s_i_n.




    Let b_max = (√(ψ(β)^2 + 2β^2) - ψ(β))/β^2 (recall it appears in the condition (<ref>) of Lemma <ref>).
    We choose n_0 > p/b_max.
    Now note that it is enough to prove (<ref>) for n=n_0: indeed, adding elements to I can only decrease the left-hand side of (<ref>) and increase the right-hand side (note that p^2β^2/2 + p(1 - 3β^2/2) ≥ 0).    
    So let n = n_0.
    Since n > p/b_max, there exists b ∈ (0,b_max) such that n = ⌈ p/b ⌉ and p/b ∉.
    Then, by Lemma <ref> and Lemma <ref>,
    
    __2β[ ( ∑_i ∈ I^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-p| Π, X_ξ]    ≤ C ( ∏_i ∈ I ∖{i_n}^β X_ξ(s_i) - ψ(β)s_i)^-b( ^β X_ξ(s_i_n) - ψ(β)s_i_n)^-p + b(n-1)
       = C ∏_j = 1^i_n( ^β (X_ξ(s_j) - X_ξ(s_j-1)) - ψ(β)(s_j - s_j-1))^-α_j,

    with α_j = p - b #(I ∩{1, …, j-1}).
    Recall that, under _2β, the spine trajectory X_ξ is a Brownian motion with drift 2β, independent of Π.
    Then,
    
    __2β[ ( ∑_i ∈ I^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-p| Π]    ≤ C ∏_j = 1^i_n^(α_j^2 β^2/2 + α_j (1 - 3β^2/2))(s_j - s_j - 1)
       ≤ C ∏_j = 1^i_n^(p^2 β^2/2 + p (1 - 3β^2/2))(s_j - s_j - 1),

    using in the last inequality that α_j ∈ [0,p] and that the function x ∈ [0,p] ↦ x^2 β^2/2 + x (1 - 3β^2/2) reaches its maximum at x=p, because β <√(2) and p≥ 1.
    This gives the desired inequality by telescoping.



    For every β∈ (0, √(2)), the process (W_t(β)^-2)_t ≥ 0 is bounded in L^p(_2β) for some p > 1.




    Let β∈ (0, √(2)) and p > 1, which will be chosen close enough to 1 depending on β later on.
    Let n ≥ 1 be given by Lemma <ref> applied to β and p.
    Let t ≥ 0 and set A = A_n, t = {s_n ≤ t}.
    On the one hand, 
    
    __2β[ W_t(β)^-2p1_A ] ≤__2β[ ( ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β) )^-2p1_A ] ≤ C __2β[ ^(2p^2β^2 + 2p(1 - 3β^2/2)) s_n],

    by Lemma <ref>.
    If we choose p close enough to 1, we have 2p^2β^2 + 2p(1 - 3β^2/2) < 2 and then the above bound is finite (recall s_n is Γ(n,2) distributed under _2β).
    On the other hand, keeping only the contribution of the spine to W_t(β), we have
    
    __2β[ W_t(β)^-2p1_A^c] ≤__2β[ ^-2p β X_ξ(t) + 2p ψ(β)t] _2β(s_n > t) 
            ≤^(2 - 3β^2 + 2pβ^2)pt C t^n-1^-2t.

    If we choose p close enough to 1, we have 2 - 3β^2 + 2pβ^2 < 0 and the above quantity converges to 0 as t →∞.
    Combining the two previous bounds shows the result.



    By Lemma <ref>, for every β∈ (0, √(2/3)), the random variable W_∞(β) is _2β-positive and finite.
    In particular, __2β[W_∞(β)^-2] > 0.
    Besides, W_t(β)^-2 converges _2β-to 
    W_∞(β)^-2 as t →∞ and is uniformly integrable for _2β, by Lemma <ref>.
    This implies that
    
    __2β[W_t(β)^-2] __2β[W_∞(β)^-2] < ∞,

    which concludes the proof.


It remains to prove Lemma <ref>.
We will use the following concentration inequality.


    Let β∈ [0, 1) and p ∈ [1, 2]. There exists C>0 such that, for any 0 ≤ s ≤ t ≤∞,
    
    | W_t(β) - W_s(β)/W_s(β)|^p≤ C ^-(1 - β^2)ps/2.




    We have, using decomposition (<ref>),
    
    (W_t(β) - W_s(β))^2 
        = ∑_u ∈(s)^2β X_u(s) - 2ψ(β)s (W_t - s^(u, s)(β) - 1)^2 + ∑_u, v ∈(s) 
     u ≠ v^β(X_u(s) + X_v(s)) - 2ψ(β)s (W_t - s^(u, s)(β) - 1)(W_t - s^(v, s)(β) - 1),

    where, conditionally on _s, the variables W_t - s^(u,s)(β) for u ∈(s) are independent with mean 1 and are bounded in L^2() because β <1 (see Lemma <ref>).
    Then, by applying successively Hölder's inequality and taking a conditional expectation given _s, we obtain
    
    | W_t(β) - W_s(β)/W_s(β)|^p≤(W_t(β) - W_s(β))^2/W_s(β)^2^p/2≤ C ^-(1 - β^2)s W_s(2β)/W_s(β)^2^p/2,

    since ψ(2β) - 2ψ(β) = -(1 - β^2) and since the additive martingale is bounded in L^2.
    The result follows from Lemma <ref>, which implies that [W_s(2β) W_s(β)^-2] = __2β[W_s(β)^-2] converges to a finite limit and hence is bounded.



