Spectral Measures of Spiked Random Matrices
Nathan Noiry

[NO \title GIVEN]
    [NO \author GIVEN]
    December 30, 2023
======================



We study two spiked models of random matrices under general frameworks corresponding respectively to additive deformation of random symmetric matrices and multiplicative perturbation of random covariance matrices. In both cases, the limiting spectral measure in the direction of an eigenvector of the perturbation leads to old and new results on the coordinates of eigenvectors.



MSC 2010 Classification: 60B20.

Keywords: Spiked random matrices, spectral measures, BBP phase transition, overlaps.




§ INTRODUCTION


The study of deformed models of random matrices has been the subject of tremendous amount of works in the last decades. In this paper, we study two of them. The first one corresponds to an additive perturbation of a symmetric random matrix (or Wigner matrix):

    W_n = 1/√(n)X_n + A_n

where


  * X_n is a random symmetric matrix of size n × n, whose entries are, up to symmetry, i.i.d. centered and reduced;

  * A_n is a deterministic symmetric matrix of size n × n (or random, independent of X_n).

The second one is a multiplicative deformation of a random covariance matrix (or Wishart matrix):

    S_n = 1/nΣ_n^1/2 X_n X_n^T Σ_n^1/2,

where


  * X_n is a random matrix of size n × n, m/n →α∈ (0,∞), whose entries are i.i.d. centered and reduced;

  * Σ_n is a deterministic symmetric matrix (or random, independent of X_n) having non-negative eigenvalues.

The spectra of these models have been well studied. Let Spec(W_n) (resp. Spec(S_n)) be the set of eigenvalues of W_n (resp. S_n) counted with multiplicities. The empirical spectral measures of W_n and S_n are:

    μ_W_n = 1/n∑_λ∈Spec(W_n) δ_λ  and  μ_S_n = 1/n∑_λ∈Spec(S_n) δ_λ .

Under mild assumptions, they converge respectively towards probability measures μ_sc⊞μ_A and μ_MP,α⊠μ_Σ properly defined in Subsections <ref> and <ref>.

In both models, we define an outlier as an eigenvalue that does not lie in the neighborhood of the support of the limiting spectrum. Many works identified necessary and sufficient conditions on the spectrum of A_n (resp. Σ_n) for the appearance of outlier in the spectrum of W_n (resp. S_n). The seminal paper is due to Baik, Ben Arous and Péché <cit.>, who identified a phase transition for the existence of an outlier in the covariance setting with Σ_n = Diag(θ,1, …,1), θ≥ 1. The most recent results can be found in <cit.> and we refer to the survey <cit.> for an extensive bibliography. In all previous approaches, two main techniques were used: a clever identity on determinants (first remarked in <cit.>), and a precise analysis of the empirical spectral measure.

The goal of this paper is to bring into focus the possible use of the spectral measures in the study of deformed models of random matrices. Let us introduce them. Let θ be an eigenvalue of A_n (or Σ_n) which we consider as atypical in that it may be responsible for the existence of an outlier. Denote v_1^(n) the associated eigenvector. We will call θ a spike of W_n (resp. S_n) and v_1^(n) the direction of the spike. The spectral measures in the direction of the spike are respectively defined by:

    μ_(W_n,v_1^(n)) := ∑_λ∈Spec(W_n) | ⟨ϕ_λ , v_1^(n)⟩|^2 δ_λ  and  μ_(S_n,v_1^(n)) := ∑_λ∈Spec(S_n) | ⟨ϕ_λ , v_1^(n)⟩|^2 δ_λ,

where ϕ_λ is a normalized eigenvector associated to eigenvalue λ. Note that unlike empirical spectral measures, these probability measures contain information on the eigenvectors of W_n and S_n. Following the well-known observation that outliers have associated eigenvectors which are localized in the direction of the spike, their influence should be present in μ_(W_n,v_1^(n)) and μ_(S_n,v_1^(n)) at a macroscopic level. Moreover, they can be easily studied as their Stieltjes transforms are given by the generalized entries of the resolvent (⟨ v_1^(n), (W_n-z)^-1 v_1^(n)⟩ and ⟨ v_1^(n), (S_n-z)^-1 v_1^(n)⟩), for which many results already exist. In particular, as stated in Corollaries <ref> and <ref>, μ_(W_n,v_1^(n)) and μ_(S_n,v_1^(n)) converge weakly towards deterministic probability measures denoted μ_sc,A,θ and μ_α,Σ,θ. We are going to present two applications of the spectral measures.

The first one recovers a classical result concerning the value of an outlier and the norm of its associated eigenvector projection in the direction of the spike.

The second one is concerned with the behavior of the projection of non-outlier eigenvectors in the direction of the spike. Namely, in the setting of an additive perturbation, if f_sc,A and f_sc,A,θ are the respective densities of μ_sc⊞μ_A and μ_sc,A,θ and if x is in the support of μ_sc⊞μ_A, we prove the following convergence in probability

    n/| {λ∈Spec(W_n),   |λ - x| ≤ε_n  }| ∑_λ∈Spec(W_n),   |λ -x| ≤ε_n |   ⟨ϕ_λ, v_1^(n)⟩|^2   n → + ∞⟶f_sc,A,θ(x)/f_sc,A(x),

for any sequence n^-1/2 <  < ε_n <  < 1. In other words, the left-hand side, which is an average in the vicinity of x of the square-projections of eigenvectors in the direction of the spike, converges towards a deterministic profile. A similar result holds in the covariance setting. These are the content of Theorems <ref> and <ref>. Our proof is inspired by the work of Benaych-Georges, Enriquez and Michaïl <cit.> and uses local laws estimates recently obtained by Knowles and Yin in <cit.>. When θ belongs to the support of the asymptotic spectrum of A_n (resp. Σ_n), Theorems <ref> and <ref> are microscopic confirmations of the results of Allez and Bouchaud <cit.> (in the Wigner setting) and Ledoit and Péché <cit.> (in the Wishart setting), who derived the asymptotic behavior of the overlaps | ⟨ϕ_λ, v_γ⟩|^2 by taking the average over eigenvectors ϕ_λ associated to eigenvalues of W_n (resp. S_n) belonging to a macroscopic part of Supp( μ_sc⊞μ_A ) (resp. Supp(μ_MP,α⊠μ_Σ)) and the average over eigenvectors v_γ of A_n (resp. Σ_n) belonging to a macroscopic part of the asymptotic spectrum of A_n (resp. Σ_n). When θ does not belong to the support of the asymptotic spectrum of A_n (resp. S_n), such a macroscopic average is not available whereas the spectral measure approach still works.

Interestingly, in rank one perturbation cases, that is when A_n (resp. Σ_n) has only one nonzero (resp. non one) eigenvalue θ, all the computations are explicit. This is because μ_sc⊞μ_A = μ_sc and μ_MP,α⊠μ_Σ = μ_MP,α are explicit, respectively equal to the semicircle law and to the Marchenko-Pastur law with parameter α:

    μ_sc(dx)    = √(4-x^2)/2 π1_|x| ≤ 2dx 
    μ_MP,α(dx)    = √((b-x)(x-a))/2 π x1_(a,b)(x) dx  + 1_α<1 (1-α) δ_0(dx),

where a,b = (1 ±√(α))^2. In particular, we obtain the following formulas for the spectral measures in the direction of the spike:

    μ_sc,θ(dx)    = √(4-x^2)/2 π (θ^2 + 1 - θ x)1_|x| ≤ 2dx + 1_|θ|>1( 1 - 1/θ^2) δ_θ + 1/θ (dx)  
    μ_MP,α,θ(dx)    = θ√((b-x)(x-a))/2 π x ( x(1-θ) + θ( αθ - α + 1) )1_(a,b)(x) dx + c_α,θδ_0(dx) + d_α,θ1_|θ-1|> 1/√(α)δ_ x_α,θ (dx),

where c_α,θ, d_α,θ and x_α,θ are explicit constants, see Propositions <ref> and <ref>. These two measures belong to the class of the so-called free Meixner laws which appear in a Gaussian context in the work of Lenczewski <cit.>.

Hence, by (<ref>), the limiting profiles for the averaged square projections of non-outlier eigenvectors are also explicit. We give numerical simulations that agree with our predictions, see Subsections <ref> and <ref>. In the covariance setting, Bloemendal, Knowles, Yau and Yin proved that individual square projections of non-outlier eigenvectors that  are associated to eigenvalues in the vicinity of the edge (of b) converge towards a chi squared random variable with given variance (see <cit.>). Although it requires an averaging step, our result completes the picture as it is concerned with eigenvectors associated to any fixed location of the bulk of the spectrum. We believe that the convergence still holds for smaller averaging windows and provide numerical simulations supporting this conjecture at the end of Section <ref>.

Let us finally say a few words about previous use of spectral measures in the literature. Benaych-Georges, Enriquez and Michaïl, in <cit.>, obtained informations on the eigenvectors of a diagonal deterministic matrix perturbed by a random symmetric matrix. In a series of works (the most recent being <cit.>), Gamboa, Nagel and Rouault studied spectral measures of some classical ensembles of random matrix theory and their connections with sum rules. More related to our setting, in <cit.>, Bai, Miao and Pan studied the spectral measure at the first vector of the canonical basis e_1 in general covariance cases, in the absence of spike.

We emphasize that our method could apply to other deformed models such as multiplicative perturbation of Wigner matrices or information plus noise matrices, but we chose to restrict our scope so that the present paper remains short and comprehensive.




  
*Notations and organization of the paper.
The random matrices that we study are built from an i.i.d. collection of real random variables (X^(n)_ij), n ≥ 1, 1 ≤ i,j ≤ n. Let X be a generic random variable with the same law. We suppose that [X]=0, [X^2]=1 and that X has moments of all order. Notice that the complex case could also be treated, replacing each transposed matrix A^T by its transposed-conjugate A^*, and making the hypothesis that [|X|^2]=1.

