# Self-repellent Brownian bridges
in an interacting Bose gas

Erwin Bolthausen111Institut für Mathematik, Universität Zürich,
Winterthurerstrasse 190, Zurich 8057<EMAIL_ADDRESS>Wolfgang König222TU
Berlin and WIAS Berlin, Mohrenstraße 39, 10117 Berlin<EMAIL_ADDRESS>and Chiranjib Mukherjee333Universität Münster, Einsteinstraße 62, 48149
Münster<EMAIL_ADDRESS>
††AMS Subject Classification: 60F10; 60J65; 82B10; 81S40.††Keywords: Self-
repellent Brownian bridges, interacting Bose gas, Dirichlet boundary
condition, condensation phase transition, random geometric permutation, random
partition, large deviations, lace expansion.

University of Zurich, WIAS Berlin and TU Berlin, Universität Münster

14 May, 2024

> Abstract: We consider a model of $d$-dimensional interacting quantum Bose
> gas, expressed in terms of an ensemble of interacting Brownian bridges in a
> large box and undergoing the influence of all the interactions between the
> legs of each of the Brownian bridges. We study the thermodynamic limit of
> the system and give an explicit formula for the limiting free energy and a
> necessary and sufficient criterion for the occurrence of a condensation
> phase transition. For $d\geq 5$ and sufficiently small interaction, we prove
> that the condensate phase is not empty. The ideas of proof rely on the
> similarity of the interaction to that of the self-repellent random walk, and
> build on a lace expansion method conducive to treating paths undergoing
> mutual repellence within each bridge.

##  1\. Introduction and main results

###  1.1. Background.

The well-known interacting Bose gas can be written, using the Feynman–Kac
formula, as a random ensemble of many Brownian bridges of random and unbounded
lengths ($=$ particle numbers) with mutually repellent interactions between
any two legs of any of the bridges. In the thermodynamic limit, the transition
from absence to emergence of macroscopically many particles in long bridges in
this ensemble as the density increases is a prominent open problem that is
strongly linked with the famous phase transition called Bose–Einstein
condensation (BEC), if not equal to that.

The starting observation of the present paper is the following: Each of these
bridges, under the influence of all the interactions between its legs alone,
is a transformation of a Brownian bridge that should lie in the class of the
well-known self-repellent walk (or weakly self-avoiding walk). This class of
random motions is prominent in statistical physics and has been studied a lot
since several decades, although a lot of problems are left widely open.

However, to the best of our knowledge, a substantial connection between the
Bose gas and the self-repellent walk has not been made. In this paper, we
establish this connection to study the influence of the interactions between
its legs and characterize the free energy of the system explicitly in terms of
a variational formula and find criteria for the presence of a condensate phase
transition in this model. These criteria lie at the heart of the critical
properties of the self-repellent Brownian bridges.

Below in Section 1.2 we will introduce the mathematical layout of the model,
in Section 1.3 we will explain our purposes and formulate our main results in
Section 1.4. In Section 2 we will discuss the necessary background, outline
the method of proof and draw conclusions from our main results. The proofs
will be given in the subsequent sections.

###  1.2. The model.

Let us introduce the model that we are interested in. As will be explained in
Section 2.1, it is indeed a version of the interacting Bose gas. For the sake
of simplicity, we prefer to formulate it as a model of a random partition of a
positive integer $N\in\mathbb{N}$.

Indeed, denote the set of partitions of $N\in\mathbb{N}$ by

$\mathfrak{P}_{N}=\Big{\\{}l=(l_{k})_{k\in\mathbb{N}}\in\mathbb{N}_{0}^{\mathbb{N}}\colon\sum_{k\in\mathbb{N}}kl_{k}=N\Big{\\}}.$
(1.1)

With a parameter $\beta\in(0,\infty)$, introduce the following probability
measure on $\mathfrak{P}_{N}$:

${\tt
P}^{{\scriptscriptstyle{({N}})}}_{\beta,\Lambda}(l)=\frac{1}{Z_{N}^{{\scriptscriptstyle{({{\rm
bc}}})}}(\beta,\Lambda)}\prod_{k\in\mathbb{N}}\frac{\big{[}|\Lambda|\Gamma^{{\scriptscriptstyle{({\rm
bc}})}}_{\Lambda,k}\big{]}^{l_{k}}}{l_{k}!\,k^{l_{k}}},\qquad
l=(l_{k})_{k\in\mathbb{N}}\in\mathfrak{P}_{N},$ (1.2)

where

$\displaystyle Z_{N}^{{\scriptscriptstyle{({{\rm
bc}}})}}(\beta,\Lambda)=\sum_{l\in\mathfrak{P}_{N}}\prod_{k\in\mathbb{N}}\frac{\big{[}|\Lambda|\Gamma^{{\scriptscriptstyle{({\rm
bc}})}}_{\Lambda,k}\big{]}^{l_{k}}}{l_{k}!\,k^{l_{k}}}$ (1.3)

is the partition function of the model (the normalization). It is well-known
that $N/\prod_{k\in\mathbb{N}}l_{k}!k^{l_{k}}$ is equal to the number of
permutations of $1,\dots,N$ that have precisely $l_{k}$ cycles for any
$k\in\mathbb{N}$. Thus, (1.1)- (1.3) indeed define a model of a random
permutation with multiplicative weight
$|\Lambda|\Gamma_{\Lambda,k}^{{\scriptscriptstyle{({\rm bc}})}}$ for any cycle
of length $k$; see Section 2.4 for further details.

Let us now explain the terms appearing in (1.1) – (1.3). First,
$\Lambda\subset\mathbb{R}^{d}$ denotes a centered box, and
$\Gamma^{{\scriptscriptstyle{({\rm bc}})}}_{\Lambda,k}$ is defined as an
integral of an exponential interaction with respect to a Brownian bridge of
length $k$ in $\Lambda$ (cf. (1.5) below). If we denote by
$\mu_{x,y}^{{\scriptscriptstyle{({\rm{bc},\beta}})}}(\cdot)$ the canonical
Brownian bridge measure on the time interval $[0,\beta]$ from $x$ to $y$
subject to the boundary condition ‘bc’ in the box $\Lambda$ (the boundary
condition to be specified later), we set

$\mu_{\Lambda}^{{\scriptscriptstyle{({{\rm bc},\beta}})}}({\rm
d}f)=\frac{1}{|\Lambda|}\int_{\Lambda}{\rm
d}x\,\mu^{{\scriptscriptstyle{({{\rm bc},\beta}})}}_{x,x}({\rm d}f).$ (1.4)

Thus, $\mu_{\Lambda}^{{\scriptscriptstyle{({{\rm bc},\beta}})}}(\cdot)$ is a
Brownian bridge measure on ${\mathcal{C}}={\mathcal{C}}_{1}$, where
${\mathcal{C}}_{k}$ denotes the set of continuous functions
$[0,k\beta]\to\mathbb{R}^{d}$, with uniform starting-termination site in the
centered box $\Lambda$. If we write $\mu(f)$ for the integral $\int f{\rm
d}\mu$ of a function $f$ with respect to a measure $\mu$, the weight
$\Gamma^{{\scriptscriptstyle{({\rm bc}})}}_{\Lambda,k}$ is defined as

$\Gamma^{{\scriptscriptstyle{({\rm
bc}})}}_{\Lambda,k}=\mu_{\Lambda}^{{\scriptscriptstyle{({{\rm
bc},k\beta}})}}\Big{[}{\rm e}^{-\sum_{1\leq i<j\leq k}V(B_{i},B_{j})}\Big{]},$
(1.5)

where, for any two continuous functions $f,g\in\mathcal{C}$, we write

$V(f,g)=\int_{0}^{\beta}v(|f(s)-g(s)|)\,{\rm d}s,$ (1.6)

and $v\colon[0,\infty)\to[0,\infty)$ is a bounded measurable pair interaction
potential having compact support. In (1.5), the Brownian bridge $B$ in
${\mathcal{C}}_{k}$ has been decomposed into its $k$ legs, defined as

$B_{j}=(B_{j}(s))_{s\in[0,\beta]}=(B((j-1)\beta+s))_{s\in[0,\beta]}\in{\mathcal{C}},\qquad
j\in[k]=\\{1,\dots,k\\}.$ (1.7)

Hence, the exponential interaction term in (1.12) is the pair-interaction sum
of all the legs of a Brownian bridge of length $k$. Therefore,
$\Gamma^{{\scriptscriptstyle{({\rm bc}})}}_{\Lambda,k}$ is the partition
function of what can be called a variant of the self-repellent (or weakly
self-avoiding) Brownian bridge, where the usual $\delta$-interaction between
any two $B(s)$ and $B(t)$ is replaced by a smooth interaction between any two
legs $B_{i}$ and $B_{j}$. The starting site of this motion is not the origin
as usual, but is uniformly distributed over a large box, and the endpoint is
conditioned to be the initial site. We will be studying this model in the
limit $N\to\infty$ with the box $\Lambda=\Lambda_{N}$ having volume $\sim
N/\rho$ for some $\rho\in(0,\infty)$, i.e., in the thermodynamic limit.

By the Feynman–Kac formula, the above model can also be represented as an
ensemble of $N$ Brownian bridges in $\Lambda$ with time-horizon $[0,\beta]$
under some symmetrization condition, where each leg has an interaction with
each other leg. Due to the symmetrization, one can decompose the ensemble into
closed loops (bridges); see Section 2.1 for further details.