    Let β∈ (0, √(2/3)).
    In order to take advantage of Lemma <ref>, we will work on the event A = A_n, a't = {|ξ(a't)| ≥ n}, where a' ∈ (0, a) is “close enough” to a and n ≥ 1 is “large enough”.
    However, in view of applying Lemma <ref>, it will be convenient to deal with a slightly smaller event that is measurable w.r.t. _∞.
    Let us define A̅ = A̅_n, a't = {∀ u ∈(a't), |u| ≥ n}.
    
    First, let us show that
    
    [ (F_t W_at(β)^-2 - F_t W_t(β)^-2) 1_A̅^c]  0.

    By conditioning on _at, we obtain
    
    [F_t W_at(β)^-21_A̅^c] = [W_at(2β) W_at(β)^-21_A̅^c] = __2β[W_at(β)^-21_A̅^c].

    Note that 1_A̅^c converges to 0 as t →∞.
    In addition, a direct consequence of Lemma <ref>, Lemma <ref> and Scheffé's lemma is that W_at(β)^-2 converges in L^1(_2β).
    In particular, this process is uniformly integrable, and so is W_at(β)^-21_A̅^c. Hence, the right-hand side of (<ref>) converges to 0 as t →∞.
    Besides,
    
    [F_t W_t(β)^-21_A̅^c] = ^(1 - β^2)at[ ν_β, t([a, 1]) 1_A̅^c] ≤^(1 - β^2)at(A̅^c).

    Applying the union bound, the change of measure (<ref>) and Lemma <ref>, we obtain
    
    (A̅^c) ≤∑_u∈(a't) 1_{|u| ≤ n} = __0[ ∑_u ∈(a't)1_{|u| ≤ n}/W_a't(0)] = e^a't_0(|ξ(a't)| ≤ n) ≤ C t^n e^-a' t,

    using in the last inequality that, under _0, the spinal generation |ξ(s)| has Poisson distribution with parameter 2s.
    Then,
    
    [F_t W_t(β)^-21_A̅^c] ≤ C t^n ^(1 - β^2)at - a't.

    By choice of a', this bound converges to 0 as t →∞.
    Hence (<ref>).
    
    Now, let us show that
    
    [(F_t W_at(β)^-2 - F_t W_t(β)^-2) 1_A̅]  0.

    Recalling that _2β, at|__∞ has Radon–Nykodim derivative W_at(2β) w.r.t. |__∞, the left-hand side of  (<ref>) equals
    
    __2β, at[ F_t/W_at(2β) (W_t(β) - W_at(β)) ( 1/W_t W_at(β)^2 + 1/W_t(β)^2 W_at(β)) 1_A̅] 
    
            = __2β, at[ W_t - at^(ξ(at), at)(β)^2 (W_t(β) - W_at(β)) ( 1/W_t(β) W_at(β)^2 + 1/W_t(β)^2 W_at(β)) 1_A̅],

    where we applied Lemma <ref> to rewrite
    
    F_t/W_at(2β) = ∑_u ∈(at)_2β, at( ξ(at) = u | _t ) W_t - at^(u, at)(β)^2.

    Let us abbreviate W_s^(ξ(at), at)(β) = W_s^(ξ, at)(β).
    Decomposing the process |W_t(β) - W_at(β)| along the spine and applying the triangle inequality, we bound it by
    
    ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i |W_t - s_i^(i)(β) - W_at - s_i^(i)(β)| + | ∑_i > n 
     s_i ≤ at^β X_ξ(s_i) - ψ(β)s_i (W_t - s_i^(i)(β) - W_at - s_i^(i)(β)) | 
    
            + ^β X_ξ(at) - ψ(β)at |W_t - at^(ξ, at)(β) - 1| = T_1(t) + T_2(t) + T_3(t).

    Besides,
    
    1/W_t(β) W_at(β)^2 + 1/W_t(β)^2 W_at(β)≤1/W_at(β)^3 + 2/W_t(β)^2 W_at(β).

    Therefore, and since A̅⊂ A, to obtain (<ref>), it suffices to show that, for i = 1, 2 and γ = 1, a,
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_i(t)/W_γ t(β)^2 W_at(β)1_A ]  0.


    Control of T_1.
    Let us show (<ref>) for i = 1.
    Let γ = 1  or  a.
    We have
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_1(t)/W_γ t(β)^2 W_at(β)1_A ] ≤[ W_t - at(β)^2 ] 
    ×∑_i = 1^n __2β[ ^β X_ξ(s_i) - ψ(β)s_i |W_t - s_i^(i)(β) - W_at - s_i^(i)(β)|/W_at(β)( ∑_j = 1 
     j ≠ i^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ].

    Let us denote by V_1^(i)(t) the process appearing in the last expectation.
    Since the additive martingale is bounded in L^2(), it suffices to get the convergence in mean of V_1^(i)(t) to 0 as t →∞, for i = 1, …, n.
    Using that W_at(β) ≥^β X_ξ(s_i) - ψ(β)s_i W_at - s_i^(i)(β) and then applying Lemma <ref>, we obtain
    
    __2 β[ V_1^(i)(t) ]    ≤__2 β[ |W_t - s_i^(i)(β) - W_at - s_i^(i)(β)|/W_at - s_i^(i)(β)( ∑_j = 1 
     j ≠ i^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ] 
       ≤ C __2 β[ ^-(1-β^2)(at-s_i)/2( ∑_j = 1 
     j ≠ i^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ].

    We can assume n ≥ n_0+1 where n_0 is given by Lemma <ref> applied to β and p = 2.
    This yields
    
    __2 β[ V_1^(i)(t) ] ≤ C __2 β[ ^-(1-β^2)(at-s_i)/2^(2 - β^2) s_n1_A ].

    By definition of the event A, we have s_i ≤ a't < at.
    The above expectation is then bounded by
    
    C ^-(1-β^2)(a-a')t/2__2 β[ ^(2 - β^2) s_n]  0,

    since s_n is Γ(n, 2) distributed under _2β and 2-β^2 < 2.
    Hence (<ref>) for i = 1.