For a complex z ∈𝐂, we will denote (z) and (z) the real part and imaginary part of z.

For a probability measure ν, we always denote

    s_ν(z) := ∫_𝐑dν(x)/x-z

its Stieltjes transform that maps the upper half-plane to itself.

In Subsections <ref> and <ref> we give general results concerning additive perturbation of a Wigner matrix and multiplicative perturbation of a Wishart matrix. Subsections <ref> and <ref> provide explicit computations for rank-one deformation cases. The proofs are done in Sections <ref> and <ref>.



§ ADDITIVE PERTURBATION OF A WIGNER MATRIX



 §.§ The general framework

In this section we consider for each n ≥ 1 the following Wigner matrix:

    X_n := [
    [ X^(n)_11 X^(n)_12        ⋯ X^(n)_1n; X^(n)_12 X^(n)_22        ⋯ X^(n)_2n;        ⋮        ⋮        ⋱        ⋮; X^(n)_1n X^(n)_2n        ⋯ X^(n)_nn ]].

We also consider A_n a deterministic matrix (or random matrix independent of X_n) whose eigenvalues are γ^(n)_1=θ, γ^(n)_2, …, γ^(n)_n, with associated eigenvectors v^(n)_1, …, v^(n)_n. We suppose that there exists a probability measure μ_A such that

    1/n∑_i=1^n δ_γ^(n)_i n → + ∞⟶μ_A

in the sense of weak convergence. Moreover, let us assume that there exists Γ > 0 such that, for all n ≥ 1, sup_1 ≤ i ≤ n |γ_i^(n)| ≤Γ, namely that the eigenvalues of the perturbation remain bounded.

 

We will study following additive perturbation model:

    W_n := 1/√(n)X_n + A_n.
 

Let λ^(n)_1 ≥⋯≥λ^(n)_n be the eigenvalues of W_n and ϕ^(n)_1, …, ϕ^(n)_n the associated normalized eigenvectors. Under assumption (<ref>), it is known that the empirical spectral measure μ_W_n converges towards a deterministic probability measure which is the free convolution μ_sc⊞μ_A between the semicircle law μ_sc(dx) = (2 π)^-1√(4-x^2)1_|x| ≤ 2dx and μ_A. Its Stieltjes transform is characterized by

    s_μ_sc⊞μ_A(z) = ∫_𝐑dμ_A(λ)/λ - s_μ_sc⊞μ_A(z) - z .

This has first been shown by Pastur in <cit.>. The study of (<ref>) provides information on the probability measure μ_sc⊞μ_A. In particular, Biane proved that it has a smooth density with respect to the Lebesgue measure, see <cit.>.

The parameter θ is considered as a spike which may create an outlier in the spectrum, that is an eigenvalue that does not lie in Supp( μ_sc⊞μ_A). Following the heuristic that an outlier in the spectrum creates a localized eigenvector, we study the spectral measure in the direction of the spike:


    μ_(W_n,v^(n)_1) = ∑_i=1^n |  ⟨ϕ^(n)_i, v^(n)_1 ⟩ |^2 δ_λ^(n)_i .


The Stieltjes transform of μ_(W_n,v^(n)_1) is given by ⟨ v^(n)_1, (W_n-z)^-1 v^(n)_1 ⟩ which is sometimes called a generalized entry of the resolvent and has already been studied in the literature. The most recent result is the local law recently obtained by Knowles and Yin [16]. It consists in a uniform estimation of ⟨ v,(W_n-z)^-1 w ⟩ for any vectors v and w and for any complex z in a domain of the upper half plane that is allowed to approach the real axis as n tends to infinity. Since it is one ingredient of the proof of the forthcoming Theorem <ref>, we provide a precise statement. 

Let us first introduce some notations. For all n ≥ 1, writing z=E+iη, we define:

    G_n(z) = ( W_n - z  )^-1, 
    Π_n(z) = ( A_n - z - s_μ_sc⊞μ_A (z)  )^-1, 
    ψ_n (z) = √(( s_μ_sc⊞μ_A  (z) ) /n η) + 1/n η.

For all x ∈𝐑, c>0 and τ>0, we also consider:

    𝒟_n^(τ)(x,c) := { z ∈𝐂,   x-c ≤ E ≤ x+c,   n^-1 + τ≤η≤τ^-1}.


The local law we will use consists in a uniform control between the generalized entries of G_n and Π_n in the spectral domain 𝒟_n^(τ)(x,c) whenever the density of μ_sc⊞μ_A is bounded away from 0 on the interval [x-c,x+c].

 <cit.>   
Let x ∈𝐑 and c>0. Suppose that:

    inf_ t ∈ [x-c,x+c] d(μ_sc⊞μ_A)(t)/dt > 0.

Then, for any τ >0, uniformly in all vectors v,w and uniformly in z ∈𝒟_n^(τ)(x,c), for all ε >0, there exists D >0 such that 

    ( | ⟨ v, G_n(z)w ⟩ - ⟨ v, Π_n(z) w ⟩| ≥ n^εψ_n (z) |  | v |  |   |  | w | | ) ≤1/n^D.


Theorem <ref> is a direct consequence of the work of Knowles and Yin <cit.>. More precisely, Theorem 12.2 of <cit.> provides a local law of the form (<ref>) uniformly in a spectral domain 𝐒_n ⊂𝐂_+ provided that an entrywise local law (meaning that v and w are vectors of the canonical basis in (<ref>)) has been proved in the particular case where A_n is diagonal, uniformly in the same spectral domain 𝐒_n. Such a result has indeed been established by Lee, Schnelli, Stetler and Yau in <cit.>.

Let us comment on hypothesis (<ref>). Together with the assumption that the eigenvalues γ_i^(n)'s remain bounded, it implies that there exists a constant C>0 such that, uniformly in z ∈𝒟_n^(τ)(x,c), for all 1 ≤ i ≤ n,

    1/C≤| γ_i^(n) - z - s_μ_sc⊞μ_A(z)   | ≤ C.
 
Equation (<ref>) is sometimes referred to as the stability assumption. Going back to the proofs of the local laws, it can be checked that, whenever (<ref>) is satisfied on a spectral domain 𝐒_n, then (<ref>) can be proved on 𝐒_n. In particular, since (<ref>) can hold without assumption (<ref>), the local law (<ref>) is usually proved on larger domains that 𝒟_n^(τ)(x,c). We choose to state it on 𝒟_n^(τ)(x,c) because we only need this weaker version in the proof of Theorem <ref>.



The non-local counterpart of Theorem <ref> is the pointwise convergence of ⟨ v, (W_n - z)^-1 w ⟩ in the domain (z) >0. Taking v=w=v_1^(n) yields the following Corollary. 


The spectral measure μ_(W_n,v^(n)_1) converges in probability towards a deterministic probability measure μ_sc,A,θ whose Stieltjes transform is given by

    s_μ_sc,A,θ(z) = 1/θ - s_μ_sc⊞μ_A (z) - z .




Equation (<ref>) could allow to retrieve the limit of 1/n[ Tr( (W_n - z)^-1 g(A_n)   ) ], obtained in <cit.> by Allez and Bouchaud, for any measurable function g. Indeed, this quantity can be rewritten 

    1/n[ ∑_i,j=1^n  | ⟨ϕ_i^(n), v_j^(n)⟩ |^2 /λ_i^(n) - z g( γ_j^(n))  ],

which is nothing but the average of the Stieltjes transforms of the pushforward of the spectral measures of W_n by g. In particular, it converges to a non-degenerate limit only when the support of g is contained into a macroscopic part of Supp(μ_A), due to the renormalization by n. When g is non-null on a microscopic part of Supp(μ_A), the study of the spectral measures allows to obtain the limit of [ Tr( (W_n - z)^-1 g(A_n)   ) ] whereas 1/n[ Tr( (W_n - z)^-1 g(A_n)   ) ] brings no information as it converges to zero.


Note that such a macroscopic result can be obtained using more simple arguments than the local law of Theorem <ref>. See for example <cit.> in the case where v and w are vectors of the canonical basis.

We provide two applications of the asymptotic behavior of the spectral measure of W_n in the direction of v_1^(n). 

The first one is concerned with outliers and the projection in the direction of the spike of their associated eigenvectors and relies on the following observation: unlike the empirical spectral measure which contains information on outliers only at the order 1/n, the spectral measure in the direction of the spike already contains it at a macroscopic order. For all x ∈𝐑∖Supp( μ_sc⊞μ_A), let us introduce 

    w(x) := x + s_μ_sc,A,θ(x).

If there exists x such that w(x) = θ, it is easy to deduce the existence of outliers for W_n as explained in the following Corollary. Although it is already-known in random matrix theory, our approach is new and relatively simple.

Suppose that there exists x_θ∉Supp(μ_sc⊞μ_A) such that w(x_θ) = θ. Then, x_θ is an outlier of W_n. More precisely, set δ >0 such that [x_θ - δ, x_θ + δ ] ∩Supp(μ_sc⊞μ_A) = ∅ and define k_n to be the number of eigenvalues of W_n inside [x_θ - δ, x_θ + δ ]. There exists 1 ≤ i_n ≤ n such that these eigenvalues satisfy

    x_θ + δ≥λ_i_n + 1^(n)≥λ_i_n + 2^(n)≥⋯≥λ_i_n + k_n^(n)≥ x_θ - δ.

Then, k_n ≥ 1 for n sufficiently large and:


  * Both λ_i_n + 1^(n) and λ_i_n + k_n^(n) converge in probability towards x_θ;

  * ∑_p=1^k_n | ⟨ϕ_i_n + p^(n), v_1^(n)⟩ |^2 converges in probability towards 1/w'(x_θ).