We remark that we have kept all features that come from the Bose gas in a
large box, in particular the fact that the particles are confined to a box
with certain boundary conditions. Most of the existing works on the Bose gas
consider periodic boundary conditions, and focus on models of random
permutations that dispense with considerations of boundary conditions.
However, we find it important to keep them in the model, since they are
physically relevant and since they might lead to additional effects. Indeed,
for periodic or Dirichlet zero boundary conditions, the asymptotics of the
weight $\Gamma^{{\scriptscriptstyle{({\rm bc}})}}_{\Lambda,k}$ in (1.5) for
large $k$ and boxes $\Lambda$ of volume $\asymp N$ are clearly different for
$k\ll N^{2/d}$ and for $k\gg N^{2/d}$.

###  1.3. Our purpose.

As mentioned above, we will be interested in the thermodynamic limit of the
model described above. Namely, we fix the inverse temperature
$\beta\in(0,\infty)$, a density $\rho\in(0,\infty)$, take the box
$\Lambda=\Lambda_{N}$ of volume $|\Lambda_{N}|=\frac{N}{\rho}$ and define the
corresponding limiting free energy per volume as

$f(\beta,\rho):=-\lim_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log
Z_{N}^{{\scriptscriptstyle{({\rm
bc}})}}(\beta,\Lambda_{N}),\qquad\beta,\rho\in(0,\infty).$ (1.8)

In our first main result, Theorem 1.1, we will see that this limit exists, is
independent of boundary conditions and can be expressed in terms of a
characteristic variational formula that describes the statistics of all the
lengths of the Brownian bridges. Next, in this result we prove a large-
deviation principle for these statistics and identify the minimizer of the
rate function. This implies in particular that the statistics of the cycle
lengths converge in distribution towards that minimizer. Now, the
characterization of the minimizer involves a possible phase transition in the
density $\rho$ – that is, a non-analyticity of the map

$\rho\mapsto f(\beta,\rho)$ (1.9)

emerges at some explicit critical threshold $\rho_{\rm c}(\beta)$, which may
or may not be finite in general. This phase transition underlines a loss of
mass in the statistics of the finite-size cycle lengths in the spirit of the
well-known effect in the free Bose gas; it is therefore a signal for a
condensation phase transition in a vicinity of the Bose–Einstein condensation.

So far, the above statements are valid for a wide choice of the weights
$\Gamma_{\Lambda,k}^{{\scriptscriptstyle{({\rm bc}})}}$ in (1.3) as well as
arbitrary $d\in\mathbb{N}$ and $\beta\in(0,\infty)$. Now for our model,
$\rho_{\rm c}(\beta)$ is shown to be equal to the Green’s function at the
origin of a transformed self-interacting Brownian bridge, geometrically
weighted with a crucial parameter, the radius of convergence. As a
consequence, finiteness of the Green’s function implies the existence of the
aforementioned non-analyticity of the map (1.9) at $\rho_{\rm c}(\beta)$.

Theorem 1.2 is our second main result, where we determine circumstances under
which the aforementioned phase transition does occur. Indeed, we show that for
$d\geq 5$ and the interaction potential $v$ remaining sufficiently small, the
critical density $\rho_{c}(\beta)$ is actually finite. The key idea here is
the observation that the self-interacting Brownian bridge lies in the same
universality class as the famous self-avoiding walk (SAW), the uniform
distribution on $N$-step nearest-neighbour random walks on $\mathbb{Z}^{d}$
starting at zero and hitting no site twice. Indeed, since the pair functional
$v$ is non-negative, the interaction $\sum_{1\leq i<j\leq k}V(B_{i},B_{j})$
repels each two legs from each other, in the spirit of a weak version of the
SAW, the self-repellent random walk. One main difference to the present model
of the interacting quantum Bose gas is that here it is paths that undergo a
mutual repellence, not endpoints.

The restrictions to $d\geq 5$ and small $\beta$ are due to a continuous
version of the lace expansion method that we develop and employ in the proof
of Theorem 1.2. This part is a modification of the method developed in
[BHK18]. We refer to Section 2.3 for a short survey on this method, and to
Section 2.2 for precise conjectures about the behaviour of the SAW and models
in its universality class. Surprisingly, these conjectures imply the
finiteness of the Green’s function (i.e., the occurrence of the phase
transition in our model) even in dimension $d=2$, not only in dimensions
$d\geq 3$, being aligned with conjectures about the BEC phase transition.

###  1.4. Main results.

The formulation of our results will require setting up some further notation,
which will subsequently be used in the entire sequel. As before, $\mathcal{C}$
will denote the space of continuous functions $[0,\beta]\to\mathbb{R}^{d}$.
The canonical Brownian bridge measure is defined for any measurable
$A\subset{\mathcal{C}}$ as

$\mu^{{\scriptscriptstyle{({\beta}})}}_{x,y}(A)=\frac{\mathbb{P}_{x}(B\in
A;B_{\beta}\in{\rm d}y)}{{\rm d}y}.$ (1.10)

Its total mass is given by

$\displaystyle\mu_{x,y}^{{\scriptscriptstyle{({\beta}})}}({\mathcal{C}})=\varphi_{\beta}(x,y)$
$\displaystyle:=\frac{\mathbb{P}_{x}(B_{\beta}\in{\rm d}y)}{{\rm
d}y}=(2\pi\beta)^{-d/2}{\rm e}^{-\frac{1}{2\beta}|x-y|^{2}},$ (1.11)

where $B=(B_{t})_{t\in[0,\beta]}$ is a Brownian motion in $\mathbb{R}^{d}$
with generator $\frac{1}{2}\Delta$, starting from $x$ under $\mathbb{P}_{x}$.
Hence, the total mass of $\mu_{0,0}^{{\scriptscriptstyle{({k\beta}})}}$ is
equal to $(2\pi\beta k)^{-d/2}$.

We fix a bounded mesurable function $v\colon[0,\infty)\to[0,\infty)$ with
compact support. Let us set

$\displaystyle\Gamma_{k}$
$\displaystyle=\mu_{0,0}^{{\scriptscriptstyle{({k\beta}})}}\Big{[}{\rm
e}^{-\sum_{1\leq i<j\leq k}V(B_{i},B_{j})}\Big{]}$ (1.12)
$\displaystyle=\mathbb{E}_{0}\Big{[}{\rm e}^{-\sum_{1\leq i<j\leq
k}V(B_{i},B_{j})}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{B(k\beta)\in{\rm
d}x\\}\Big{]}\Big{/}{\rm d}x\Big{|}_{x=0},$

with $V(\cdot,\cdot)$ defined in (1.6), compare to (1.5). Then the sequence
$(\Gamma_{k}(2\pi\beta k)^{d/2})_{k\in\mathbb{N}}$ is sub-multiplicative.
Consequently, by Fekete’s lemma, the limit (known as the connective constant)

$\displaystyle\lambda_{\rm
c}(\beta)=\lim_{k\to\infty}\Gamma_{k}^{-1/k}\quad\mbox{exists and satisfies }$
(1.13) $\displaystyle\lambda_{\rm c}(\beta)^{k}\Gamma_{k}\geq(2\pi\beta
k)^{-d/2}\quad\forall k\in\mathbb{N}.$

$\lambda_{c}(\beta)$ is the radius of convergence of the power series with
coefficients $\Gamma_{k}$. It will turn out to capture the limiting behaviour
of the coefficients $\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
bc}})}}$ of the model. Finally, introduce

$I(p)=\sum_{k\in\mathbb{N}}p_{k}\log\frac{p_{k}k}{\Gamma_{k}{\rm e}},\qquad
p=(p_{k})_{k\in\mathbb{N}}\in[0,\infty)^{\mathbb{N}}.$ (1.14)

The function $I$ will turn out to play the role of a large-deviation rate
function – it is the relative entropy of $p$ with respect to the sequence
$(\Gamma_{k}/k)_{k\in\mathbb{N}}$ plus the sum of the $\Gamma_{k}/k$.

Here is the identification of the free energy $f(\beta,\rho)$ defined in
(1.8), which is the first main result of our article.

######  Theorem 1.1 (Free energy and cycle lengths).

Fix $\beta,\rho\in(0,\infty)$ and consider the model defined in (1.3) and
(1.2) in the centred box $\Lambda_{N}$ with volume $N/\rho$ with Dirichlet
boundary conditions ${\rm bc}\in\\{{\rm Dir}\\}$. Then the following hold.