    Control of T_2.
    Let us show (<ref>) for i = 2.
    Let γ = 1  or  a.
    We have
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_2(t)/W_γ t(β)^2 W_at(β)1_A ] ≤[ W_t - at(β)^2 ] 
    ×__2β[ | ∑_i > n 
     s_i ≤ at^β X_ξ(s_i) - ψ(β)s_i (W_t - s_i^(i)(β) - W_at - s_i^(i)(β))/W_at(β)| ( ∑_j = 1^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-21_A ].

    Let us denote by V_2(t) the process appearing in the last expectation and show that it converges in mean to 0 as t →∞.
    To do so, we show that it converges _2β-to 0 and that it is bounded in L^p(_2β) for some p > 1.
    
    By Lemma <ref>, the processes W_at(β) and W_γ t - s_j^(j)(β) for j ≥ 1 converge _2β-to some positive limits as t →∞.
    Then, the _2β-convergence of V_2(t) to 0 will follow from
    
    ∑_i ≥ 1^β X_ξ(s_i) - ψ(β)s_i (W_t - s_i^(i)(β) - W_at - s_i^(i)(β)) 1_s_i ≤ at 0,   _2β-

    Each term of the above series converges _2β-to 0 and is dominated by 2 ^β X_ξ(s_i) - ψ(β)s_isup_t ≥ 0 W_t^(i)(β).
    By Doob's maximal inequality, this domination is summable.
    Hence (<ref>).
    
    Now fix p ∈ (1, 2) such that α_p = 2p^2β^2 + 2p(1 - 3β^2/2) < 2.
    We can assume n ≥ n_0 where n_0 is given by Lemma <ref> applied to β and 2p.
    Using that W_at(β) ≥^β X_ξ(s_i) - ψ(β)s_i W_at - s_i^(i)(β), applying Lemmas <ref> together with Lemma <ref>, and finally Lemma <ref>, we obtain
    
    __2 β[ V_2(t)^p ]    ≤__2 β[ | ∑_i > n 
     s_i ≤ atW_t - s_i^(i)(β) - W_at - s_i^(i)(β)/W_at - s_i^(i)(β)|^p ( ∑_j = 1^n ^β X_ξ(s_j) - ψ(β)s_j W_γ t - s_j^(j)(β) )^-2p1_A ] 
       ≤ C __2 β[ ∑_i > n 
     s_i ≤ at__2 β[ | W_t - s_i^(i)(β) - W_at - s_i^(i)(β)/W_at - s_i^(i)(β)|^p | Π] ^α_p s_n1_A ] 
       ≤ C __2 β[ ∑_i > n 
     s_i ≤ at^-(1 - β^2)p(at - s_i)/2^α_p s_n1_A ].

    We then decompose the sum over the spinal branching times located in [0, a't] and those in (a't, at].
    First,
    
    __2 β[ ∑_i > n 
     s_i ≤ a't^-(1 - β^2)p(at - s_i)/2^α_p s_n1_A ] ≤^-(1 - β^2)p(a - a')t/2__2 β[ #(Π∩ [0, a't]) ^α_p s_n].

    Note that #(Π∩ [0, a't]) has Poisson distribution with parameter 2a't and then its moment of order k ≥ 1 is an O(t^k).
    Thanks to Hölder's inequality, we see that the last expectation factor is an O(t) since α_p < 2.
    In particular, the above quantity is bounded.
    Now, by independence,
    
    __2 β[ ∑_i > n 
     a't < s_i ≤ at^-(1 - β^2)p(at - s_i)/2^α_p s_n1_A ]    = __2 β[ ∑_a't < s_i ≤ at^-(1 - β^2)p(at - s_i)/2] __2 β[ ^α_p s_n1_A ] 
       ≤( ∫_a't^at^-(1 - β^2)p(at - s)/2 2ṣ) __2 β[ ^α_p s_n],

    which is also bounded since α_p < 2.
    Finally, the process V_2(t) is bounded in L^p(_2β) and then its convergence to 0 implies its convergence in mean.
    Hence (<ref>) for i = 2.

    Control of T_3.
    Let us show (<ref>) for i = 3.
    Let γ = 1  or  a.
    We have
    
    __2β, at[ W_t - at^(ξ, at)(β)^2 T_3(t)/W_γ t(β)^2 W_at(β)1_A ] 
    ≤| W_t - at(β)^3 - W_t - at(β)^2 | __2β[ ^β X_ξ(at) - ψ(β) at/W_at(β)( ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i W_γ t - s_i^(i)(β) )^-21_A ].

    Let us denote by V_3(t) the process appearing in the last expectation.
    Since the additive martingale is bounded in L^3(), it suffices to show that V_3(t) converges in mean to 0 as t →∞.
    But it converges _2β-to 0 since
    
        
  * the processes W_at(β) and W_γ t - s_i^(i)(β) for i ≥ 1 converge to some positive limits as t →∞, by Lemma <ref>,
        
  * β X_ξ(s) - ψ(β)s = β B_s - (1 - 3β^2/2)s, where B_s is a standard Brownian motion, and 1 - 3β^2/2>0.
    