Let x_θ∉Supp(μ_sc⊞μ_A) be such that w(x_θ)=θ. The value of μ_sc,A,θ({x_θ}) is given by the residue of s_μ_sc,A,θ at x_θ:

    (x_θ-z)s_μ_sc,A,θ(z) = x_θ-z/w(x_θ)-w(z) z → x^+ ⟶1/w'(x_θ) > 0.

Since μ_(W_n,v_1^(n)) converges towards μ_sc,A,θ by Proposition <ref>, the Corollary is proved.
 
In particular, when W_n is known to be a rank-one perturbation of a matrix W_n' whose empirical spectral measure converges towards μ_sc⊞μ_A and which contains no outlier, the interlacing property implies that W_n has a unique outlier, namely that k_n=1 in Corollary <ref>. Therefore, in that case, the unique outlier converges towards x_θ and the square projection in the direction of the spike of its associated eigenvector converges towards 1/w'(x_θ).

Before stating our the second result, which represents the main novelty of this paper, and is also an illustration of the use of the spectral measure, we need the following observation:

 μ_sc,A,θ is absolutely continuous with respect to the Lebesgue measure on Supp( μ_sc⊞μ_A ).


In <cit.>, Biane proved that μ_sc⊞μ_A is absolutely continuous with respect to the Lebesgue measure. Therefore, the inverse formula 

    dμ_sc,A,θ(x)/d x =  1/πlim_ t → 0^+(s_μ_sc,A,θ(x+it) )

and Equation (<ref>) imply that μ_sc,A,θ is also absolutely continuous with respect to the Lebesgue measure at any x ∈Supp( μ_sc⊞μ_A ).
 

We will denote f_sc,A and f_sc,A,θ the respective densities of μ_sc⊞μ_A and μ_sc,A,θ on Supp( μ_sc⊞μ_A ) (these are well-defined quantities by Proposition <ref>). It turns out that the averaged square-projections of the non-outlier eigenvectors associated to eigenvalues in the vicinity of x ∈Supp( μ_sc⊞μ_A) converges towards the ratio of these two densities.

Let x ∈Supp(μ_sc⊞μ_A) be such that f_sc,A(x) > 0. Let ε_n be a sequence that satisfies n^δ / √(n) << ε_n << 1 for some 0<δ<1/2. Then, for every t>0, if ℐ^(n)_ε_n(x) = { 1 ≤ i ≤ n:   | λ_i^(n) - x | ≤ε_n   }:

    ( |  n/|ℐ^(n)_ε_n(x)|∑_i ∈ℐ^(n)_ε_n(x)| ⟨ϕ^(n)_i,v^(n)_1 ⟩|^2 - f_sc,A,θ(x)/f_sc,A(x)| > t    )  n → + ∞⟶ 0.



By taking g the indicator of an interval contained in Supp(μ_A) into the statistic introduced in Remark <ref>, Allez and Bouchaud <cit.> obtained the asymptotic behavior of the overlaps |⟨ϕ_i^(n), v_j^(n)⟩ |^2 after taking average over eigenvectors ϕ_i^(n)'s (resp. v_j^(n)'s) with associated eigenvalues λ_i^(n)'s belonging to a macroscopic proportion of Supp(μ_sc⊞Supp(μ_A)) (resp. μ_A). When θ∈Supp( μ_A), Theorem <ref> confirms their result at a microscopic scale. Indeed, denoting respectively a and b the real and imaginary parts of 1/πlim_t → 0^+    s_μ_sc⊞μ_A (x+it), one can rewrite, using the inverse formula (<ref>): 

    f_sc,A,θ(x)/f_sc,A(x) = 1/( θ -x - a)^2 + b^2.

When θ∉Supp(μ_A), the approach of <cit.> provides no information on the overlap because it only gives access to n^-1s_μ_(W_n,v_1^(n))(z) which converges to zero, whereas the spectral measure approach still works.




 §.§ The rank-one perturbation

In the special case where γ^(n)_2 = ⋯ = γ^(n)_n = 0 for all n≥ 1, W_n is a rank-one perturbation of a classical Wigner matrix. The limiting spectrum of the perturbation is μ_A=δ_0 and almost surely, μ_W_n weakly converges towards the semicircle distribution. In this setting, we provide explicit computations. Proposition <ref> has now the more explicit formulation:

In probability, μ_(W_n,v^(n)_1) converges towards:

    μ_sc,θ(dx) := √(4-x^2)/2 π (θ^2 + 1 - θ x)1_ |x| ≤ 2dx + 1_|θ|>1( 1 - 1/θ^2) δ_θ + 1/θ (dx).


Remark that W_n is a rank-one perturbation of n^-1/2 X_n. Therefore, since λ_1(n^-1/2X_n) → 2 and λ_n(n^-1/2X_n) → -2 in probability (see <cit.>), W_n has a single outlier whose location is given by the atom of μ_sc,θ and whose associated eigenvector has a square projection in the direction of the spike given by the mass of this atom.
 The following holds:


  * If θ > 1, then, in probability, λ_1(W_n)n → + ∞⟶θ + 1/θ >- 2 and |⟨ϕ^(n)_1,v^(n)_1 ⟩| n → + ∞⟶√( 1 - 1/θ^2).

  * If θ < -1, then, in probability, λ_n(W_n) n → + ∞⟶θ + 1/θ < -2 and |⟨ϕ^(n)_n,v^(n)_1 ⟩| n → + ∞⟶√( 1 - 1/θ^2).


Finally, the averaged square-projections have also an explicit form, which is just the inverse of a linear function in that case:

Let x∈ (-2,2). Let ε_n be a sequence that satisfies n^δ / √(n) << ε_n <  < 1 for some δ >0. Then, for every t>0,

    ( |  n/|ℐ^(n)_ε_n(x)|∑_i ∈ℐ^(n)_ε_n(x)| ⟨ϕ^(n)_i,v^(n)_1 ⟩|^2 - 1/θ^2-θ x + 1| > t   )  n → + ∞⟶ 0.









§ MULTIPLICATIVE PERTURBATION OF A WISHART MATRIX



 §.§ The general framework

Let m=m(n) be a sequence of integers such that m/n →α >0 as n → + ∞. For all n ≥ 0, we consider the following random rectangular matrix:

    X_n := [
    [ X^(n)_11 X^(n)_12        ⋯ X^(n)_1m; X^(n)_21 X^(n)_22        ⋯ X^(n)_2m;        ⋮        ⋮        ⋱        ⋮; X^(n)_n1 X^(n)_n2        ⋯ X^(n)_nm ]].

Let also Σ_n be a general covariance matrix of size n × n, with eigenvalues given by γ^(n)_1 =θ, γ^(n)_2, …, γ^(n)_n and associated eigenvectors v^(n)_1, …, v^(n)_n. We suppose that there exists a probability measure μ_Σ such that

    1/n∑_i=1^n δ_γ^(n)_in → + ∞⟶μ_Σ
 
in the sense of weak convergence. Moreover, let us assume that there exists Γ>0 such that for all ≥ 1, sup_1 ≤ i ≤ n |γ_i^(n)| ≤Γ.
We study the following multiplicative perturbation model:

    S_n := 1/nΣ_n^1/2 X_n X^T_n Σ^1/2_n.

The matrix S_n can be considered as the sampled covariance matrix of m i.i.d. vectors in 𝐑^n having covariance matrix Σ_n. Let λ^(n)_1 ≥⋯≥λ^(n)_n be the eigenvalues of S_n and ϕ^(n)_1, …, ϕ^(n)_n the associated normalized eigenvectors.

The empirical spectral distribution of S_n converges towards the free product μ_MP,α⊠μ_Σ whose Stieltjes transform is characterized by:

    s_μ_MP,α⊠μ_Σ(z)  =  ∫_𝐑dμ_Σ(t)/t(α-1-z s_μ_MP,α⊠μ_Σ(z) ) - z .

This is a consequence of the work of Silverstein <cit.>. A later work of Choï and Silverstein <cit.> proved that μ_MP,α⊠μ_Σ is absolutely continuous with respect to the Lebesgue measure at x, for any x>0.

The study of S_n is intimately linked to the study of S_n:=1/nX_n^T Σ_n X_n. Indeed, the spectra of S_n and S_n only differ by the number of zero eigenvalues. More precisely, denoting μ_S_n and μ_S_n the respective empirical spectral measures of S_n and S_n:

    μ_S_n = 1/n∑_λ∈Spec(S_n) δ_λ + m-n/nδ_0.

In term of the Stieltjes transforms, this relation translates into s_μ_S_n(z) = s_μ_S_n(z) - (α -1)/z. Therefore, when n tends to infinity, the empirical spectral measure of S_n converges towards a deterministic measure whose Stieltjes transform, denoted s, is given by:

    s(z) = s_μ_MP,α⊠μ_Σ(z) - α-1/z.


(Scaling conventions).
In the literature of random covariance matrices, many authors are using different scaling than this paper. Namely, they study Wishart matrices of the form B_N = 1/N Y_N Y_N^T where Y_N is of size p × N where p/N → y >0 as N tends to infinity. In this context, the scaling factor is the dimension of the rows vectors of Y_N whereas in our case, we scale by the dimension of the columns vectors. In order to facilitate the comparison between the two models, let us describe their differences. First, note the following correspondence between the parameters: n=p, m=N and α = 1/y. Writing B_N = (p/N) 1/pY_N Y_N^T, it is easy to see that the empirical spectral measures of the two models satisfy the following equality in law:

    μ_B_N = Λ_p/N( μ_S_n) = Λ_n/m( μ_S_n),

where Λ_ξ (·) denotes the pushforward by the dilatation x ↦ξ x. Hence, μ_B_N converges towards the pushforward of μ_MP,α⊠μ_Σ by Λ_1/α. In the particular case where Σ_n is the identity, the limiting measure is given by the pushforward of μ_MP,α which, after computations, is given by

    √(((1+√(y))^2-x)(x-(1-√(y))^2))/2 π y x1_x ∈ ( (1-√(y))^2, (1+√(y))^2 ) dx + (1 - 1/y) δ_0(dx).