  1. (i)

With $\lambda_{(}{\rm c}\beta)$ defined in (1.13), the limiting free energy
$f(\beta,\rho)$ defined in (1.8) exists and is identified as

$f(\beta,\rho)=\inf_{p\in[0,\infty)^{\mathbb{N}}\colon\sum_{k}kp_{k}\leq\rho}\Big{[}I(p)+\Big{(}\rho-\sum_{k\in\mathbb{N}}kp_{k}\Big{)}\log\lambda_{\rm
c}(\beta)\Big{]}.$ (1.15)

  2. (ii)

Let $(L_{k})_{k\in\mathbb{N}}$ be a random variable under ${\tt
P}^{{\scriptscriptstyle{({N}})}}_{\beta,\Lambda_{N}}$, then the distribution
of $(L_{k}/|\Lambda_{N}|)_{k\in\mathbb{N}}$ satisfies a large-deviations
principle (LDP) 2††2 This means the following: A sequence
$(X_{N})_{N\in\mathbb{N}}$ of random variables taking values in a topological
space $\mathcal{X}$ satisfies an LDP on the scale $\gamma_{N}$ with rate
function $J\colon\mathcal{X}\to[0,\infty]$ if $J$ is lower-semicontinuous and
for any open subset $G$ and any closed subset $F$ of $\mathcal{X}$,
$\liminf_{N\to\infty}\frac{1}{\gamma_{N}}\log\mathbb{P}(X_{N}\in
G)\geq-\inf_{G}J\qquad\mbox{and}\qquad\limsup_{N\to\infty}\frac{1}{\gamma_{N}}\log\mathbb{P}(X_{N}\in
F)\leq-\inf_{F}J.$ If $J$ has a unique minizer $x^{*}\in\mathcal{X}$, then it
follows that $X_{N}$ converges weakly towards $x^{*}$. We refer to [DZ98] for
the general theory of large deviations. on the set
$\mathcal{X}_{\rho}=\\{p\in[0,\infty)^{\mathbb{N}}\colon\sum_{k\in\mathbb{N}}kp_{k}\in[0,\rho]\\}$
on the scale $|\Lambda_{N}|$ with rate function $J$ given by

$J(p)=I(p)+\Big{(}\rho-\sum_{k\in\mathbb{N}}kp_{k}\Big{)}\log\lambda_{\rm
c}(\beta)-f(\beta,\rho).$ (1.16)

  3. (iii)

Put

$\rho_{\rm c}(\beta)=\sum_{k\in\mathbb{N}}\lambda_{\rm
c}(\beta)^{k}\Gamma_{k}\in[0,\infty].$ (1.17)

Then the unique minimizer $p^{*}$ of the formula on the right-hand side of
(1.15) is given as

$p^{*}_{k}=\frac{1}{k}\lambda_{\rm
c}(\beta)^{k}\Gamma_{k}\times\begin{cases}{\rm e}^{-c(\rho)k},&\mbox{if
}\rho\leq\rho_{\rm c}(\beta),\\\ 1&\mbox{if }\rho>\rho_{\rm
c}(\beta),\end{cases}\qquad k\in\mathbb{N},$ (1.18)

where $c(\rho)\in[0,\infty)$ is such that

$\sum_{k\in\mathbb{N}}\lambda_{\rm c}(\beta)^{k}\Gamma_{k}{\rm
e}^{-c(\rho)k}=\rho.$

In particular, $\rho\mapsto c(\rho)$ is strictly decreasing with $c(\rho_{\rm
c}(\beta))=0$. Furthermore,
$\sum_{k\in\mathbb{N}}kp^{*}_{k}=\rho\wedge\rho_{\rm c}(\beta)$ for any
$\rho\in(0,\infty)$.

  4. (iv)

$f(\beta,\rho)=\rho\log\lambda_{\rm c}(\beta)-\begin{cases}\rho
c(\rho)+\sum_{k\in\mathbb{N}}\frac{1}{k}\Gamma_{k}\lambda_{\rm
c}(\beta)^{k}{\rm e}^{-c(\rho)k}&\mbox{if }\rho\leq\rho_{\rm c}(\beta),\\\
\sum_{k\in\mathbb{N}}\frac{1}{k}\Gamma_{k}\lambda_{\rm c}(\beta)^{k}&\mbox{if
}\rho>\rho_{\rm c}(\beta).\end{cases}$ (1.19)

In particular, $f(\beta,\cdot)$ has a phase transition (non-analyticity) in
$\rho_{\rm c}(\beta)$ if this point is finite.

The proof of Theorem 1.1 is provided in Section 3. Let us make some remarks
about the assertions appearing there. First note that the rate function of the
LDP there has two terms, the entropic term $I$, which describes the statistics
of the finite cycle lengths, and the energy term,
$(\rho-\sum_{k}kp_{k})\log\lambda_{\rm c}(\beta)$, which is the contribution
from the condensate part. Unlike in the free Bose gas (see, e.g., [KVZ23]),
the condensate yields here an explicit contribution on the leading scale, but
not an entropic one. While each cycle of length $k$ comes with an energetic
term $\Gamma_{k}$, the condensate comes with the logarithm of the connective
constant of the sequence $(\Gamma_{k})_{k\in\mathbb{N}}$. The convergence of
$(L_{k}/\Lambda_{N})_{k\in\mathbb{N}}$ towards the minimizer $p^{*}$ shows
that the model has a phase transition at the critical density $\rho_{\rm
c}(\beta)$ if this quantity is finite. In (1.17) we see that $\rho_{\rm
c}(\beta)$ is the Green’s function of the transformed Brownian motion
geometrically weighted with parameter $\lambda=\lambda_{\rm c}(\beta)$, taken
at the origin, and $\lambda_{\rm c}(\beta)$ is the radius of convergence. If
the Green’s function is finite at this point, then the free energy
$f(\beta,\cdot)$ is not analytic here, since all the coefficients in the
series in (1.15) are positive.

Now the question of course arises, under what circumstances this phase
transition occurs. We give a positive answer in our next main result.

######  Theorem 1.2 (Phase transition in $d\geq 5$).

Assume that $d\geq 5$ and that the interaction potential $v$ (cf. (1.5) and
(1.6)) is bounded and continuous and has a bounded support. Let $\rho_{\rm
c}(v,\beta)=\rho_{\rm c}(\beta)$ be the critical density defined in (1.17).
Then, for any $\beta\in(0,\infty)$, there is $\alpha_{\beta}>0$ such that
$\rho_{\rm c}(\alpha v,\beta)<\infty$ for any $\alpha\in(0,\alpha_{\beta}]$.

The proof of Theorem 1.2 is spanned through Section 4 – Section 7. As we will
explain in Section 2, the restriction to $d\geq 5$ is related to the question
that lies at the heart of a famous and fundamental question that is
notoriously difficult to answer in dimensions $d\in\\{2,3,4\\}$: the question
whether or not the Green’s function of the weakly self-avoiding walk is finite
at the critical point.

The organization of the remainder of this paper is as follows. In Section 2 we
discuss several aspects of our main results, in Section 3 we prove Theorem
1.1, in Section 4 we give the proof of Theorem 1.2, subject to the proof of
two crucial results that we prove in Sections 5 and 6, respectively. In
Section 7, we prove some technical estimates, which are used in the two
sections preceding to that.

##  2\. Background and discussion

In this section we discuss the background, proof method and conclusions of our
main results. More concretely, in Section 2.1 we explain the relation between
the well-known interacting Bose gas and the model that we study in this paper,
in Section 2.2 we summarize what is known and what is conjectured about the
weakly self-avoiding walk, and we draw conjectural conclusions about the
occurrence of the phase transition in those dimensions that we do not handle
in this paper. Some elements and outline of our main proof method for Theorem
1.2 is presented and commented on in Section 2.3. Finally, we comment on our
model from the viewpoint of random permutations in Section 2.4, and in Section
2.5 we comment on the role of the boundary conditions carried in our model.

###  2.1. The interacting Bose gas.

The model that we are interested in and described in Section 1.2 is strongly
inspired by an investigation of an interacting quantum gas at positive
temperature $1/\beta\in(0,\infty)$ in the thermodynamic limit, the setup for
which can be briefly described as follows. As before, let
$\Lambda\subset\mathbb{R}^{d}$ be a centered box and

${\mathcal{H}}^{{\scriptscriptstyle{({\rm{bc}}})}}_{N,\Lambda}=-\frac{1}{2}\sum_{i=1}^{N}\Delta_{i}+\sum_{1\leq
i<j\leq N}v(|x_{i}-x_{j}|),\qquad x_{1},\dots,x_{N}\in\Lambda,$

is the Hamiltonian operator for $N$ particles located at $x_{1},\dots,x_{N}$
in $\Lambda\subset\mathbb{R}^{d}$ with a prescribed boundary condition. Again
$v\colon[0,\infty)\to[0,\infty)$ is some pair-interaction functional, which is
assumed to be bounded with bounded support. We are interested in bosons, and
in this vein a key object of interest is the symmetrized trace

$\mathbf{Z}_{N}^{{\scriptscriptstyle{({\rm bc}})}}(\beta,\Lambda):={\rm
Tr}_{+}\big{(}{\rm
e}^{-\beta{\mathcal{H}}^{{\scriptscriptstyle{({\rm{bc}}})}}_{N,\Lambda}}\big{)},\qquad\beta\in(0,\infty),$
(2.1)

where the index $+$ stands for symmetrization – that is, application of the
projection operator on the set of all permutation invariant wave functions.

We are going to rewrite this trace in terms of many interacting Brownian
bridges, which is a well-known manipulation since the early 1970s and builds
on Feynman’s intuition from 1953. We refer to [ACK11, Lemma 2.1] for a proof
and further details for the following, and to [U06b] for a non-technical and
physics-related introduction to the rewrite of the Bose gas in terms of
Brownian bridges.