    In addition, the process V_3(t) is bounded in L^p(_2β).
    Indeed, noting that ^β X_ξ(at) - ψ(β) at≤ W_at(β),
    
    __2β[ V_3(t)^p ] ≤__2β[ ( ∑_i = 1^n ^β X_ξ(s_i) - ψ(β)s_i W_γ t - s_i^(i)(β) )^-2p],

    and we see that the right-hand side is bounded thanks to Lemma <ref>.
    Finally, the process V_3(t) converges in mean to 0.
    Hence (<ref>) for i = 3.
    This concludes the proof of Lemma <ref>.




 §.§ Lower temperature


We aim here at proving Theorem <ref>.<ref>, which covers the case β∈ (√(2/3), √(2)).
We tackle separately the lower bound and the upper bound.



  §.§.§ Lower bound


Our goal is to find a constant c > 0 such that, for every t ≥ 1,

    ν_β, t([a, 1])≥ c t^-3/2^-(2 - β^2)^2 at/8β^2.

It will be useful for the proof of Theorem <ref>.<ref> to note that the arguments below work for β = √(2/3) as well, so we only assume β∈ [√(2/3), √(2)).

We can restrict ourselves to the event Γ = Γ_at = {sup_s ∈ [0, at] (X_ξ(s) - v(β) s + f_at(s)) ≤ 1}, where v(β) = 1/β + β/2, f_t(s) = s^α∧ (t-s)^α and α is an arbitrary number in (0, 1/2).
By (<ref>), the definition of _2β,at, and then Lemma <ref>,

    ν_β, t([a, 1])   = ^(β^2 - 1) at__2β, at[ 1/W_t(β)^2∑_u ∈(at)^2β X_u(at) - ψ(2β)at/W_at(2β) W_t - at^(u, at)(β)^2 ] 
       = ^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-2] ≥^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_Γ].

Then, to prove (<ref>), it suffices to get

    __2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_Γ] ≥ c t^-3/2^-(2β - v(β))^2at/2.

Decomposing W_t(β) along the spine, we can rewrite the left-hand side of (<ref>) as

    __2β, at[ W_t - at^(ξ, at)(β)^2 (U_t + V_t W_t - at^(ξ, at)(β))^-21_Γ],

where

    U_t = ∑_i ≥ 1 
     s_i ≤ at^β X_ξ(s_i) - ψ(β)s_i W_t - s_i^(i)(β)   and   V_t = ^β X_ξ(at) - ψ(β)at.

For all U, V, W > 0,

    W^2 (U + V W)^-2≥ W^2 (V + V W)^-2∧ W^2 (U + U W)^-2≥ W^2 (1 + W)^-2 (U + V)^-2.

Thus, the left-hand side of (<ref>) admits the following lower bound,

    [ W_t - at(β)^2 (1 + W_t - at(β))^-2] __2β[ (U_t + V_t)^-21_Γ] ≥[ W_∞(β)^2 (1 + W_∞(β))^-2] __2β[ (U_t + V_t)^-21_Γ],

by Fatou's lemma.
Since W_∞(β) > 0 , to prove (<ref>), it suffices to get

    __2β[ (U_t + V_t)^-21_Γ] ≥ c t^-3/2^-(2β - v(β))^2at/2.

On the event Γ, since β v(β) = ψ(β),

    U_t ≤^β∑_i ≥ 1 
     s_i ≤ at^-β f_at(s_i) W_t - s_i^(i)(β)   and   V_t ≤^β.

Then,

    __2β[ (U_t + V_t)^-21_Γ] ≥^-2 β__2β[ ( ∑_i ≥ 1 
     s_i ≤ at^-β f_at(s_i) W_t - s_i^(i)(β) + 1 )^-2] _2β( Γ).

By Jensen's inequality and recalling the s_i's are the points of a PPP(2 ṣ),

    __2β[ ( ∑_i ≥ 1 
     s_i ≤ at^-β f_at(s_i) W_t - s_i^(i)(β) + 1 )^-2] ≥( ∫_0^at^-β f_at(s) 2 ṣ + 1 )^-2≥ c > 0.

Besides, recalling that, under _2β, (X_ξ(s))_s≥ 0 is a Brownian motion with drift 2β, it follows from Girsanov theorem that

    _2β( Γ) 
        = [ ^(2β - v(β))B_at-(2β - v(β))^2 at/21_{sup_s ∈ [0, at] (B_s + f_at(s)) ≤ 1}],

where (B_s)_s≥ 0 denotes a standard Brownian motion.
We then restrict ourselves to the event { B_at∈ [0,1] } and use that 2β - v(β) ≥ 0 to get

    _2β( Γ) 
        ≥^-(2β - v(β))^2 at/2( sup_s ∈ [0, at] (B_s + f_at(s)) ≤ 1, B_at∈ [0, 1] ) 
        ≥ c ^-(2β - v(β))^2 at/2 t^-3/2,

by (<ref>).
By (<ref>), (<ref>) and (<ref>), the inequality (<ref>) holds.
Hence, it proves (<ref>) and the lower bound in Theorem <ref>.<ref>.



  §.§.§ Upper bound


Our goal is to show that

    ν_β, t([a, 1]) = O(t^-3/2^-(2 - β^2)^2 at/8β^2).

We work with β∈ [√(2/3),√(2)), except in the last step where we specify that β>√(2/3), so that the beginning of the argument can be used to prove the upper bound in Theorem <ref>.<ref> as well.