We think of θ as a spike, that is an atypical eigenvalue compared to the sequence γ^(n)_i, 2 ≤ i ≤ n. We are interested in its influence on the apparition of outliers for S_n. To that purpose, we study the spectral measure in the direction of the spike:

    μ_(S_n,v_1^(n)) := ∑_i=1^n | ⟨ϕ^(n)_i, v^(n)_1   ⟩ |^2 δ_λ^(n)_i.


The Stieltjes transform of μ_(S_n,v^(n)_1) is given by ⟨ v^(n)_1, (S_n-z)^-1 v^(n)_1 ⟩ and has already been studied in the literature. As in the Wigner case, the most recent result is the local law obtained by Knowles and Yin <cit.>. It consists in a uniform estimation of ⟨ v , (S_n-z)^-1 w ⟩ for any vectors v and w and for any complex z in a domain of the upper half plane that is allowed to approach the real axis as n tends to infinity. Since it is an ingredient of the proof of Theorem <ref>, we provide a precise statement. 

Let us first introduce some notations. For all n ≥ 1, writing z=E+iη, we define:

    G_n(z) = ( S_n - z  )^-1, 
    Π_n(z):= - (z(1 + s(z)Σ_n ))^-1, 
    ψ_n (z) = √(( s_μ_MP,α⊠μ_Σ (z) ) /n η) + 1/n η,

where we recall that s is defined in Equation (<ref>). For all x ∈𝐑, c>0 and τ>0, we also consider:

    𝒟_n^(τ)(x,c) := { z ∈𝐂,   x-c ≤ E ≤ x+c,   n^-1 + τ≤η≤τ^-1}.


The local law we will use consists in a uniform control between the generalized entries of G_n and Π_n in the spectral domain 𝒟_n^(τ)(x,c) whenever the density of μ_MP,α⊠μ_Σ is bounded away from 0 on the interval [x-c,x+c].

<cit.>
Let x >0 and c>0. Suppose that:

    inf_ t ∈ [x-c,x+c] d(μ_MP,α⊠μ_Σ)(t)/dt > 0.

Then, for any τ >0, uniformly in all vectors v,w and uniformly for in any z ∈𝒟_n^(τ), for all ε >0, there exists D >0 such that 

    ( | ⟨ v, Σ_n^-1/2(  G_n(z) - Π_n(z) ) Σ_n^-1/2  w ⟩| ≥ n^εψ_n (z) |  | v |  |   |  | w | | ) ≤1/n^D.



Theorem <ref> is a direct consequence of the work of Knowles and Yin <cit.>. More precisely, they first prove <cit.> that (<ref>) holds in the particular case where v and w are vectors of the canonical basis, and when Σ_n is diagonal. Then, using a comparison argument, they extend this result to the general setting (see <cit.>).

It is possible to comment hypothesis (<ref>) as in the Wigner case (see the paragraphs below Theorem <ref>). Together with the assumption that the eigenvalues γ_i^(n)'s remain bounded, it implies that there exists a constant C>0 such that, uniformly in z ∈𝒟_n^(τ)(x,c), for all 1 ≤ i ≤ n,

    1/C≤| 1 +  γ_i^(n)s(z)   | ≤ C.
 
Equation (<ref>) is sometimes referred to as the stability assumption. Going back to the proofs of the local laws, it can be checked that, whenever (<ref>) is satisfied on a spectral domain 𝐒_n, then (<ref>) can be proved on 𝐒_n. In particular, since (<ref>) can hold without assumption (<ref>), the local law (<ref>) is usually proved on larger domains that 𝒟_n^(τ)(x,c). We choose to state it on 𝒟_n^(τ)(x,c) because we only need this weaker version in the proof of Theorem <ref>.



The non-local counterpart of Theorem <ref> is the pointwise convergence of ⟨ v, (S_n - z)^-1 w ⟩ in the domain (z) >0. Taking v=w=v_1^(n) yields the following Corollary.


In probability, μ_(S_n,v^(n)_1) weakly converges towards a probability measure μ_α,Σ,θ whose Stieltjes transform is given by:

    s_μ_α,Σ,θ(z) = 1/θ(α-1) - z θ s_μ_MP,α⊠μ_Σ(z) - z.




Equation (<ref>) could allow to retrieve the limit of 1/nTr( (S_n - z)^-1 g(Σ_n)   ), obtained in <cit.> by Ledoit and Péché, for any measurable function g. Indeed, this quantity can be rewritten 

    1/n∑_i,j=1^n  | ⟨ϕ_i^(n), v_j^(n)⟩ |^2 /λ_i^(n) - z g( γ_j^(n)),

which is nothing but the average of the Stieltjes transform of the pushforward of the spectral measures of S_n by g. In particular, it converges to a non-degenerate limit only when the support of g is contained into a macroscopic part of Supp(μ_Σ), due to the renormalization by n. When g is non-null on a microscopic part of Supp(μ_Σ), the study of the spectral measures yields the limit of Tr( (S_n - z)^-1 g(Σ_n)   ) whereas 1/nTr( (S_n - z)^-1 g(Σ_n)   ) brings no information as it converges to zero.


Of course, such a macroscopic result can be obtained using more simple arguments than the local law of Theorem <ref>. We refer for example to <cit.>.

In what follows, we provide two applications of the asymptotic behavior of the spectral measure of S_n in the direction of v_1^(n). 

The first one recovers a classical result on outliers of S_n and the projection in the direction of the spike of their associated eigenvectors. The proof we propose is simple and based on the following observation: unlike the empirical spectral measure which contains information on outliers only at the order 1/n, the spectral measure in the direction of the spike already contains it at a macroscopic order. Before stating our result, let us introduce, for all x>0 such that x^-1∈𝐑∖Supp(μ_MP,α⊠μ_Σ):

    F(x) = α x - x - s_μ_MP,α⊠μ_Σ(1/x).
 
If there exists x>0 such that 1/F(1/x) = θ, we easily obtain the existence of an outlier as explained in the following Corollary.

Suppose that there exists x_α,θ∉Supp(μ_MP,α⊠μ_Σ) such that 1/F(1/x_α,θ)= θ. Then, x_α,θ is an outlier of S_n. More precisely, set δ >0 such that [x_α,θ - δ, x_α,θ + δ ] ∩Supp(μ_sc⊞μ_A) = ∅ and define k_n to be the number of eigenvalues of S_n inside [x_α,θ - δ, x_α,θ + δ ]. There exists 1 ≤ i_n ≤ n such that these eigenvalues satisfy

    x_α,θ + δ≥λ_i_n + 1^(n)≥λ_i_n + 2^(n)≥⋯≥λ_i_n + k_n^(n)≥ x_α,θ - δ.

Then, k_n ≥ 1 for n sufficiently large and:


  * Both λ_i_n + 1^(n) and λ_i_n + k_n^(n) converge in probability towards x_α,θ;

  * ∑_p=1^k_n | ⟨ϕ_i_n + p^(n), v_1^(n)⟩ |^2 converges in probability towards x_α,θF(1/x_α,θ)/ F'(1/x_α,θ).



Let x_α,θ>0 be such that x_α,θ∉Supp(μ_MP,α⊠μ_Σ) and 1/F(1/x_α,θ)=θ. The value of μ_α,Σ,θ( {x_α,θ}) is given by the residue of s_μ_α,Σ,θ at x_α,θ:

    (x_α,θ-z)s_μ_α,Σ,θ(z) = (x_α,θ-z) F(1/x_α,θ) /z (F(1/z) - F(1/x_α,θ))  z → x_α,θ^+ ⟶x_α,θF(1/x_α,θ)/ F'(1/x_α,θ) > 0.

Since μ_(S_n,v_1^(n)) converges towards μ_α, Σ, θ, the Corollary is easily deduced.
 
A particular case is when S_n is a rank-one perturbation of a matrix S_n' which has no outlier and whose empirical spectral measure converges towards μ_MP,α⊠μ_Σ. In that setting, the interlacing property implies that k_n=1 for all n sufficiently large in Corollary <ref>, meaning that S_n has only one outlier which converges towards x_α,θ and whose associated eigenvector has a square projection in the direction of the spike which converges towards x_α,θF(1/x_α,θ)/ F'(1/x_α,θ).

Before stating our second result, which is concerned with the projection of non-outlier eigenvectors onto the direction of the spike, we need the following Proposition.


μ_α,Σ,θ is absolutely continuous with respect to the Lebesgue measure on Supp(μ_MP,α⊠μ_Σ ) ∖{0}.


Let x ∈Supp(μ_MP,α⊠μ_Σ ) ∖{0}. The work of Choï and Silverstein <cit.> ensures that μ_MP,α⊠μ_Σ is absolutely continuous with respect to the Lebesgue measure at x. Then, the inverse formula for the Stieltjes transform:

    dμ_sc,A,θ(x)/d x =  1/πlim_ t → 0^+(s_μ_sc,A,θ(x+it) ),
 
combined with Equation (<ref>), implies the Proposition.


Let f_α,Σ and f_α,Σ,θ be the respective densities of μ_MP,α⊠μ_Σ and μ_α, Σ, θ on Supp(μ_MP,α⊠μ_Σ). It turns out that the averaged square projections of the non-outlier eigenvectors associated to eigenvalues in the vicinity of x ∈Supp(μ_MP,α⊠μ_Σ ) converges towards the ratio of these two densities.