Using well-known trace formulas [G70], (2.1) can be rewritten in probabilistic
terminology using Brownian bridges as

$\mathbf{Z}_{N}^{{\scriptscriptstyle{({\rm
bc}})}}(\beta,\Lambda)=\frac{1}{N!}\sum_{\sigma\in\mathfrak{S}_{N}}\int_{\Lambda^{N}}{\rm
d}x_{1}\cdots{\rm
d}x_{N}\bigotimes_{i=1}^{N}{\mu_{x_{i},x_{\sigma(i)}}^{{\scriptscriptstyle{({\rm
bc,\beta}})}}}\Big{[}\exp\Big{\\{}-\sum_{1\leq i<j\leq
N}V(B^{{\scriptscriptstyle{({i}})}},B^{{\scriptscriptstyle{({j}})}})\Big{\\}}\Big{]}.$
(2.2)

Next, it is well-known that each permutation $\sigma\in\mathfrak{S}_{N}$ can
be decomposed into cycles – that is, each $\sigma$ can be represented by a
partition of $N$. For each $\sigma$ there is an $l\in{\mathfrak{P}}_{N}$ such
that $\sigma$ consists of precisely $l_{k}$ cycles of length $k$, for any $k$.
Using the Markov property of the Brownian motion (or, equivalently, the
semigroup property of the family of Brownian bridge measures), we can identify
each sequence of legs that lie within one cycle of length $k$ as one Brownian
bridge starting and ending at the same site and having the time interval
$[0,\beta k]$. Hence, for any $k\in\mathbb{N}$, $\sigma$ gives rise to $l_{k}$
bridges of length $k$, labeled

$B^{{\scriptscriptstyle{({k,1}})}},\dots,B^{{\scriptscriptstyle{({k,l_{k}}})}},$
(2.3)

and any bridge $B^{{\scriptscriptstyle{({k,i}})}}$ has the legs

$B^{{\scriptscriptstyle{({k,i}})}}_{j}=(B_{j}^{{\scriptscriptstyle{({k,i}})}}(s))_{s\in[0,\beta]}=(B^{{\scriptscriptstyle{({k,i}})}}((j-1)\beta+s))_{s\in[0,\beta]}.$
(2.4)

Then, writing $[m]=\\{1,\dots,m\\}$, our ensemble of legs is the collection

$\bigg{\\{}(B_{j}^{{\scriptscriptstyle{({k,i}})}})_{k,i,j}\colon
k\in[N],\,\,i\in[l_{k}],\,\,j\in\ [k]\bigg{\\}}$

sampled under the measure
$\bigotimes_{k\in\mathbb{N}}(|\Lambda|\mu_{\Lambda}^{{\scriptscriptstyle{({{\rm
bc},k\beta}})}})^{\otimes l_{k}}$. Furthermore, observe that, for any
$l\in\mathfrak{P}_{N}$, the number of permutations of $1,\dots,N$ such that
$l_{k}$ is the number of its cycles of length $k$ for any $k\in\mathbb{N}$ is
equal to $N!/\prod_{k\in\mathbb{N}}l_{k}!\,k^{l_{k}}$ [C02, Th. 12.1]. As a
result, the partition function of the interacting Bose gas may be written

$\sum_{l\in\mathfrak{P}_{N}}\Big{(}\prod_{k\in\mathbb{N}}\frac{|\Lambda|^{l_{k}}}{l_{k}!\,k^{l_{k}}}\Big{)}\Big{[}\bigotimes_{k\in\mathbb{N}}\big{(}\mu_{\Lambda}^{{\scriptscriptstyle{({{\rm
bc},k\beta}})}}\big{)}^{\otimes l_{k}}\Big{]}\big{[}{\rm
e}^{-{G}_{N,\beta}}\big{]},$ (2.5)

where the entire interaction can be written as

${G}_{N,\beta}=\sum_{(k_{1},i_{1},j_{1})\neq(k_{2},i_{2},j_{2})}V\big{(}B^{{\scriptscriptstyle{({k_{1},i_{1}}})}}_{j_{1}},B^{{\scriptscriptstyle{({k_{2},i_{2}}})}}_{j_{2}}\big{)}.$
(2.6)

Hence, in the standard form of the interacting Bose gas, every leg of every
bridge interacts with any other leg of a bridge. If we drop all interactions
between two legs of different bridges, that is, if we replace ${G}_{N,\beta}$
on the right-hand side of (2.5) by the interaction

$H_{N,\beta}=\sum_{k=1}^{N}\sum_{i=1}^{l_{k}}\sum_{1\leq j_{1}<j_{2}\leq
k}V\big{(}B^{{\scriptscriptstyle{({k,i}})}}_{j_{1}},B^{{\scriptscriptstyle{({k,i}})}}_{j_{2}}\big{)},$
(2.7)

then, for this interaction, the expectation over the product of the
$\mu_{\Lambda}^{{\scriptscriptstyle{({{\rm bc},k\beta}})}}$ decomposes into a
product of expectations, and we see that in (2.5), the partition function of
our model, $Z_{N}^{{\scriptscriptstyle{({\rm bc}})}}(\beta,\Lambda)$, defined
in (1.3) arises. This is our motivation to study this model. Although
different cycles do not interact and it is a simplification compared to the
scenario of the interacting Bose gas, the self-interactions within each cycle
already manifest into an interesting phase transition, as highlighted by
Theorem 1.1 and Theorem 1.2, confirming the conjecture that we will now be
discussing in Section 2.2.

###  2.2. Self-repellent random motions.

In our model (1.3) of a partially interacting Bose gas, it turned out in
Theorem 1.1 that the occurrence of a condensation phase transition is
equivalent to the summability of $\lambda_{\rm c}(\beta)^{k}\Gamma_{k}$ on
$k\in\mathbb{N}$. This in turn is equivalent to the finiteness of the Green’s
function at zero for the corresponding self-repellent random path model in the
critical point. This model is of the type of the weakly self-avoiding walk,
however, for a Brownian bridge instead of a free (i.e., open-end) path, and
with a different type of mutually repellent interaction, which has not yet
been studied in the literature on self-interacting path models. Indeed, the
interaction $\sum_{1\leq i<j\leq k}V(B_{i},B_{j})$ for a Brownian motion $B$
is in the same spirit as the interaction $H_{k}=\sum_{1\leq i<j\leq
k}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{S_{i}=S_{j}\\}$
(the self-intersection local time) for a random walk
$(S_{i})_{i\in\mathbb{N}_{0}}$ in $\mathbb{Z}^{d}$, since $V>0$. The two
differences between these two models are that (1) precise intersections do
almost surely not appear for the Brownian-motion model, and (2) the entire leg
$B_{i}\colon[0,\beta]\to\mathbb{R}^{d}$ is involved in the interaction, not
only its endpoint $B_{i}(\beta)$. But we find it obvious that both models
should lie in the same universality class of transformed path measures. Hence
let us briefly summarize what is known for the weakly self-avoiding walk.

See [S11] for a brief survey on the weakly self-avoiding walk, the random walk
measure on $k$-step nearest-neighbour paths $(S_{0},S_{1},\dots,S_{k})$ in
$\mathbb{Z}^{d}$ starting from the origin with probability weight
$\exp\\{-\alpha\sum_{1\leq i<j\leq
k}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{S_{i}=S_{j}\\}\\}$
and partition function $Z_{k}$. We restrict to dimensions $d\geq 2$. The main
conjecture is that the expected end-to end distance $|S_{k}-S_{0}|$ after $k$
steps should be of size $\approx k^{\nu_{d}}$, with
$\nu_{d}\in[\frac{1}{2},1]$ a critical power that is believed to be
$\nu_{2}=\frac{3}{4}$, $\nu_{3}\approx 0.5876$, and $\nu_{d}=\frac{1}{2}$ in
$d\geq 4$, with logarithmic corrections in $d=4$. Furthermore, $Z_{k}$ is
conjectured to be $\asymp\lambda^{-k}k^{\mu_{d}-1}$ for some $\mu_{d}$. The
powers $\nu_{d}$ and $\mu_{d}$ are believed to be universal, i.e., not to
depend on details of the underlying random walk. Their conjectured values are
$\mu_{2}=\frac{43}{32}$ and $\mu_{3}\approx 1.1568$ and $\mu_{4}=1$ and
$\mu_{d}=3-d$ for $d\geq 5$. This implies that the partition function in the
critical point should satisfy $Z_{k}\approx\lambda_{\rm c}^{-k}k^{\mu_{d}-1}$
as $k\to\infty$, where $\lambda_{\rm c}$ is the critical parameter (the
connective constant, defined analogously to (1.13)).

Conjectures about the finiteness of the Green’s function in the critical point
can be deduced from the above conjectures. Recall that the prediction is that
$|S_{k}|\approx k^{\nu_{d}}$ and $Z_{k}\approx\lambda^{-k}k^{\mu_{d}-1}$ as
$k\to\infty$, where we remark that $Z_{k}$ is the experctation of ${\rm
e}^{-\alpha H_{k}}$ for the random walk with a free end, while $\Gamma_{k}$ is
the same for the random walk bridge. Assuming even that $S_{k}k^{-\nu_{d}}$
converges in distribution under the (free-end) weakly self-avoiding walk model
towards some non-degenerate variable (and neglecting the logarithmic
corrections that are expected in $d=4$), we would then obtain that

$\Gamma_{k}\lambda^{k}\approx
k^{\mu_{d}-1}\frac{1}{Z_{k}}\mathbb{E}\big{[}{\rm e}^{-\alpha
H_{k}}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{S_{k}k^{-\nu_{d}}\in{\rm
d}x\\}}\big{]}/{\rm d}x|_{x=0}\approx k^{\mu_{d}-1-d\nu_{d}}.$

That is, if the right-hand side is summable, then the model with bridge
instead of free end should have a finite Green’s function in the critical
point. This is the case precisely if $\mu_{d}<d\nu_{d}$. Surprisingly, this
should be true in any dimension $d\in\\{2,3,4\\}$, also for $d=2$.