Consider an arbitrary integer n ≥ 1 and define A̅ = A̅_2n, at = {∀ u ∈(at), |u| ≥ 2n}.
As seen in (<ref>), we have

    ν_β, t([a, 1]) 1_A̅^c≤A̅^c≤ C t^2n-1^-at = o(t^-3/2^-(2 - β^2)^2 at/8β^2),

using that (2 - β^2)^2/8β^2 ≤ 1/3 < 1 for β∈ [√(2/3), √(2)).
Concerning the contribution of the event A̅, we can rewrite, similarly to (<ref>),

    ν_β, t([a, 1]) 1_A̅ = ^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_A̅] ≤^(β^2 - 1) at__2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_A ],

where A = A_2n, at = {|ξ(at)| ≥ 2n}.
By (<ref>) and (<ref>), to prove (<ref>), it suffices to get

    __2β, at[ W_t - at^(ξ, at)(β)^2 W_t(β)^-21_A ] = O(t^-3/2^-(2β - v(β))^2 at/2),

where v(β) = 1/β + β/2.
Then we decompose W_t(β) = U_t + V_t W_t - at^(ξ, at)(β) with U_t and V_t defined in (<ref>). We also change measures, first using that _2β, at has density ^2β X_ξ(at)-2β^2at w.r.t. _0, at, and then applying Girsanov theorem under _0, at to trade the factor ^v(β)X_ξ(at)-v(β)^2at/2 for a drift v(β) added to the spine trajectory, so that U_t and V_t become (recall ψ(β) = β v(β))

    U_t' = ∑_i ≥ 1 
     s_i ≤ at^β X_ξ(s_i) W_t - s_i^(i)(β)   and   V_t' = ^β X_ξ(at).

Therefore, we get that (<ref>) is equivalent to

    __0, at[ ^(2β - v(β)) X_ξ(at) W_t - at^(ξ, at)(β)^2 (U_t' + V_t' W_t - at^(ξ, at)(β))^-21_A ] = O(t^-3/2).


For all U, V, W > 0 and any α∈ [0, 2],

    W^2 (U + V W)^-2≤ W^2 U^-α (V W)^-2 + α = U^-α V^-2 + α W^α.

Thus, the left-hand side of (<ref>) is bounded by

    __0, at[ ^(2β - v(β))X_ξ(at) (U_t')^-α (V_t')^-2 + α W_t - at^(ξ, at)(β)^α1_A ] = [ W_t - at(β)^α] __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A ].

Let α∈ [0, 2] ∩ [1, 2/β^2) so that the additive martingale is bounded in L^α() by Lemma <ref>.
To prove (<ref>), it suffices to get

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A ] = O(t^-3/2).

Now, decompose the process over the following events, indexed by integers k, ℓ≥ 0,

    E_k, ℓ = E_k, ℓ(X_ξ(s), 0 ≤ s ≤ at) = {sup_[0, at] X_ξ∈ [k, k+1], sup_[0, at] X_ξ - X_ξ(at) ∈ [ℓ, ℓ+1] }.

Fix k, ℓ and denote s_k, n the n-th point of Π after time τ_k = inf{s ≥ 0 : X_ξ(s) = k}.
Let us first treat the case s_k, n≤ at.
On the event E_k, ℓ∩{s_k, n≤ at},

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α| Π, X_ξ] 
    ≤ C ^-v(β)k - (αβ - v(β))ℓ__0[ ( ∑_i ≥ 1 
    τ_k ≤ s_i ≤ s_k, n^β(X_ξ(s_i) - X_ξ(τ_k)) W_t - s_i^(i)(β) )^-α| Π, X_ξ] 
    ≤ C ^-v(β)k - (αβ - v(β))ℓ^αβsup_[τ_k, s_k, n] (X_ξ(τ_k) - X_ξ)__0[ ( ∑_i ≥ 1 
    τ_k ≤ s_i ≤ s_k, n W_t - s_i^(i)(β) )^-α| Π, X_ξ].

By Lemma <ref> and Lemma <ref>, we can choose n so that the last conditional expectation is bounded, which yields

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A 1_E_k, ℓ∩{s_k, n≤ at}] 
    ≤ C ^-v(β)k - (αβ - v(β))ℓ__0[ ^αβsup_[τ_k, s_k, n] (X_ξ(τ_k) - X_ξ)1_E_k, ℓ∩{s_k, n≤ at}].

By using time reversal invariance of Brownian motion and Poisson point process, we can treat the case s_k, n > at similarly.
Indeed, on the event A, if s_k, n > at, then there are at least n spinal branching times between 0 and τ_k.
Define s_i' = at - s_i, Π' the corresponding point process and X_ξ'(s) = X_ξ(at - s) - X_ξ(at). Denote s_ℓ, n' the n-th point of Π' after time τ_ℓ' = inf{s ≥ 0 : X_ξ'(s) = ℓ} if it exists, and set s_ℓ, n' = +∞ otherwise.
Note that, on E_k,ℓ, we have τ_k ≤ at-τ_ℓ', so on the event E_k,ℓ∩ A ∩{ s_k, n > at }, there are at least n point of Π' after τ_ℓ', so we get

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A 1_E_k, ℓ∩{s_k, n > at}] 
       ≤__0[ ^-(αβ - v(β)) X_ξ'(at)( ∑_i ≥ 1 
    τ_ℓ' ≤ s_i' ≤ s_ℓ, n'^β(X_ξ'(s_i') - X_ξ'(at)) W_t - (at - s_i')^(i)(β) )^-α1_E_ℓ, k(X_ξ'(s), 0 ≤ s ≤ at) ∩{s_ℓ, n' ≤ at}] 
       = __0[ ^-(αβ - v(β)) X_ξ(at)( ∑_i = 1^n ^β(X_ξ(s_ℓ, i) - X_ξ(at)) W_t - (at - s_ℓ, i)^(i)(β) )^-α1_E_ℓ, k(X_ξ(s), 0 ≤ s ≤ at) ∩{s_ℓ, n≤ at}] 
       ≤ C ^-v(β)k -(αβ - v(β))ℓ__0[ ^αβsup_[τ_ℓ, s_ℓ, n] (X_ξ(τ_ℓ) - X_ξ)1_E_ℓ, k∩{s_ℓ, n≤ at}],

by Lemma <ref> and Lemma <ref> as before.
Using α≤ 2 if β<1 and α < 2/β^2 otherwise, we have αβ < 2, which allows us to apply Lemma <ref> below.
This together with (<ref>) and (<ref>) lead to