Let x∈Supp(μ_MP,α⊠μ_Σ )  ∖{0 } be such that f_α, Σ(x) > 0. Let ε_n be a sequence that satisfies n^δ / √(n) << ε_n << 1 for some δ>0. Then, for every t>0, if ℐ^(n)_ε_n(x) = { 1 ≤ i ≤ n:   | λ_i^(n) - x | ≤ε_n   }:

    ( |  n/|ℐ^(n)_ε_n(x)|∑_i ∈ℐ^(n)_ε_n(x)| ⟨ϕ^(n)_i,v^(n)_1 ⟩|^2 - f_α,Σ,θ(x)/f_α,Σ(x)| > t    )  n → + ∞⟶ 0.



As in the Wigner case, Theorem <ref> can be seen as a generalization of a result of <cit.>, where the authors obtained the asymptotic behavior of the overlaps |⟨ϕ_i^(n), v_j^(n)⟩ |^2 after taking average over eigenvectors ϕ_i^(n)'s (resp. v_j^(n)'s) with associated eigenvalues λ_i^(n)'s belonging to a macroscopic proportion of Supp(μ_MP,α⊠μ_Σ) (resp. μ_Σ), by taking taking functions of the type g= 1_[γ,+∞) in the statistics introduced in Remark <ref>. Indeed, when θ∈Supp( μ_Σ ), Theorem <ref> is a microscopic version of the result of <cit.>. To obtain their formula it suffices to remark that, if a and b are the real and imaginary parts of 1 - α - zs_μ_MP,α⊠μ_Σ(z), one can rewrite, using Equation (<ref>):

    f_α,Σ,θ(x)/f_α,Σ(x) = x θ/(a θ - x)^2 + θ^2 b^2,

When θ∉Supp(μ_Σ), the techniques of <cit.> provide no information on the overlaps as it only gives access to n^-1s_μ_(S_n,v_1^(n))(z), which converges to zero, whereas the spectral measure approach still works.



 §.§ Rank-one perturbation of the Marchenko-Pastur law

In the peculiar case where γ^(n)_2 = ⋯ = γ^(n)_n = 1 for all n≥ 1, S_n is a rank-one perturbation of a classical Wishart matrix. The limiting spectrum of the perturbation is μ_Σ=δ_1 and almost surely, μ_S_n weakly converges towards the Marchenko-Pastur law μ_MP,α. All previous results have now a more explicit formulation. First, the limit of the spectral measure in the direction of the spike can be identified.

In probability, μ_(S_n,v^(n)_1) converges towards:

    μ_MP,α,θ(dx) = θ√((b-x)(x-a))/2 π x(x(1-θ) + θ (αθ - α + 1))1_(a,b)(x)dx + c_α1_α<1δ_0(dx) + d_α,θ1_ |θ-1|>1/√(α)δ_x_α,θ(dx),

where c_α = 1-α/α(θ-1)+1, d_α,θ =   1- 1/α(θ-1)^2/  1 + 1/α(θ-1) and x_α,θ = θ(αθ - α + 1)/θ - 1.

A consequence of <cit.> is that n^-1X_nX_n^T has no outlier. Since S_n is a rank one perturbation of this matrix, the discussion following Corollary <ref> implies that S_n has a single outlier.
 The following holds:


  * If θ > 1 + 1/√(α), then, in probability, λ_1(S_n)n → + ∞⟶ x_α,θ > b and |⟨ϕ^(n)_1,v^(n)_1 ⟩| n → + ∞⟶√(d_α,θ).

  * If θ < 1 - 1/√(α), then, in probability, λ_n(S_n) n → + ∞⟶ x_α,θ < a and |⟨ϕ^(n)_n,v^(n)_1 ⟩| n → + ∞⟶√(d_α,θ).


The ratio of the density of μ_MP,α,θ and μ_MP,α is explicit and we obtain the following Theorem.

Let x∈ (a,b). Let ε_n be a sequence that satisfies n^δ / √(n) << ε_n <  < 1 for some δ >0. Then, for every t>0,

    ( |  n/|ℐ^(n)_ε_n(x)|∑_i ∈ℐ^(n)_ε_n(x)| ⟨ϕ^(n)_i,v^(n)_1 ⟩|^2 - θ/x(1-θ) + θ(αθ - α +1)| > t    )  n → + ∞⟶ 0.



When x tends to b, the limiting profile becomes θ / (1+ √(α)(1-θ)^2), in accordance with <cit.> where the authors obtain a convergence of individual square-projections onto the direction of the spike towards chi-squared random variables with expectation θ / (1+ √(α)(1-θ)^2). A natural question would be to study an analog convergence in law in the bulk of the spectrum (for any x ∈ (a,b)). We do not pursue this issue here.





§ IDENTIFICATION OF THE LIMITING LAWS IN RANK-ONE PERTURBATION CASES

In this section we prove Proposition <ref> and <ref>. We use the following branch of the complex square-root:

    √(z) = sign(  (z) ) |z| + z/√(2(|z| +  (z))).



The Stieltjes transform of the semicircle law is given by:

    s_μ_sc(z) = -z + √(z^2-4)/2 .

Therefore, using Equation (<ref>) and the fact that μ_A is in this case δ_0:

    s_μ_sc,θ(z)    = 2/2 θ - z - √(z^2 -4)
       = √(z^2 -4) +2 θ -  z / 2 (θ^2 - θ z + 1) .

The absolutely continuous part of μ_sc,θ is given by

    dμ_sc,θ (x) /dx = 1/πlim_z → x^+ (s_ μ_sc,θ(z)) = √(4-x^2)/2 π (θ^2 + 1 - θ x)1_ |x| ≤ 2dx .

The atom at θ + 1/θ is given by the corresponding residue of s_μ_sc,θ:

    - lim_z → (θ + 1/θ)^+  (z-θ-1/θ) s_μ_sc,θ(z).

By our choice of square-root, lim_z →θ + 1/ θ√(z^2 - 4) = |θ - 1 |/θ and one easily deduces:

    μ_sc,θ({θ + 1/ θ}) =  {[           0  if |θ| ≤ 1;   1 - 1/θ^2 if |θ| > 1. ].






Recall the expression of the Stieltjes transform of the Marchenko-Pastur law μ_MP,α = μ_α,1:

    s_μ_MP,α(z) = α-z-1 + √((z-b)(z-a))/2z.

Substituting in Equation (<ref>), we get

    s_μ_MP,α,θ(z)    = -2/2z - 2θ(α-1) + θ(α-1) - θ z + θ√((z-b)(z-a))
       = -2( 2z - θ (α - 1) - θ z - θ√((z-b)(z-a)))/( (2-θ)z - θ(α-1)  )^2 - θ^2(z-b)(z-a)
       = θ√((z-b)(z-a)) + z(θ-2) +θ(α-1)/2z( z(1-θ) + θ( αθ - α +1)  ).

This expression will allow us to obtain an explicit formula for μ_MP,α,θ, through classical inversion results.

The absolutely continuous part of μ_MP,α,θ is given by:

    dμ_MP,α,θ/dx(x) =  1/πlim_ z → x^+ (s_μ_MP,α,θ(z)) = θ√((b-x)(x-a))/2 π x ( x(1-θ) + (αθ - α + 1)  )1_(a,b)(x) .

The atom of μ_MP,α,θ at zero is given by:

    -lim_ε→ 0^+ iε s_μ_MP,α,θ(iε) = -lim_ε→ 0^+√((iε-b)(iε-a)) + (α-1)/2(αθ - α +1).


By our choice of square-root that preserves the upper-half plane, √((iε-b)(iε-a))→ -|ab| = -|α-1| as ε→ 0^+. Therefore:

    μ_MP,α,θ({0}) = |α - 1| - (α - 1)   /2( αθ - α + 1 ) = {[              0        if α≥ 1; 1-α/α(θ-1) + 1        if α<1. ].

Finally, let us compute the atom at x_θ=θ( αθ - α + 1)/θ - 1. It is given by:

    -lim_ z → x_θ^+  (z-x_θ) s_μ_MP,α,θ(z) = √( (x_θ-b)(x_θ-a)) + x_θ(θ-2) + θ(α-1)/2(θ-1)x_θ .

We use the following relations which are easily verified.


  * x_θ - b = α( (θ - 1) - 1/√(α))^2/θ - 1,

  * x_θ - a = α( (θ - 1) + 1/√(α))^2/θ - 1.

As for the computation of the atom at zero:

    lim_ε→ 0^+√((x_θ+iε-b)(x_θ+iε-a))   = sign(θ-1) α/|θ-1|| (θ-1)^2 - 1/α|  
       = α/θ-1| (θ-1)^2 - 1/α|.

Besides:

    x_θ(θ-2) + θ(α-1)    = θ(αθ - α + 1) (θ - 2) + θ (α -1)(θ-1)/θ - 1
       = αθ/θ-1( (θ-1)^2 - 1/α).

One obtains:

    μ_MP,α,θ( { x_α,θ}) = αθ| (θ-1)^2 - 1/α| + ( (θ-1)^2 - 1/α) /2 θ (θ - 1) (αθ - α + 1) =  {[                                    0                  if |θ - 1| ≤ 1/√(α); α( (θ-1)^2 - 1/α)/(θ- 1)(αθ - α + 1)                 if |θ - 1| > 1/√(α). ].





§ CONVERGENCE OF THE AVERAGED SQUARE PROJECTIONS


We only focus on the proof of Theorem <ref> concerning the convergence of averaged square-projections into the direction of the spike in the Wishart setting. The proof of Theorem <ref>, which concerns the Wigner setting, would follow the same reasoning, the only difference being the use of the local law of Theorem <ref> instead of the local law of Theorem <ref>. For the rest of this section, we fix 0<δ<1/2 and (ε_n)_n ≥ 1 a sequence of real numbers such that n^δ / √(n) <  < ε_n <  < 1. We also fix x_0 >0 such that μ_α, Σ,θ has a positive density at x_0. 