Hence, we conjecture that the condensation phase transition appears in our
model in (1.3) also in $d=2$. This is surprising, since in the original
interacting quantum Bose gas (see Section 2.1) it is generally conjectured
that the phase transition exists only in dimensions $d\geq 3$, but not in
$d=2$. For the two-dimensional free (i.e., non-interacting) quantum Bose gas,
the non-existence of the phase transition relies on the non-summability of
$1/k$, which is ‘just at the boundary’ of the validity of this statement.

###  2.3. Lace expansion.

Let us comment on the method that we use in the proof of Theorem 1.2, the lace
expansion. This is a diagrammatic expansion of the exponential interaction
term ${\rm e}^{-\alpha\sum_{1\leq i<j\leq
k}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{S_{i}=S_{j}\\}}$
in terms of a procedure that reminds of the inclusion-exclusion principle from
basic probability theory. Since diagrammatic illustrations of the method
remind of the laces on the edge of a tablecloth, this method got its name
“lace expansion”. It was introduced in the mid-eighties in statistical
mechanics and was first applied to the (weakly) self-avoiding walk in [HS92]
and was several times extended and improved, see [vdHS02, HvdHS03] and, more
recently [S22].

The method is perturbative and needs a simple, but non-trivial model to expand
around, which is most often simple random walk. Since (at least conjecturally)
the weakly self-avoiding walk is similar to the simple random walk only in
dimensions $d\geq 5$, the method has been successful only in these dimensions,
and only for sufficiently small values of $\alpha$. For these values of
$\alpha$, rather strong assertions could be proved, among which are the
following: the endpoint satisfies a central limit theorem (even a local one),
i.e., the distribution of $S_{k}/\sqrt{k}$ under the transformed measure
converges towards a Gaussian distribution with an appropriate non-trivial
variance. Furthermore, the Green’s function is finite in the critical
parameter value $\lambda_{\rm c}$.

Besides many other extensions, in recent years the lace expansion was extended
to models in continuous space, where self-intersections almost surely do not
occur. In [ABR16], which gave us the leading idea for the proof in the present
paper, the following model is successfully handled in this vein. Here the
energy $V(f,g)$ defined in (1.6) is replaced by

$V_{\rm
SAW}(f,g)=\alpha{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{|f(\beta)-g(\beta)|\leq
r\\},\qquad f,g\in{\mathcal{C}},$ (2.8)

where $r,\alpha>0$ are parameters, with a free end of the motion. Then the
model is the weakly self-avoiding walk for a Gaussian random walk on
$\mathbb{R}^{d}$, whose steps have mean zero and variance $\beta$. In [ABR16],
it was proved that the local central limit theorem holds:

$\lim_{k\to\infty}\frac{1}{Z_{k}}\mathbb{E}_{0}\Big{[}{\rm
e}^{-\alpha\sum_{1\leq i<j\leq
k}V(S_{i},S_{j})}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{k^{-1/2}S_{k}\approx
x\\}\Big{]}=g_{D}(0,x),\qquad x\in\mathbb{R}^{d}\setminus U,$ (2.9)

for sufficiently small $\alpha>0$ and some neighbourhood $U$ of $0$, where
$Z_{k}$ is the partition function, $g_{D}$ is the Gaussian transition kernel
with $D=D_{\alpha}>0$ a suitable diffusion constant, and “$\approx$” needs to
be adjusted to a proper transition from discrete to continuous space. Along
the proof, it was also shown that $Z_{k}\sim C\lambda^{-k}$ for some
$C,\lambda>0$, implying of course that $\lambda=\lambda_{\rm c}(\gamma)$ is
the critical parameter. For $x=0$, we have only “$\leq$” in (2.9), which says
that $\Gamma_{k}\leq O(\lambda_{k}^{-k}k^{-d/2})$, which implies that
$\Gamma_{k}\lambda^{k}$ is summable, i.e., the Green’s function in the
critical point is finite. This implies that Theorem 1.2 holds for this model
for sufficiently small interaction parameter $\alpha$.

The novelty of the present paper is to extend it to the interaction $V$
defined in (1.6). This is a substantial non-trivial step since the interaction
depends now on the entire leg $B_{i}$, not only on its endpoint
$B_{i}(\beta)$. The new ideas needed in the present context, for which we will
draw on the lace expansion method using Banach algebras introduced in [BHK18],
will be discussed along the way of proving Theorem 1.2.

###  2.4. Random permutations.

The model that we study in this paper is a probability measure on the set
$\mathfrak{P}_{N}$ of all integer partitions of $N$ of the form

${\tt
P}_{N}(l)=\frac{1}{Z_{N}}\prod_{k\in\mathbb{N}}\frac{[\theta^{{\scriptscriptstyle{({N}})}}_{k}]^{l_{k}}}{l_{k}!\,k^{l_{k}}},\qquad
l=(l_{k})_{k\in\mathbb{N}}\in\mathfrak{P}_{N}.$ (2.10)

Here $Z_{N}$ is the normalisation constant that turns ${\tt P}_{N}$ into a
probability measure, and
$\theta_{k}^{{\scriptscriptstyle{({N}})}}\in(0,\infty)$ are positive weights.
Such measures are also known as models of random permutations, i.e., as a
probability measure on the set $\mathfrak{S}_{N}$ of all permutations $\sigma$
of $1,\dots,N$, giving the weight $\theta_{k}^{{\scriptscriptstyle{({N}})}}$
to any cycle of $\sigma$ with length $k$, independently over all cycles. Note
that one can freely multiply $\theta_{k}^{{\scriptscriptstyle{({N}})}}$ by a
factor ${\rm e}^{\alpha k}$ for fixed $k$ without changing ${\tt P}_{N}$,
since this factor drops out in the normalisation.

For $\theta_{k}^{{\scriptscriptstyle{({N}})}}=\theta_{k}$ not depending on
$N$, such distributions have been studied in recent years for many choices of
the weights $\theta_{k}$, and various limiting regimes and limiting
distributions have been identified and studied in great detail, producing a
rich picture. This line of research goes under the name random (spatial resp.
geometric) permutations and was initiated in [BU09] as a simplified model for
the interacting Bose gas (it appears after interchanging the logarithm with a
certain sum in the partition function). See [RZ20] for one of the latest
contributions and a host of references on this line of research, which has
departed considerably from the spirit of the interacting Bose gas in the last
decade.

The weights that we study in this paper, are roughly of the form

$\theta_{k}^{{\scriptscriptstyle{({N}})}}=N(\gamma_{N,k}+o(1)),\qquad
N\to\infty,k\in\mathbb{N},$ (2.11)

with some $\gamma_{N,k}\in(0,\infty)$, whose behaviour as $N\to\infty$ is a
priori not easy to determine. Indeed, for fixed $k$, it converges, as
$N\to\infty$ towards $\gamma_{k}=\frac{1}{\rho}\Gamma_{k}$ defined in (1.12),
and this is also true for $k\ll N^{2/d}$, but for $k\gg N^{2/d}$, other
effects emerge in the asymptotics, and a proof of the necessary asymptotics
for deriving an LDP are not so easy to get, and actually in the case of
periodic boundary condition we do not. Note that $\Gamma_{k}$ is exponentially
fast in $k$, but this exponential terms drops out of the question about the
phase transition, and the second term is decisive. For the free Bose gas,
$\gamma_{k}=\frac{1}{\rho}(2\pi\beta k)^{-d/2}$ with $\rho,\beta\in(0,\infty)$
are the particle density and inverse temperature.

###  2.5. Influence of boundary conditions.

The technical difficulties in deriving an LDP in Theorem 1.1 stem from the
precise form of $\gamma_{N,k}$ in (2.11), which depend on the boundary
conditions that we use in our interacting Bose gas model.

The by far most often used condition (in the standard interacting Bose gas and
in practically all variants) is the periodic one, which is a mathematical
idealisation that gives simpler formulas and less technicalities. Many papers
on the free Bose gas (e.g. [Sü93, Sü02, BCMP05]) concentrate either on this
choice or on the even simpler free boundary conditions, where the starting and
termination sites of the bridges stay inside the box, but not necessarily the
entire path (this does not come from a quantum mechanical model). However, for
zero boundary condition, (2.11) is actually not satisfied in the free Bose gas
for $k$ coupled with $N$, since for $k\gg N^{2/d}$, the probability to stay in
the box with diameter $\asymp N^{1/d}$ decays to zero exponentially in
$kN^{-2/d}$. To the best of our knowledge, this seems to be a fact that has
not been noticed nor handled in the study of the free Bose gas before 2023,
even though this case is highly relevant from the physics point of view. In
the recent paper [KVZ23], this gap is closed by deriving asymptotics for the
reduced one-particle matrix of the free Bose gas for all these well-known
boundary conditions, also including Neumann conditions, to prove off-diagonal
long-range order, which is considered to be the equivalent to the occurrence
of BEC.