    __0[ ^(αβ - v(β)) X_ξ(at) (U_t')^-α1_A ] ≤ C t^-3/2∑_k = 0^∞∑_ℓ = 0^∞^-v(β)k - (αβ - v(β))ℓ (k + ℓ + 1) ^- c ℓ^2/t.

We finally assume that β∈ (√(2/3),√(2)), so that v(β)/β < 2 ∧ 2/β^2 and we can thus choose α > v(β)/β. This ensures that the summand in (<ref>) decreases exponentially fast in ℓ, and be can simply bound ^- c ℓ^2/t≤ 1.
Then, the double sum in (<ref>) is a O(1), which proves (<ref>) and therefore (<ref>).
This concludes the proof of Theorem <ref>.<ref>.


    For any γ∈ [0,2), there exist C, c > 0 such that, for all t ≥ 1 and k, ℓ≥ 0,
    
    __0[ ^γsup_[τ_k, s_k, n] (X_ξ(τ_k) - X_ξ)1_E_k, ℓ(X_ξ(s), 0 ≤ s ≤ t)1_{s_k, n≤ t}] ≤ C (k + ℓ + 1) t^-3/2^-c k^2/t - c ℓ^2/t.




    The exponential factor in the bound (<ref>) has not been used to prove (<ref>), because we bounded the factor ^- c ℓ^2/t in (<ref>) by 1.
    However, it will play a key role in the proof of the upper bound for Theorem <ref>.<ref>, which is also based on (<ref>).


To prove Lemma <ref>, we will need the following intermediate lemma.


    Let σ_n be a random variable with distribution Γ(n, 2) and (B_t)_t ≥ 0 be a standard Brownian motion independent of σ_n.
    For any γ < 2, there exist C, c > 0 such that, for any t > 0,
    
    [ ^γsup_[0, σ_n] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1], σ_n ≤ t}] ≤
                C (t + 1)^-3/2   if ℓ = 0  or  1, 
    
                C ℓ t^-3/2^-c ℓ^2/t   if ℓ≥ 2.




    First consider ℓ = 0 or 1.
    The proof of the case ℓ≥ 2 will be similar although more technical.
    We have
    
    [ ^γsup_[0, σ_n] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1], σ_n ≤ t}] = C ∫_0^t [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ.

    Bounding the indicator function by 1 and using [e^γsup_[0,s] B] ≤ 2 ^γ^2s/2, we obtain
    
    ∫_t/2^t [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ   ≤ C ∫_t/2^t s^n - 1^-(2 - γ^2/2)sṣ
       ≤ C t^n ^-(2 - γ^2/2)t/2.

    Concerning the integration over s ∈ [0, t/2], we use Markov property at time s and Lemma <ref> (recall ℓ = 0 or 1) to get 
    
    [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] 
               ≤[ ^γsup_[0, s] B1_{B_s ≥ -1}( C (B_s + 2) (t - s)^-3/2∧ 1 ) ] 
       ≤ C [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 2) (t - s + 1)^-3/2].

    Note that (t - s + 1)^-3/2≤ C(t + 1)^-3/2 since s ≤ t/2.
    Moreover, (B_s, sup_[0, s] B) has the following density (see <cit.> or <cit.>),
    
    (x, y) ↦√(2/π) s^-3/2 (2y - x) ^-(2y - x)^2/2s1_x ≤ y, 0  ≤ yx̣ỵ.

    We deduce that, up to a multiplicative constant, (<ref>) is bounded by
    
    1/(t + 1)^3/2 s^3/2∫_0^∞ (y + 1)^2 ^γ y∫_-1^y ^-(2y - x)^2/2sx̣ỵ≤C/(t + 1)^3/2 s∫_-∞^∞ (y + 1)^2 ^γ y - y^2/2sỵ = P(s) ^γ^2 s/2/(t + 1)^3/2√(s),

    for some polynomial P.
    By integrating this against s^n-1^-2s1_s ∈ [0, t/2]ṣ, we obtain
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C (t + 1)^-3/2.

    By (<ref>) and (<ref>), the bound (<ref>) holds for ℓ = 0 or 1.

    Now consider ℓ≥ 2.
    By Hölder's inequality, for all p, q > 1 such that 1/p + 1/q = 1,
    
    [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] ≤[ ^p γsup_[0, s] B]^1/pB_t ≥ℓ - 1^1/q≤ C ^p γ^2 s/2^-(ℓ - 1)^2/2qt.

    Taking p close enough to 1 so that pγ^2 < 4 and integrating (<ref>) against s^n-1^-2s1_s ∈ [t/2, t]ṣ, we obtain
    
    ∫_t/2^t [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ^-c t - c ℓ^2/t.

    The integration over s ∈ [0, t/2] is more intricate.
    If t ≤ 2, then (<ref>) implies that
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ^-cℓ^2/t≤ C ℓ (t/2)^-3/2^-cℓ^2/t.

    Now assume that t > 2 so that t - s > 1.
    By using Markov property and Lemma <ref>,
    
    [ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] 
    ≤ C [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 1) ℓ (t - s)^-3/2( ^-(B_s - ℓ - 1)^2/2(t-s) + ^-(B_s - ℓ + 1)^2/2(t-s)) ].