Let us explain the heuristic behind Theorem <ref>. We will denote I_ε_n(x_0) := [x_0 -ε_n, x_0 + ε_n]. Recall that f_α, Σ and f_α, Σ, θ are the respective densities of μ_MP,α⊠μ_Σ and μ_α,Σ,θ on Supp( μ_MP,α⊠μ_Σ). Then,

    ∫_I_ε_n(x_0)dμ_(S_n,v^(n)_1)(x) ≈∫_I_ε_n(x_0)dμ_MP,α,θ(x) + o_1(1) ≈ 2 ε_n f_α, Σ, θ(x_0) + o_1(1).

On the other hand, if μ_S_n = n^-1∑_1 ≤ i ≤ nδ_λ^(n)_i denotes the empirical spectral measure of S_n:

    ∫_I_ε_n(x_0)dμ_(S_n,v^(n)_1)(x) = ∑_i ∈ℐ^(n)_ε_n(x_0)| ⟨ϕ_i,e_1 ⟩|^2    = ( n/|ℐ^(n)_ε_n(x_0)|∑_i ∈ℐ^(n)_ε_n(x_0)| ⟨ϕ^(n)_i,v^(n)_1 ⟩|^2 ) ×∫_I_ε_n(x_0)dμ_S_n(x) 
       ≈( n/|ℐ^(n)_ε_n(x_0)|∑_i ∈ℐ^(n)_ε_n(x_0)| ⟨ϕ_i,e_1 ⟩|^2 ) ×( 2 ε_n f_α, Σ(x_0) + o_2(1) ),

where we recall that ℐ^(n)_ε_n(x_0) = { 1 ≤ i ≤ n:   | λ^(n)_i -x_0 | ≤ε_n }. Theorem <ref> would be proved if the errors o_1(1) and o_2(1) were explicit and of a smaller order than ε_n. The understanding of these errors is precisely the purpose of the so-called local laws that have been recently developed in random matrix theory. In the Wishart setting, it is given by the local law of Knowles and Yin stated in Theorem <ref>. 

The rest of this section makes the above heuristic rigorous. It combines a local law on the Stieltjes transform of μ_(S_n,v_1^(n)) together with an approximation argument which allows to estimate a term of the form μ_(S_n,v_1^(n))(I_ε_n(x_0)). The idea of the latter is to bound the indicator of I_ε_n(x_0) by two smooth analytic functions which we now introduce. Let (ω_n)_n≥ 1 be a sequence of real numbers such that 

    n^δ / √(n) <  < ω_n <  < ε_n.

Let Ψ be a smooth decreasing function such that Ψ(x) ≡ 1 on (-∞,0] and Ψ(x) ≡ 0 on [1,+∞). For all n ≥ 1, we define:

    ϕ_n^-(x) = Ψ( 1 +  x - x_0 - ε_n + ω_n/ω_n) Ψ(  1 - x-x_0 + ε_n - ω_n/ω_n),
    ϕ_n^+(x) = Ψ(  x - x_0 - ε_n - ω_n/ω_n) Ψ( - x-x_0 + ε_n + ω_n/ω_n).

With these definitions, it is easy to check the following properties.


  * the support of ϕ^-_n (resp. ϕ^+_n) is included in [x_0-ε_n+ ω_n,x_0+ε- ω_n] (resp. [x_0 -ε_n- 2ω_n,x_0 +ε_n+ 2ω_n]);

  * ϕ^-_n (resp. ϕ^+_n) is constant equal to 1 on [x_0 -ε_n + 2 ω_n, x_0 + ε_n - 2 ω_n] (resp. [x_0 -ε_n-ω_n,x_0 +ε_n+ω_n]);

  * the supports of (ϕ_n^-)' and (ϕ_n^-)” (resp. (ϕ_n^+)' and (ϕ_n^+)”) are included into [x_0-ε_n+ω_n,x_0-ε_n+2 ω_n] ∪ [x_0+ε_n-2ω_n,x_0+ε_n - ω_n] (resp. [x_0-ε_n- 2ω_n,x_0-ε_n - ω_n] ∪ [x_0+ε_n + ω_n,x_0+ε_n + 2ω_n]);

  * ||(ϕ^-_n)'||_∞ = || (ϕ^+_n)' ||_∞ = O(1/ω_n) and ||(ϕ^-_n)”||_∞ = || (ϕ^+_n)” ||_∞ = O(1/ω_n^2).

By construction,

    ∫_𝐑ϕ^-_n(λ) dμ_(S_n,v^(n)_1)(λ) ≤∫_I_ε_n(x_0)dμ_(S_n,v^(n)_1)(λ) ≤∫_𝐑ϕ^+_n(λ) dμ_(S_n,v^(n)_1)(λ), 
    ∫_𝐑ϕ^-_n(λ) dμ_S_n(λ) ≤∫_I_ε_n(x_0)dμ_(S_n,v^(n)_1)(λ) ≤∫_𝐑ϕ^+_n(λ) dμ_S_n(λ).

The main result of this section is an estimate on both sides of the above inequalities.


Let ϵ∈{-,+}. There exists D>0 such that, with probability at least 1-n^-D,

    | ∫_𝐑ϕ_n^ϵ(λ) d( μ_(S_n,v_1^(n)) - μ_α,Σ,θ) (λ)| = O( ω_n )

and

    | ∫_𝐑ϕ_n^ϵ(λ) d( μ_S_n - μ_MP,α) (λ)| = O( ω_n ).

 
Before giving the proof of Lemma <ref>, let us explain how it leads to Theorem <ref>.

Combining estimates (<ref>) and (<ref>) with Equation <ref>, for any ε >0, there exists D>0 such that with probability at least 1 - n^-D:

    ∫_ I_ε_n(x_0) dμ_(S_n,v_1^(n))(x) = 2 ε_n f_α, Σ, θ (x_0) + O(  ω_n  )

and

    ∫_ I_ε_n(x_0) dμ_S_n(x) = 2 ε_n f_α, Σ (x_0) + O(  ω_n ).

Therefore, since ε_n >  > n^-1/2, for all ε>0, there exists D>0 such that with probability at least 1-n^-D,

    n/| ℐ_ε_n(x_0) |∑_ i ∈ℐ_ε_n(x_0)| ⟨ϕ_i , e_1 ⟩|^2  
       =  ∫_ I_ε_n(x_0) dμ_(S_n,v_1^(n))(λ) /∫_ I_ε_n(x_0) dμ_S_n(λ)  
       = f_α, Σ, θ (x_0)/f_α, Σ (x_0) + O(  ω_n/ε_n).

This ends the proof of Theorem <ref> since ε_n >  > ω_n.


We now turn to the proof of Lemma <ref>. We will use a local law on the Stieltjes transform of μ_(S_n,v_1^(n)) which can be obtained by taking v=w=v_1^(n) in Equation (<ref>). Recall the following definitions:

    𝒟_n^(τ)(x,c) = { z ∈𝐂,   x-c ≤ E ≤ x+c,   n^-1 + τ≤η≤τ^-1}

and

    ψ_n (z)= √(( s_μ_MP,α⊠μ_Σ (z) ) /n η) + 1/n η .

Since x_0 is such that f_α, Σ(x_0)>0, there exists c>0 small enough such that Equation (<ref>) is satisfied. We fix such a c for the rest of this section. Then, Theorem <ref> translates into the following result.

<cit.> 
For any τ >0, uniformly in z ∈𝒟_n^(τ)(x_0,c), for any ε > 0, there exists D >0 such that

    ( | s_(S_n,v_1^(n))(z) - s_μ_α,Σ,θ(z)  | ≥ n^εψ(z)  ) ≤1/n^D.



In the proof of Lemma <ref>, we will use the following classical estimate on the error function.


For all x∈𝐑 such that f_α,Σ(x)>0,

    |ψ_n(x+iy)| = O( √(( s_μ_MP,α⊠μ_Σ (x+iy) ) /n y)).

In particular, for all 1 ≤ |y| ≤ 2, 

    |ψ_n(x+iy)| = O( n^-1/2).



Let x ∈𝐑 be such that f_α,Σ(x)>0. Then, since

    f_α,Σ(x) = 1/πlim_t → 0^+( s_μ_MP,α⊠μ_Σ (x+it) ),

the leading term in the expression of ψ_n(x+iy) is √(( s_μ_MP,α⊠μ_Σ (x+iy) ) /n y), which proves (<ref>). This latter term is of order n^-1/2 whenever y is bounded away from 0, which implies (<ref>).



We are now ready to prove Lemma <ref>. The strategy is based on the Helffer-Sjöstrand formula, which allows to translate an estimate of the form (<ref>) into an estimate on sufficiently regular functions integrated against μ_(S_n,v^(n)_1). Although the argument is standard and can be found for the empirical spectral measure of a Wigner matrix in the survey of Benaych-Georges and Knowles <cit.>, we choose to provide the details as it has not been done for the spectral measures. A similar argument is also present in the work of Benaych-Georges, Enriquez and Michaïl <cit.>. 

 
We first prove estimate (<ref>) that corresponds to the spectral measure and then explain how to adapt the proof for the second inequality (<ref>) which corresponds to the empirical spectral measure. We only focus on the case ϵ = + because the case ϵ = - follows from the same argument. In order to lighten notations, we denote ϕ_n =ϕ_n^+.