In our present paper, however, the technicalities arising in the proof of the
LDP turned out serious for the case of periodic boundary condition and we
decided to complete the proof only for Dirichlet boundary condition. Indeed,
we were not able to prove the upper bound
$\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm per}})}}\leq\lambda_{\rm
c}(\beta)^{k}(1+\varepsilon)^{k}$ for any $k\in\\{1,\dots,N\\}$ and all
sufficiently large $N$, precisely for the reason mentioned above (the
asymptotics for $k\gg N^{2/d}$ are difficult to handle). We elaborate on this
point in Remark 1 towards the end of Section 3.1.

##  3\. Proof of Theorem 1.1

The proof of Theorem 1.1 is split into three steps: In Section 3.1 we will
show that the limiting free energy exists as $N\to\infty$ and is independent
of the Dirichlet boundary conditions imposed. The assertions (i) and (ii) of
Theorem 1.1 will be shown in Section 3.2, while assertions (iii) and (iv) will
be proved in Section 3.3.

###  3.1. Approximation of $\Gamma^{{\scriptscriptstyle{({\rm
bc}})}}_{\Lambda_{N},k}$ with $\Gamma_{k}$.

Recall from (1.12) and from (1.5) that

$\displaystyle\Gamma_{k}$ $\displaystyle=$
$\displaystyle\mathbb{E}_{0}\Big{[}{\rm
e}^{-H_{k}(B)}\,{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{B(k\beta)\in{\rm
d}x\\}\Big{]}\Big{/}{\rm d}x\Big{|}_{x=0},$ (3.1)
$\displaystyle\Gamma^{{\scriptscriptstyle{({\rm Dir}})}}_{\Lambda_{N},k}$
$\displaystyle=$ $\displaystyle\frac{1}{|\Lambda_{N}|}\int_{\Lambda_{N}}{\rm
d}x\,\mathbb{E}_{x}\Big{[}{\rm
e}^{-H_{k}(B)}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{B(s)\in\Lambda_{N}\,\forall
s\in[0,\beta
k]\\}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{B(k\beta)\in{\rm
d}x\\}\Big{]}\Big{/}{\rm d}x,$ (3.2)

where we put

$H_{k}(B)=\sum_{1\leq i<j\leq k}V(B_{i},B_{j})=\sum_{1\leq i<j\leq
k}\int_{0}^{\beta}{\rm d}s\,v\big{(}|B_{i}(s)-B_{j}(s)|\big{)}$

and recall that $B_{i}=(B((i-1)+s))_{s\in[0,\beta]}$ is the $i$-th leg of the
Brownian motion $B$ on $[0,k\beta]$.

In our proof of the LDP in Section 3.2 below, we will need upper bounds for
$\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm Dir}})}}$ in terms of
$\Gamma_{k}$ for any $k\in\\{1,\dots,N\\}$, but lower bounds only for any
$k\leq k_{N}$ for some $k_{N}\to\infty$. In this section, we formulate and
prove these assertions in (3.3) and (3.4).

For convenience, we put
$\Lambda_{N}=[-\frac{1}{2}L_{N},\frac{1}{2}L_{N})^{d}$. For
$x\in\mathbb{R}^{d}$, we define $z(x)\in\mathbb{Z}^{d}$ by $x\in z(x)L_{N}$
and $[x]=x-z(x)L_{N}\in\Lambda_{N}$.

First we note that

$\Gamma_{k}\geq\Gamma^{{\scriptscriptstyle{({\rm
Dir}})}}_{\Lambda_{N},k},\qquad N,k\in\mathbb{N},$ (3.3)

which follows from dropping the indicator on the event
$\\{B(s)\in\Lambda_{N}\,\forall s\in[0,\beta k]\\}$ in (3.2) and noting the
shift-invariance of $H_{k}(B)$.

We now turn to a lower bound for $\Gamma^{{\scriptscriptstyle{({\rm
Dir}})}}_{\Lambda_{N},k}$ for $k$’s that are fixed or diverging slowly. Pick
some integer $k_{N}\to\infty$ such that $k_{N}\leq o(N^{1/d})$. Then we claim
that for any $\varepsilon\in(0,1)$, there is $N_{0}$ such that

$\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
Dir}})}}\geq\Gamma_{k}(1-\varepsilon),\qquad N\geq
N_{0},k\in\\{1,\dots,k_{N}\\}.$ (3.4)

Indeed, with a small $\eta\in(0,1)$, split the integration in (3.2) into
$(1-\eta)\Lambda_{N}$ and $\Lambda_{N}\setminus(1-\eta)\Lambda_{N}$ to see
that

$\displaystyle\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm Dir}})}}$
$\displaystyle\geq\Gamma_{k}-\frac{1}{|\Lambda_{N}|}\int_{(1-\eta)\Lambda_{N}}{\rm
d}x\,\mu_{x,x}^{{\scriptscriptstyle{({\beta k}})}}\Big{[}{\rm
e}^{-H_{k}(B)}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{B(s)\in\Lambda_{N}^{\rm
c}\text{ for some }s\in[0,\beta k]\\}\Big{]}$
$\displaystyle\qquad-\frac{1}{|\Lambda_{N}|}\int_{\Lambda_{N}\setminus(1-\eta)\Lambda_{N}}{\rm
d}x\,\mu_{x,x}^{{\scriptscriptstyle{({\beta k}})}}\Big{[}{\rm
e}^{-H_{k}(B)}\Big{]}$ $\displaystyle\geq\Gamma_{k}-{\rm
e}^{-CN^{2/d}/k}-\frac{|\Lambda_{N}\setminus(1-\eta)\Lambda_{N}|}{|\Lambda_{N}|}\Gamma_{k}$
$\displaystyle\geq\Gamma_{k}-{\rm e}^{-CN^{2/d}/k}-C\eta\Gamma_{k},$

where we first used an estimate for the probability that a Brownian bridge on
the time interval $[0,k\beta]$ reaches a site that is $\eta N^{1/d}$ away
(with some suitable $C>0$, not depending on $N$) and then used the maximal
growth of $k$. Now use that $\Gamma_{k}=\lambda_{\rm c}(\beta)^{-k(1+o(1))}$
has a nontrivial exponential behaviour in $k$ to see that the latter bound can
be lower estimated against $\Gamma_{k}(1-\varepsilon)$ for any $N\geq N_{0}$,
if $\eta$ and $N_{0}$ are picked suitably. This proves (3.4).

###### Remark 1

For the periodic boundary conditions, we are not able to derive the
corresponding bounds for the following reason: Indeed, by shift-invariance of
the interaction

$H_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
per}})}}(B)=\sum_{z\in\mathbb{Z}^{d}}\sum_{1\leq i<j\leq
k}\int_{0}^{\beta}v\big{(}|B_{i}(s)-B_{j}(s)+zL_{N}|\big{)}\,{\rm d}s,$

we see that

$\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
per}})}}=\sum_{z\in\mathbb{Z}^{d}}\mathbb{E}_{0}\Big{[}{\rm
e}^{-H_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
per}})}}(B)}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}\\{B(k\beta)\in{\rm
d}y\\}\Big{]}\Big{|}_{y=zL_{N}}.$

Now we can invoke the lower bound
$\Gamma_{\Lambda,k}^{{\scriptscriptstyle{({\rm
per}})}}\geq\Gamma_{\Lambda,k}^{{\scriptscriptstyle{({\rm Dir}})}}$ and use
our lower-bound proof for the latter. We can also estimate
$H_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm per}})}}(B)\geq H_{k}(B)$.
However, the problem that we encounter now is to upper-estimate the
expectation density on $\\{B(k\beta)\in L_{N}\mathbb{Z}^{d}\\}$ against the
one on $\\{B(k\beta)=0\\}$. In fact, this seems to be a rather deep issue,
since the behavior of the self-repellent walk depends on the spread of the
path in a complicated manner that is currently not understood. However, under
the assumptions that we imposed in Theorem 1.2, an extension of that proof
might very well be sufficient to derive suitable asymptotics, but we have
refrained from pursuing this here.

###  3.2. Proof of the LDP.

In this section we will prove assertions (i) and (ii) of Theorem 1.1. Recall
the set of partitions,
$\mathfrak{P}_{N}=\\{l=(l_{k})_{k\in\mathbb{N}}\in\mathbb{N}_{0}^{\mathbb{N}}\colon\sum_{k}kl_{k}=N\\}$
and the state space
$\mathcal{X}_{\rho}=\\{p\in[0,\infty)^{\mathbb{N}}\colon\sum_{k}kp_{k}\in[0,\rho]\\}$.
We are going to prove that, for any $p\in\mathcal{X}_{\rho}$,

$\lim_{\delta\downarrow 0}\lim_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log{\tt
P}^{{\scriptscriptstyle{({N}})}}_{\beta,\Lambda_{N}}\Big{(}(L_{k}/|\Lambda_{N}|)_{k\in\mathbb{N}}\in
B_{\delta}(p)\Big{)}=-J(p),$ (3.5)

where the rate function $J(p)$ is introduced in (1.16), and $B_{\delta}(p)$ is
the $\delta$-ball around $p$ in some metric that induces the product topology
on $[0,\infty)^{\mathbb{N}}$. Since the state space $\mathcal{X}_{\rho}$ is
compact, from (3.5) the LDP in (ii) and the identification of the free energy
in (i) of Theorem 1.1 follow along some standard arguments from LDP theory.