    Let ℓ' = ℓ± 1.
    Since (B_s, sup_[0, s] B) has density (<ref>),
    
    [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 1) ^-(B_s - ℓ')^2/2(t-s)] 
    ≤C/s^3/2∫_0^∞ (y + 1)^2 ^γ y∫_-1^y ^-(x - ℓ')^2/2(t-s) - (2y - x)^2/2sx̣ỵ.

    The function x ↦√(t/2 π s(t - s))^-(x - ℓ')^2/2(t-s) - (2y - x)^2/2s is the density at time s of a Brownian bridge from (0, 2y) to (t, ℓ').
    It achieves its maximum at x_max = 2y (t-s)/t + ℓ' s/t, which is larger than y since ℓ' > 0 and s ≤ t/2.
    We deduce that (<ref>) is bounded by
    
    C/s∫_0^∞ (y + 1)^2 ^γ y - (y - ℓ')^2/2(t-s) - y^2/2sỵ = C/s^γ^2 s (t-s)/2t + γℓ' s/t - ℓ'^2/2t∫_0^∞ (y + 1)^2 ^-(y - y_max)^2 t/2s(t-s)ỵ,

    where y_max = ℓ' s/t + γ s(t-s)/t.
    Subsequently,
    
    [ ^γsup_[0, s] B1_{B_s ≥ -1} (B_s + 1) ^-(B_s - ℓ')^2/2(t-s)]    ≤ C (y_max + 1)^2 + s(t-s)/t/√(s)^γ^2 s (t-s)/2t + γℓ' s/t - ℓ'^2/2t
       ≤P(s)/√(s)^γ^2 s/2 + γℓ' s/t - cℓ'^2/t,

    for some polynomial P, where we have used that (ℓ' s/t)^2 ^-ℓ'^2/2t≤ s (ℓ'^2/t) ^-ℓ'^2/2t≤ C s ^-c ℓ'^2/t.
    Hence,
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{B_s ≥ -1}(B_s + 1) ℓ/(t - s)^3/2^-(B_s - ℓ')^2/2(t-s)] s^n - 1^-2sṣ
    ≤C ℓ/t^3/2^-c ℓ'^2/t∫_0^t/2P(s) s^n - 1/√(s)^(-2 + γ^2/2 + γℓ'/t)sṣ,

    Let ε∈ (0, 1).
    
        
  * If t ≥ (1 + ε) 2 γ/4 - γ^2ℓ', then
        
    -2 + γ^2/2 + γℓ'/t≤-4 + γ^2/2 + 4 - γ^2/2 (1 + ε) < 0.

        In this case, (<ref>) is bounded by C ℓ t^-3/2^-c ℓ'^2/t.
        
  * If t < (1 + ε) 2 γ/4 - γ^2ℓ', then we can directly use Hölder's inequality (<ref>) to bound
        
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ^-cℓ^2/t≤ C ^-ct^-cℓ^2/t,

        up to a change of the constant c.
    
    In both cases, there exist C, c > 0 such that, for all t > 2,
    
    ∫_0^t/2[ ^γsup_[0, s] B1_{inf_[0, t] B ≥ -1, B_t ∈ [ℓ-1, ℓ+1]}] s^n-1^-2sṣ≤ C ℓ t^-3/2^-cℓ^2/t.

    By (<ref>), (<ref>) and (<ref>), the bound (<ref>) holds for ℓ≥ 2.



    If k = 0, then τ_k = 0 and (<ref>) is a direct consequence of Lemma <ref>, because X_ξ under _0 has the same distribution as -B.
    Assume k ≥ 1.
    In order to use Markov property, define σ_n = s_k, n - τ_k and
    
    F_ℓ(s) = ^γsup_[s, s + σ_n] (X_ξ(s) - X_ξ)1_{inf_[s, t] (X_ξ(s) - X_ξ) ≥ -1, X_ξ(s) - X_ξ(t) ∈ [ℓ - 1, ℓ + 1], σ_n ≤ t - s}.

    The left-hand side of (<ref>) is bounded by
    
    __0[ F_ℓ(τ_k) 1_{sup_[0, τ_k] X_ξ≤ k + 1, τ_k ≤ t}] = ∫_0^t __0[ F_ℓ(s) ] _0 ( sup_[0, τ_k] X_ξ≤ k + 1, τ_k ∈ṣ).

    If ℓ = 0 or 1, then by Lemma <ref> and <cit.>, up to a multiplicative constant, (<ref>) is bounded by
    
    ∫_0^t k ^-k^2/2s/(t - s + 1)^3/2 s^3/2ṣ≤2^3/2 k/t^3/2( ∫_0^t/2^-k^2/2s/s^3/2ṣ + ∫_t/2^t ^-k^2/2s/(t - s + 1)^3/2ṣ),

    which is bounded by C k t^-3/2^-k^2/2t since
    
    ∫_0^t^-k^2/2s/s^3/2ṣ≤^-k^2/4t∫_0^∞^-k^2/4s/s^3/2ṣ 
            = C ^-k^2/4t/k  and  ∫_0^∞ṣ/(s + 1)^3/2 < ∞.

    If ℓ≥ 2, then by Lemma <ref> and <cit.>, up to a multiplicative constant, (<ref>) is bounded by
    
    ∫_0^t ℓ^-c ℓ^2/(t-s) k ^-k^2/2s/(t-s)^3/2 s^3/2ṣ≤2^3/2 k ℓ/t^3/2( ^-c ℓ^2/t∫_0^t/2^-k^2/2s/s^3/2ṣ + ^-k^2/2t∫_t/2^t ^-c ℓ^2/(t-s)/(t - s)^3/2ṣ).