For all x ∈𝐑, by the the Helffer-Sjöstrand formula (see <cit.> ):

    ϕ_n(x) =  ∫_𝐂∂( ϕ̃_n(z) χ(z)  ) /x-zdz,

where:


  * χ is a smooth symmetric cutoff function that equals 1 on [-1,1] and 0 outside [-2,2];

  * ϕ̃_n is the quasi-analytic extension of degree 1 of ϕ_n, defined by ϕ̃_n(x+iy) = ϕ_n(x) + iy ϕ'_n(x);

  * ∂ = 1/2( ∂_n + i ∂_y ).

Let us define

    μ̂_n := μ_(S_n,v_1^(n)) - μ_α,Σ,θ

and its Stieltjes transform ŝ_n := s_μ_(S_n,v_1^(n)) - s_μ_α,Σ,θ. Equation (<ref>) leads to:

    ∫_𝐑ϕ_n(λ) dμ̂_n(λ) =    +bi/2 π∫_x ∈𝐑∫_y ∈𝐑ϕ”_n(x) y χ (y) ŝ_n(x+iy) dx dy 
       + i/2 π∫_x ∈𝐑∫_y ∈𝐑( ϕ_n(x) + iyϕ'_n(x)  ) χ'(y) ŝ_n(x+iy) dx dy.

Remark that the right-hand-side is real so that

    ∫_𝐑ϕ_n(λ) dμ̂_n(λ) ≤   +b-1/2 π∫_x ∈𝐑∫_|y| ≤ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy)) dx dy 
       + -1/2 π∫_x ∈𝐑∫_|y| ≥ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy) ) dx dy 
       + | 1/2 π∫_x ∈𝐑∫_y ∈𝐑( ϕ_n(x) + iyϕ'_n(x)  ) χ'(y) ŝ_n (x+iy) dx dy |  .

We now estimate all of the three terms of the right-hand side. In what follows, we fix τ > 0 and ε >0 and we argue on the event

    E_ε := {∀ z ∈𝒟_n^(τ)(x_0,c),  | s_(S_n,v_1^(n))(z) - s_μ_α,Σ,θ(z)  | ≤ n^εψ_n (z)  } .

Note that by the local law (<ref>), there exists D>0 such that (E) ≥ 1 - n^-D. 


  
*Estimation of the term (<ref>).
Recall that, from the definition of ϕ_n, given in Equation (<ref>), the support of ϕ_n” is contained in 

    I_n := [x_0-ε_n- 2ω_n,x_0-ε_n - ω_n] ∪ [x_0+ε_n + ω_n,x_0+ε_n + 2ω_n].

Moreover, since ω_n <<1, χ(y)=1 when |y| ≤ω_n. Therefore:

    | 1/1.    -1/2 π. ∫_x ∈𝐑∫_|y| ≤ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy)) dx dy  | 
       ≤1/2 π∫_x ∈ I_n∫_0 ≤ y ≤ω_n|ϕ”_n(x) y (ŝ_n (x+iy)) | dx dy + 1/2 π∫_x ∈ I_n∫_-ω_n ≤ y < 0|ϕ”_n(x) y (ŝ_n (x+iy)) | dx dy

We only treat in details the first term of (<ref>) as the second one can be analyzed similarly.

Let x ∈ I_n. The function y ↦ y (  s_μ_(S_n,v_1^(n))(x+iy) ) is non-decreasing, which implies that: 

    ∀ 0 ≤ y ≤ω_n,    y (ŝ_n (x+iy))   ≤  y ( s_μ_(S_n,v_1^(n)) (x+iy))  ≤ω_n ( s_μ_(S_n,v_1^(n)) (x+iω_n)).

By assumptions, the point x_0 ∈𝐑 is such that f_α,Σ,θ(x_0)>0. Since lim_t → 0^+( s_μ_α,Σ,θ(x_0 + it  ) =  π f_α,Σ,θ(x_0), this implies that there exists a constant C̃>0 such that, for large enough n:

    ∀ x ∈ I_n,  ∀ 0 ≤ y ≤ω_n,   ( s_μ_α,Σ,θ(x + iy)  ) ≤C̃.

Therefore, since we are working on the event E_ε introduced in (<ref>), uniformly in x ∈ I_n:

    (  s_μ_(S_n,v_1^(n)) (x+i ω_n)) ≤  n^εψ_n(x+i ω_n) + C̃.

By Equation (<ref>) of Lemma <ref>, n^εψ_n(x+i ω_n) converges to zero as n tends to infinity. Therefore, by combining Inequalities (<ref>) and (<ref>), we deduce the existence of some constant C>0 such that:

    ∫_x ∈ I_n∫_0 ≤ y ≤ω_n|ϕ”_n(x) y (ŝ_n (x+iy)) | dx dy ≤  C ω_n ∫_x ∈ I_n∫_0 ≤ y ≤ω_n |ϕ”_n(x)|  dx dy.

Finally, using that ||ϕ_n”||_∞ = O(1/ω_n^2) and ∫_I_ndx = 2 ω_n in (<ref>) yields:

    ∫_x ∈ I_n∫_0 ≤ y ≤ω_n|ϕ”_n(x) y (ŝ_n (x+iy)) | dx dy ≤ O ( ω_n ).

As already mentioned, the same argument implies that the bound (<ref>) also holds if the domain of integration was I_n ×[-ω_n,0). Hence, we proved that, on the event E_ε,

    |  -1/2 π∫_x ∈𝐑∫_|y| ≤ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy)) dx dy  | = O(ω_n).





  
*Estimation of the term (<ref>).

We first decompose the term according to the sign of y.

    -1/2 π∫_x ∈𝐑∫_|y| ≥ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy) ) dx dy 
       = -1/2 π∫_x ∈𝐑∫_y≤ -ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy) ) dx dy + -1/2 π∫_x ∈𝐑∫_y ≥ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy) ) dx dy .

The two terms on the right-hand side of (<ref>) can be analyzed in the same way, so that we only focus on the case where y ≥ω_n.

Differentiating with respect to x and y and using that ∂_x  ( ŝ_n (x+iy) ) = - ∂_y  ( ŝ_n (x+iy)), we obtain:

    -1/2 π   ∫_x ∈𝐑∫_y ≥ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy) ) dx dy 
       = -1/2 π∫_y ≥ω_n y χ(y) dy × (-1) ∫_x ∈𝐑ϕ_n'(x) ∂_x ( ŝ_n(x+iy)  ) dx 
       = -1/2 π∫_ x ∈𝐑ϕ_n'(x) dx (  ω_n ( ŝ_n(x+i ω_n)   )  - ∫_y > ω_n (y χ'(y) + χ(y))  ( ŝ_n(x+iy)   )  dy  ) 
       = -1/2 π∫_ x ∈𝐑ϕ_n'(x)  ω_n ( ŝ_n(x+i ω_n)   ) dx 
       =+  1/2 π∫_ x ∈𝐑∫_y > ω_nϕ_n'(x) y χ'(y) ( ŝ_n(x+iy)   ) dy dx 
       =+ 1/2 π∫_ x ∈𝐑∫_y > ω_nϕ_n'(x) χ(y) ( ŝ_n(x+iy)   ) dy dx  .

The first term (<ref>) can be bounded as follows, recalling that ϕ_n' is supported on I_n (defined in Equation (<ref>)):

    | -1/2 π∫_ x ∈𝐑ϕ_n'(x) ω_n ( ŝ_n(x+i ω_n)   ) dx  | 
    ≤1/2 π ||ϕ_n'||_∞ω_n ∫_I_n| ( ŝ_n(x+i ω_n)   ) | dx.

By the definition of ϕ_n, it holds that ||ϕ_n'||_∞ = O(1/ω_n). Combining this with inequality (<ref>), which holds uniformly in x ∈ I_n, we get that:

    | -1/2 π∫_ x ∈𝐑ϕ_n'(x) ω_n ( ŝ_n(x+i ω_n)   ) dx  | = O(ω_n).


The second term (<ref>) can be bounded as follows, using that χ' is supported on [-2,-1] ∪ [1,2]:

    | 1/2 π∫_ x ∈𝐑∫_ y ≥ω_nϕ_n'(x)y χ'(y)   ( ŝ_n(x+iy)   ) dy dx | ≤1/2 π ||ϕ_n'||_∞∫_ x ∈ I_n ∫_1≤  y ≤ 2 y χ'(y) ( ŝ_n(x+iy)   ) dx dy.

Since we are working on the event E_ε, |( ŝ_n(x+iy)   )| ≤ n^εψ_n(x+iy). Therefore, using Lemma <ref>:

    | 1/2 π∫_ x ∈𝐑∫_ y ≥ω_nϕ_n'(x)y χ'(y)   ( ŝ_n(x+iy)   ) dy dx | = O(  n^ε/√(n)).


Finally, we bound the last term (<ref>). Since ϕ_n' is supported on I_n, we can replace the integral over 𝐑 by an integral over I_n. Moreover, using (<ref>), we get that:

    | 1/2 π∫_ x ∈𝐑∫_y > ω_nϕ_n'(x) χ(y) ( ŝ_n(x+iy)   ) dy dx | 
       ≤1/2 π||ϕ_n'||_∞∫_I_n∫_ω_n^2 n^ε/√(n y)dy dx  
       = O ( n^ε/√(n)) .


Hence, putting together (<ref>), (<ref>) and (<ref>), we proved that, on the event E_ε:

    | -1/2 π∫_x ∈𝐑∫_|y| ≥ω_nϕ”_n(x) y χ (y) (ŝ_n (x+iy) ) dx dy | = O( max{n^ε/n^1/2, ω_n })




  
*Estimation of the term (<ref>).
Since χ' is symmetric and supported on [-2,-1] ∪ [1,2], there exists some constant C>0 such that:

    | 1/2 π∫_x ∈𝐑∫_y ∈𝐑( ϕ_n(x) + iyϕ'_n(x)  ) χ'(y) ŝ_n (x+iy) dx dy | 
    ≤C/2 π∫_x ∈ [x_0 - ε_n - 2 ω_n, x_0 + ε_n + 2 ω_n]∫_y ∈ [-2,-1] ∪ [1,2]( 1 +  ||ϕ'_n||_∞) | ŝ_n (x+iy) | dx dy .