Fix $p\in\mathcal{X}_{\rho}$. From a combinatorial argument [A76] it follows
that $\\#\mathfrak{P}_{N}={\rm e}^{o(N)}$ as $N\to\infty$. Hence, the upper
bound in (3.5) will follow from

$\displaystyle\limsup_{\delta\downarrow
0}\limsup_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\bigg{(}\sup_{l\in\mathfrak{P}_{N}\colon
l/|\Lambda_{N}|\in
B_{\delta}(p)}\prod_{k\in\mathbb{N}}\frac{\big{[}|\Lambda_{N}|\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
bc}})}}\big{]}^{l_{k}}}{l_{k}!k^{l_{k}}}\bigg{)}$ (3.6)
$\displaystyle\leq-I(p)-\Big{(}\rho-\sum_{k\in\mathbb{N}}kp_{k}\Big{)}\log\lambda_{\rm
c}(\beta),$

and the lower bound will follow from

$\displaystyle\liminf_{\delta\downarrow
0}\liminf_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\bigg{(}\prod_{k\in\mathbb{N}}\frac{\big{[}|\Lambda_{N}|\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
bc}})}}\big{]}^{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}}}{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}!k^{l_{k}^{{\scriptscriptstyle{({\delta,N}})}}}}\bigg{)}$
$\displaystyle\geq-I(p)-\Big{(}\rho-\sum_{k\in\mathbb{N}}kp_{k}\Big{)}\log\lambda_{\rm
c}(\beta),$ (3.7)

for some $l^{{\scriptscriptstyle{({\delta,N}})}}\in B_{\delta}(p)$. Then
combining (3.6) and (3.7) will imply (3.5).

Proof of (3.6): Pick $\delta>0$ and $l\in\mathfrak{P}_{N}$ such that
$l/|\Lambda_{N}|\in B_{\delta}(p)$. We need a large auxiliary parameter
$L\in\mathbb{N}$. We split the product on $k$ on the left-hand side of (3.6)
into the products on $k\leq L$ and on $k>L$. We substitute

$p_{k}^{{\scriptscriptstyle{({N,L}})}}=\frac{l_{k}}{|\Lambda_{N}|},\qquad
k\in\\{1,\dots,L\\}.$ (3.8)

Using Stirling’s formula and (3.3)-(3.4), the first partial product can be
asymptotically identified as

$\displaystyle\prod_{k=1}^{L}\Big{[}\frac{1}{l_{k}!}\Big{(}\frac{1}{k}|\Lambda_{N}|\Gamma_{\Lambda_{n},k}^{{\scriptscriptstyle{({\rm
bc}})}}\Big{)}^{l_{k}}\Big{]}$ $\displaystyle={\rm
e}^{o(|\Lambda_{N}|)}\prod_{k=1}^{L}\frac{(\Gamma_{k}/k)^{p^{{\scriptscriptstyle{({N,L}})}}_{k}|\Lambda_{N}|}}{(p^{{\scriptscriptstyle{({N,L}})}}_{k}/{\rm
e})^{p^{{\scriptscriptstyle{({N,L}})}}_{k}|\Lambda_{N}|}}$ (3.9)
$\displaystyle={\rm
e}^{o(|\Lambda_{N}|)}\exp\Big{\\{}-|\Lambda_{N}|I_{L}(p^{{\scriptscriptstyle{({N,L}})}})\Big{\\}},$

where $I_{L}(p)=\sum_{k=1}^{L}p_{k}\log\frac{p_{k}k}{\Gamma_{k}{\rm e}}$ is
the “$L$-cutoff” version of the rate function $I$ given in (1.14). Hence, we
see from the preceding argument that

$\displaystyle\limsup_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\bigg{(}\sup_{l\in\mathfrak{P}_{N}\colon
l/|\Lambda_{N}|\in
B_{\delta}(p)}\prod_{k=1}^{L}\frac{\big{[}|\Lambda_{N}|\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
bc}})}}\big{]}^{l_{k}}}{l_{k}!k^{l_{k}}}\bigg{)}$
$\displaystyle\leq-\inf_{\widetilde{p}\in B_{\delta}(p)}I_{L}(\widetilde{p}).$
(3.10)

Now the product on $k>L$ on the left-hand side of (3.6) is decomposed into the
product of the $\Gamma$-terms and the remainder. Indeed, we are going to show,
for any $\delta\in(0,1)$,

$\limsup_{L\to\infty}\limsup_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\bigg{(}\sup_{l\in\mathfrak{P}_{N}\colon
l/|\Lambda_{N}|\in
B_{\delta}(p)}\prod_{k>L}\Big{[}\frac{1}{l_{k}!}\Big{(}\frac{1}{k}|\Lambda_{N}|\Big{)}^{l_{k}}\Big{]}\bigg{)}\leq
0.$ (3.11)

Concerning the remainder product of the $\Gamma$-terms, we will show that

$\displaystyle\limsup_{\delta\downarrow
0}\limsup_{L\to\infty}\limsup_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\bigg{(}\sup_{l\in\mathfrak{P}_{N}\colon
l/|\Lambda_{N}|\in
B_{\delta}(p)}\prod_{k>L}\big{[}\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\mathrm{bc}}})}}\big{]}^{l_{k}}\bigg{)}$
$\displaystyle\leq-\Big{(}\rho-\sum_{k\in\mathbb{N}}kp_{k}\Big{)}\log\lambda_{\rm
c}(\beta).$ (3.12)

For proving (3.11), we use the Stirling bound

$\Big{(}\frac{l}{{\rm
e}}\Big{)}^{-l}c\frac{1}{\sqrt{l}}\leq\frac{1}{l!}\leq\Big{(}\frac{l}{{\rm
e}}\Big{)}^{-l}C\frac{1}{\sqrt{l}}\qquad l\in\mathbb{N},$

for some $c,C\in(0,\infty$. Then we see that the left-hand side of (3.11) can
be bounded as follows.

$\displaystyle\frac{1}{|\Lambda_{N}|}\log\prod_{k>L}\Big{[}\frac{1}{l_{k}!}\Big{(}\frac{1}{k}|\Lambda_{N}|\Big{)}^{l_{k}}\Big{]}\leq\frac{\rho}{N}\sum_{k>L}l_{k}\log\frac{N{\rm
e}}{\rho kl_{k}}+\frac{1}{N}\rho\log C$

Now use that
$\sum_{k>L}\frac{l_{k}}{N}\leq\frac{1}{L}\sum_{k>L}\frac{kl_{k}}{N}\leq\frac{1}{L}$
and apply Jensen’s inequality to the logarithm and the sum over
$l_{k}/\sum_{n>L}l_{n}$, to see that (3.11) holds.

Let us turn to a proof of (3.12). Recall from (1.13) that
$\lim_{k\to\infty}\Gamma_{k}^{-1/k}=\lambda_{\rm c}(\beta)$. Pick an
$\varepsilon>0$, then we may assume that $L$ is so large that
$\Gamma_{k}\leq(\lambda_{\rm c}(\beta)^{-1}+\varepsilon)^{k}$ for any $k>L$.
Recall that by (3.3) we have
$\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm Dir}})}}\leq\Gamma_{k}$.
Thus we can bound the left-hand side of (3.12) as follows, for any
$l\in\mathfrak{P}_{N}$ and for $N>N_{0}$.

$\displaystyle\frac{1}{|\Lambda_{N}|}\log\prod_{k>L}\big{[}\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\mathrm{bc}}})}}\big{]}^{l_{k}}$
$\displaystyle\leq\frac{1}{|\Lambda_{N}|}\sum_{k>L}l_{k}\log\Big{(}\big{[}(\lambda_{\rm
c}(\beta)^{-1}+\varepsilon)\big{]}^{k}\Big{)}$ (3.13)
$\displaystyle\leq\log\big{[}(\lambda_{\rm
c}(\beta)^{-1}+\varepsilon)\big{]}\sum_{k>L}\frac{kl_{k}}{|\Lambda_{N}|}$
$\displaystyle=\log\big{[}(\lambda_{\rm
c}(\beta)^{-1}+\varepsilon)\big{]}\Big{(}\rho-\sum_{k=1}^{L}kp_{k}^{{\scriptscriptstyle{({N,L}})}}\Big{)},$

with $p_{k}^{{\scriptscriptstyle{({N,L}})}}$ as in (3.8).