    We can use (<ref>) to bound this by C (k + ℓ + 1) t^-3/2^-c k^2/t - c ℓ^2/t, which concludes.




 §.§ Intermediate temperature


Let β = √(2/3).
In this subsection, we prove Theorem <ref>.<ref>. The ideas are closed to the one used in the case β∈ (√(2/3),√(2)) in the previous subsection, only the end of the argument has to be adapted.



  §.§.§ Lower bound


Let α be an arbitrary number in (0, 1/2).
The arguments used to prove (<ref>) also hold for β = √(2/3), but the fact of restricting ourselves to the event { B_at∈ [0,1] } in the last step (<ref>) of the proof is now too crude: there the factor ^(2β - v(β))B_at prevents B_at from being large in the negatives, whereas here this factor does not play a role because v(β)=2β.
Instead, we stop the reasoning at (<ref>) to get 

    ν_β, t([a, 1])≥ c ^-at/3sup_s ∈ [0, at] (B_s + s^α∧ (at - s)^α) ≤ 1.

By (<ref>), this quantity is larger than c ^-at/3 t^-1/2, which concludes the proof of the lower bound of Theorem <ref>.<ref>.
Note that, here, we could have replaced s^α∧ (at - s)^α with s^α since the resulting probabilities are of the same order.



  §.§.§ Upper bound


Our goal is to show that ν_β, t([a, 1]) = O(t^-1/2^-at/3).
We follow the proof of (<ref>) up to (<ref>) (recall it has been done in the case β∈ [√(2/3),√(2))) and choosing α = 2 which satisfies the condition α∈ [0, 2] ∩ [1, 2/β^2).
Noting that 2β = v(β) when β = √(2/3), (<ref>) becomes

    __0[ (U_t')^-21_A ] ≤ C t^-3/2∑_k = 0^∞∑_ℓ = 0^∞^-2 β k (k + ℓ + 1) ^-c k^2/t - c ℓ^2/t 
        = O(t^-1/2).

This implies ν_β, t([a, 1]) = O(t^-1/2^-at/3) in the same way as (<ref>) implies (<ref>).
This concludes the proof of Theorem <ref>.<ref>.




 §.§ Infinite temperature


We consider here the case β = 0 and prove Theorem <ref>.<ref>. 
Using the first expression in (<ref>) with β=0, and then that #(t) is geometrically distributed with parameter ^-t (see the first example in Section III.5 of <cit.>), we can write

    ν_0, t([a, 1]) 
    = 1/(#(t))^2∑_u∈(at) (#{ v ∈(t) : v ≥ u})^2
    = ∑_i=1^M N_i^2/(∑_i=1^M N_i)^2,

where M is geometrically distributed with parameter q = ^-at and (N_i)_i≥ 1 is a sequence of independent geometric r.v. with parameter p = ^-(1-a)t, independent of M.
Using the representation x^-2 = ∫_0^∞ u ^-uxụ for x > 0 together with Fubini's theorem, we get

    [ ν_0, t([a, 1]) ]
    = ∫_0^∞[ ∑_i=1^M N_i^2 exp( - u ∑_i=1^M N_i) ] u ụ.

Noting that 

    [ ∑_i=1^M N_i^2 exp( - u ∑_i=1^M N_i) | M ]
    = M ·[ N_1^2 ^-uN_1] ·[ ^-uN_1]^M-1.

and [M s^M-1] = q(1-(1-q)s)^-2 for any s ∈ [0,1/(1-q)), we get

    [ ν_0, t([a, 1]) ]
    = ∫_0^∞q u [ N_1^2 ^-uN_1]/(1-(1-q)[^-uN_1])^2ụ
    = pq ∫_0^∞u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ,

using [^-uN_1] = p^-u/(1-(1-p)^-u) and [N^2 ^-uN_1] = p^-u(1+(1-p)^-u)/(1-(1-p)^-u)^3.
We now split the integral on the right-hand side of (<ref>) in three pieces, the main part being the one from pq to p.
Recall q = ^-at and p = ^-(1-a)t.

For the part u ∈ [0,pq], we use 1-(1-r)^-u≥ r ^-1 with r =p and r=pq to get

    ∫_0^pqu ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ≤C/p (pq)^2∫_0^pq u ụ≤C/p.

For the part u ∈ [p,∞), we use 1-(1-r)^-u≥ 1-^-u≥ (1-^-1)(u∧1) with r =p and r=pq to get

    ∫_p^∞u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ≤∫_p^∞Cu ^-u/(u∧1)^3ụ≤∫_p^1 C/u^2ụ
    + ∫_1^∞ Cu ^-uụ≤C/p.

Finally, we consider the part u ∈ [pq,p]. We have the following asymptotic behavior, as t→∞, uniformly in u ∈ [pq,p],

    u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2 
    = u (1+O(u)) (2+O(p))/(p+O(u+p^2))(u+O(pq+u^2))^2 
    = 2/pu( 1 + O ( p + u/p + pq/u) ),

which yields 

    ∫_pq^p
    u ^-u (1+(1-p)^-u)/(1-(1-p)^-u)(1-(1-pq)^-u)^2ụ
    = 2/p( log1/q + O ( p log1/q + 1 ) ).

Combining (<ref>), (<ref>), (<ref>) together with (<ref>), we get 
[ν_0, t([a, 1])] ∼ 2q log1/q, which concludes the proof of Theorem <ref>.<ref> recalling that q = ^-at.




§ ACKNOWLEDGEMENT


We warmly thank Pascal Maillard and Bastien Mallein for helpful discussions throughout the project. We also acknowledge support from the Réseau thématique Mathématiques et Physique.

abbrv