By Lemma <ref>, uniformly in x ∈ [x_0 - ε_n - 2 ω_n, x_0 + ε_n + 2 ω_n] and y ∈ [-2,-1] ∪ [1,2], | ŝ_n (x+iy) | ≤n^ε/√(n). Therefore:

    | 1/2 π∫_x ∈𝐑∫_y ∈𝐑( ϕ_n(x) + iyϕ'_n(x)  ) χ'(y) ŝ_n (x+iy) dx dy | = O ( n^ε/√(n)).





  
*Conclusion
Putting together estimates (<ref>), (<ref>) and (<ref>), we proved that, for all ε>0, on the event E_ε:

    | ∫_𝐑ϕ_n(λ) dμ̂_n(λ)   | = O( max{n^ε/n^1/2, ω_n }) = O(ω_n) .

Now, recall from (<ref>) that ω_n >  > n^δ / √(n). This implies that for all 0 <ε≤δ:

    max{n^ε/n^1/2, ω_n } = ω_n.

Hence, there exists D>0 such that with probability at least 1-n^-D:

    | ∫_𝐑ϕ_n(λ) dμ̂_n(λ)   | = O ( ω_n ).

This ends the proof of the first part of Lemma <ref>.



It remains to explain how to adapt the above argument to obtain the second estimate (<ref>). Since it is concerned with the empirical spectral measure μ_S_n, we need a local law for this quantity, which is an analog of Theorem <ref> in this context, and is a consequence of the work of Knowles and Yin. Recall that 

    𝒟_n^(τ)(x_0,c) = { z ∈𝐂,   x_0-c ≤ E ≤ x_0+c,   n^-1 + τ≤η≤τ^-1} .

<cit.>
Let s_μ_S_n be the Stieltjes transform of μ_S_n. Let τ > 0. Then, uniformly in z  ∈𝒟_n^(τ)(x_0,c), for any ε >0, there exists D>0 such that:

    ( | s_μ_S_n(z) - s_μ_MP,α⊠μ_Σ(z) | ≥n^ε/n (z)) ≤1/n^D .


As before, we only discuss the case where ϵ = + and denote ϕ_n = ϕ_n^+. As in the case of the spectral measure, denoting μ̃_n := μ_(S_n,v_1^(n)) - μ_α,Σ,θ and s̃_n := s_μ_S_n - s_μ_MP,α⊠μ_Σ, the Helffer-Sjöstrand formula leads to:

    ∫_𝐑ϕ_n(λ) dμ̃_n(λ) ≤   +b-1/2 π∫_x ∈𝐑∫_|y| ≤ω_nϕ”_n(x) y χ (y) (s̃_n (x+iy)) dx dy 
       + -1/2 π∫_x ∈𝐑∫_|y| ≥ω_nϕ”_n(x) y χ (y) (s̃_n (x+iy) ) dx dy 
       + | 1/2 π∫_x ∈𝐑∫_y ∈𝐑( ϕ_n(x) + iyϕ'_n(x)  ) χ'(y) s̃_n (x+iy) dx dy |  .

Now, each term (<ref>), (<ref>) and (<ref>) can be treated in the same way as the previous terms (<ref>), (<ref>) and (<ref>). The only difference is that each occurence of the former error term ψ_n(z) is now replaced by the error term of Theorem <ref>, namely (n (z))^-1. 


We underline that a weaker version of Theorem <ref> (resp. Theorem <ref>) can be obtained as long as a uniform estimation of s_(S_n,v_1^(n))(z) - s_μ_α, Σ, θ(z) is available for z in a domain of the upper half-plane that is allowed to approach the real axis as n becomes larger. Indeed, if

    | s_(S_n,v_1^(n))(z) - s_μ_α, Σ, θ(z) | = O( ε_n ),
 
then, the Helffer-Sjöstrand argument that we developed during the proof of Theorem <ref> yields a convergence of the averaged-square projection onto the direction of the spike for averaging windows of size ε_n, as long as ε_n >  > n^δ / √(n) for some δ >0. This limitation corresponds to the optimal rate in the local laws of Knowles and Yin (<ref>).

One natural question would be to weaken the assumption on the size ε_n of the averaging window: do our Theorems <ref>, <ref>, <ref> and <ref> hold as soon as ε_n = o(1)? We believe that the answer is positive (see Figure <ref> for simulations). Morevoever, the results of <cit.>, which state the convergence in law of properly rescaled individual square projection of eigenvectors associated to eigenvalues in the vicinity of the edge, suggest the following natural question: does such a convergence also holds in the bulk of the spectrum?  





  
*Acknowledgment I am very thankful to Nathanaël Enriquez for many suggestions about this work. Many thanks to Maxime Février for numerous insightful discussions and for his careful reading of an earlier version of this paper. I would also like to thank Laurent Ménard for his advices.



plain
10

MR3256861
Romain Allez and Jean-Philippe Bouchaud.
Eigenvector dynamics under free addition.
Random Matrices Theory Appl., 3(3):1450010, 17, 2014.

MR2330979
Z. D. Bai, B. Q. Miao, and G. M. Pan.
On asymptotics of eigenvectors of large sample covariance matrix.
Ann. Probab., 35(4):1532–1572, 2007.

MR1617051
Z. D. Bai and Jack W. Silverstein.
No eigenvalues outside the support of the limiting spectral
  distribution of large-dimensional sample covariance matrices.
Ann. Probab., 26(1):316–345, 1998.

MR2165575
Jinho Baik, Gérard Ben Arous, and Sandrine Péché.
Phase transition of the largest eigenvalue for nonnull complex sample
  covariance matrices.
Ann. Probab., 33(5):1643–1697, 2005.

MR3729610
Serban T. Belinschi, Hari Bercovici, Mireille Capitaine, and Maxime
  Février.
Outliers in the spectrum of large deformed unitarily invariant
  models.
Ann. Probab., 45(6A):3571–3625, 2017.

benaych2018eigenvectors
Florent Benaych-Georges, Nathanaël Enriquez, and Alkéos Michaïl.
Eigenvectors of a matrix under random perturbation.
arXiv preprint arXiv:1801.10512, 2018.

MR3792624
Florent Benaych-Georges and Antti Knowles.
Local semicircle law for Wigner matrices.
In Advanced topics in random matrices, volume 53 of Panor.
  Synthèses, pages 1–90. Soc. Math. France, Paris, 2017.

MR2782201
Florent Benaych-Georges and Raj Rao Nadakuditi.
The eigenvalues and eigenvectors of finite, low rank perturbations of
  large random matrices.
Adv. Math., 227(1):494–521, 2011.

MR1488333
Philippe Biane.
On the free convolution with a semi-circular distribution.
Indiana Univ. Math. J., 46(3):705–718, 1997.

MR3449395
Alex Bloemendal, Antti Knowles, Horng-Tzer Yau, and Jun Yin.
On the principal components of sample covariance matrices.
Probab. Theory Related Fields, 164(1-2):459–552, 2016.

MR3090543
M. Capitaine.
Additive/multiplicative free subordination property and limiting
  eigenvectors of spiked additive deformations of Wigner matrices and spiked
  sample covariance matrices.
J. Theoret. Probab., 26(3):595–648, 2013.

MR3792626
Mireille Capitaine and Catherine Donati-Martin.
Spectrum of deformed random matrices and free probability.
In Advanced topics in random matrices, volume 53 of Panor.
  Synthèses, pages 151–190. Soc. Math. France, Paris, 2017.

MR637828
Z. Füredi and J. Komlós.
The eigenvalues of random symmetric matrices.
Combinatorica, 1(3):233–241, 1981.

gamboa2018sum
Fabrice Gamboa, Jan Nagel, and Alain Rouault.
Sum rules and large deviations for spectral matrix measures in the
  jacobi ensemble.
arXiv preprint arXiv:1811.06311, 2018.

MR3262497
Antti Knowles and Jun Yin.
The outliers of a deformed Wigner matrix.
Ann. Probab., 42(5):1980–2031, 2014.

MR3704770
Antti Knowles and Jun Yin.
Anisotropic local laws for random matrices.
Probab. Theory Related Fields, 169(1-2):257–352, 2017.

MR2834718
Olivier Ledoit and Sandrine Péché.
Eigenvectors of some large sample covariance matrix ensembles.
Probab. Theory Related Fields, 151(1-2):233–264, 2011.

MR3502606
Ji Oon Lee, Kevin Schnelli, Ben Stetler and Horng-Tzer Yau.
Bulk universality for deformed Wigner matrices.
Ann. Probab., 44(3):2349–2425, 2016.

MR0475502
L. A. Pastur.
The spectrum of random matrices.
Teoret. Mat. Fiz., 10(1):102–112, 1972.

MR1370408
Jack W. Silverstein.
Strong convergence of the empirical distribution of eigenvalues of
  large-dimensional random matrices.
J. Multivariate Anal., 55(2):331–339, 1995.

MR1345541
Jack W. Silverstein and Sang-Il Choi.
Analysis of the limiting spectral distribution of large-dimensional
  random matrices.
J. Multivariate Anal., 54(2):295–309, 1995.

MR3373058
Lenczewski, Romuald.
Random matrix model for free Meixner laws.
Int. Math. Res. Not. IMRN, 11:3499–3524, 2015.







Nathan Noiry :

Laboratoire Modal'X, 

UPL, Université Paris Nanterre,
 
F92000 Nanterre France