Collecting (3.10), (3.11) and (3.13), we see that, for any $L\in\mathbb{N}$
and $\delta,\varepsilon\in(0,1)$,

$\displaystyle\limsup_{N\to\infty}\frac{1}{|\Lambda_{N}|}$
$\displaystyle\log\sup_{l\in\mathfrak{P}_{N}\colon l/|\Lambda_{N}|\in
B_{\delta}(p)}\prod_{k\in\mathbb{N}}\frac{\big{[}|\Lambda_{N}|\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
bc}})}}\big{]}^{l_{k}}}{l_{k}!k^{l_{k}}}$ (3.14)
$\displaystyle\leq-\inf_{\widetilde{p}\in
B_{\delta}(p)}I_{L}(\widetilde{p})+\Big{(}\rho-\inf_{\widetilde{p}\in
B_{\delta}(p)}\sum_{k=1}^{L}k\widetilde{p}_{k}\Big{)}\log\big{[}(\lambda_{\rm
c}(\beta)^{-1}+\varepsilon)\big{]}.$

Letting $\delta\downarrow 0$ and $\varepsilon\downarrow 0$, we obtain, for any
$L\in\mathbb{N}$,

$\limsup_{\delta\downarrow 0}\big{(}{\rm l.h.s.~{}of}\
\eqref{fastfertig}\big{)}\leq-
I_{L}(p)-\Big{(}\rho-\sum_{k=1}^{L}kp_{k}\Big{)}\log\lambda_{\rm c}(\beta).$

Passing to the limit $L\to\infty$, we complete the proof of (3.6). Now let us
turn to the

Proof of (3.7). Again, we pick a large auxiliary parameter $L$ and define
$l^{{\scriptscriptstyle{({\delta,N}})}}=(l^{{\scriptscriptstyle{({\delta,N}})}}_{k})_{k\in\\{1,\dots,N\\}}$
(also depending on $L$) by

$l^{{\scriptscriptstyle{({\delta,N}})}}_{k}=\begin{cases}\lfloor|\Lambda_{N}|p_{k}\rfloor&\mbox{for
}k\leq L,\\\ A_{p,L,N}&\mbox{for }k=k_{N}=\lfloor N^{1/2d}\rfloor,\\\
0&\mbox{otherwise,}\end{cases}$

were $A_{p,L,N}$ is picked such that
$\sum_{k=1}^{N}kl^{{\scriptscriptstyle{({\delta,N}})}}_{k}=|\Lambda_{N}|\rho$.
Then $l^{{\scriptscriptstyle{({\delta,N}})}}_{k}/|\Lambda_{N}|\to p_{k}$ as
$N\to\infty$ for $k\in\\{1,\dots,L\\}$, and
$l^{{\scriptscriptstyle{({\delta,N}})}}_{k_{N}}/|\Lambda_{N}|=A_{p,L,N}/|\Lambda_{N}|=(\rho-\sum_{k=1}^{L}kp_{k})(1+o(1))\frac{1}{k_{N}}$
as $N\to\infty$. We see that
$l^{{\scriptscriptstyle{({\delta,N}})}}/|\Lambda_{N}|\in B_{\delta}(p)$ for
$N$ and $L$ sufficiently large.

As in (3.9), we see that

$\lim_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\prod_{k=1}^{L}\frac{\big{[}|\Lambda_{N}|\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\rm
bc}})}}\big{]}^{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}}}{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}!k^{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}}}\geq-
I_{L}(p)$

Furthermore, it is easy to see that

$\liminf_{L\to\infty}\liminf_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\prod_{k>L}\Big{[}\frac{1}{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}!}\Big{(}\frac{1}{k}|\Lambda_{N}|\Big{)}^{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}}\Big{]}=0.$

Finally, usin the assertion around (3.4) (which holds for both bc$=$per and
bc$=$Dir) and recalling that $\Gamma_{k}=\lambda_{\rm c}(\beta)^{-k(1+o(1))}$,
we get that

$\liminf_{L\to\infty}\liminf_{N\to\infty}\frac{1}{|\Lambda_{N}|}\log\prod_{k>L}\big{[}\Gamma_{\Lambda_{N},k}^{{\scriptscriptstyle{({\mathrm{bc}}})}}\big{]}^{l^{{\scriptscriptstyle{({\delta,N}})}}_{k}}\geq-\Big{(}\rho-\sum_{k\in\mathbb{N}}kp_{k}\Big{)}\log\lambda_{\rm
c}(\beta).$

Putting together the last three displays, we see that (3.7) holds.

Hence, (3.5) has been proved, and this finishes the proof of the LDP in the
assertions (i) and (ii) of Theorem 1.1.

###  3.3. Analysis of the rate function

Now we prove assertions (iii) and (iv) of Theorem 1.1. It is clear that the
state space $\mathcal{X}_{\rho}$ is compact and that $J$ is continuous and
strictly convex on ${\mathcal{X}}_{\rho}$. Hence, it suffices to derive the
Euler–Lagrange equations for minimisers. Abbreviate $\alpha_{k}=k/\Gamma_{k}$
and $\alpha=(\alpha_{k})_{k}$, then $J(p)=\rho\log\lambda_{\rm
c}(\beta)+\langle p,\log(p\alpha/{\rm e})-{\rm id}\lambda_{\rm
c}(\beta)\rangle$ for any $p\in{\mathcal{X}}_{\rho}$. Assume that $p$ is a
minimizer for $J$. It is easy to see that $p_{k}>0$ for any $k$, since
otherwise adding a small weight would clearly make the $J$-value smaller.
Then, for any perturbator $\delta=(\delta_{k})_{k}\in\mathbb{R}^{\mathbb{N}}$,
we have

$0=\partial_{\varepsilon}\Big{|}_{\varepsilon=0}J(p+\varepsilon\delta)=\langle\delta,\log(p\alpha/{\rm
e})-{\rm id}\lambda_{\rm c}(\beta)\rangle+\langle
p,\textstyle{\frac{\delta}{p}}\rangle=\langle\delta,\log(p\alpha)-{\rm
id}\lambda_{\rm c}(\beta)\rangle.$ (3.15)

Let us assume that the minimizer $p$ satisfies $\sum_{k}kp_{k}=\rho$. Then we
may assume that the perturbator $\delta$ satisfies
$0=\sum_{k}k\delta_{k}=\langle\delta,{\rm id}\rangle$, and the right-hand side
of (3.15) is equal to $\langle\delta,\log(p\alpha)\rangle$. Hence, we have
derived that any $\delta$ that is perpendicular to the vector ${\rm
id}=(k)_{k}$ is also perpendicular to the vector $\log(p\alpha)$. Hence, there
is some constant $C$ such that $Ck=\log(p_{k}\alpha_{k})$ for any $k$, which
reads

$kp_{k}={\rm e}^{-Ck}\Gamma_{k},\qquad k\in\mathbb{N}.$ (3.16)

Now check the constraint $\sum_{k}kp_{k}=\rho$. In the first case, where
$\sum_{k}\Gamma_{k}\lambda_{\rm c}(\beta)^{k}$ is finite, the smallest $C$
that can ever make the sum of ${\rm e}^{-Ck}\Gamma_{k}$ finite is
$C=-\log\lambda_{\rm c}(\beta)$, and the largest $\rho$ that can ever be
realized by that sum is $\rho_{\rm c}(\beta)$. Substituting ${\rm
e}^{-Ck}=\lambda_{\rm c}(\beta)^{k}{\rm e}^{-c(\rho)k}$ we arrive at the
Euler-Lagrange equations in (1.18) with $c=c(\rho)$ as in the text below
(1.18). In the case where $\sum_{k}\Gamma_{k}\lambda_{\rm
c}(\beta)^{k}=\infty$, then any $\rho>0$ can be realized by that choice of
$C=c(\rho)+\log\lambda_{\rm c}(\beta)$.

Now, if the minimizer $p$ satisfies $\sum_{k}kp_{k}<\rho$, then the
perturbator $\delta$ is arbitrary, and from (3.15) we get that
$\log(p\alpha)-{\rm id}\lambda_{\rm c}(\beta)=0$, i.e., that (3.16) holds with
${\rm e}^{-Ck}$ replaced by $\lambda_{\rm c}(\beta)^{k}$, which proves (1.18)
as well. In particular, $\sum_{k}kp_{k}=\rho_{\rm c}(\beta)$. The formula in
(iv) is proved by substituting what we have derived.

##  4\. Proof of Theorem 1.2: Estimates on the Green function

This section is devoted to the proof of Theorem 1.2. Without loss of
generality, we are going to do the proof only for $\beta=1$ (otherwise,
consider a simple rescaled version of $v$).

In Section 4.1 we introduce some notation, formulate the main steps of the
proof and prove Theorem 1.2, subject to these steps. In Section 4.2 we prove
explain the main one of them, and prove it finally in Section 4.3.

###  4.1. Some notation, and the main steps

Recall that $\varphi_{t}(x)=(2\pi t)^{-d/2}\exp[-\frac{|x|^{2}}{2t}]$ denotes
the Gaussian kernel with variance $t$, and we write $\varphi:=\varphi_{1}$. We
define

$G(x):=\sum_{n=1}^{\infty}\varphi_{n}(x),\qquad x\in\mathbb{R}^{d},$ (4.1)

which is, up to the missing summand for $n=0$ (which is often formally written
as the delta function at $x$), equal to the Green function of the random walk
with standard normal increments. (It is not to be confused with the Green
function $G_{\rm c}(x)=\int_{0}^{\infty}\varphi_{t}(x)\,{\rm d}t$ for Brownian
motion. See Lemma 7.3 and its proof for some relation between them.) In spite
of the missing summand for $n=0$, we will call $G$ the Green function from
now. Note that $G(x)$ is finite in $d\geq 3$, and $G(\cdot)$ is rotational
symmetric, bounded, and infinitely often differentiable with all derivatives
in the set ${\mathcal{C}}_{0}(\mathbb{R}^{d})$ of continuous functions
$\mathbb{R}^{d}\to\mathbb{R}$ vanishing at infinity. Recall that $v$ is a
bounded continuous function $\mathbb{R}^{+}\to\mathbb{R}^{+}$ with compact
support, i.e., we assume that there exists $L,R>0$ such that $v(r)\leq L$ for
every $r,$ and $v(r)=0$ for $r\geq R.$

Let us introduce the main object of the proof, a version of the self-repellent
Brownian motion. Recall the Hamiltonian on ${\mathcal{C}}_{N}$:

$H_{N}(B):=\sum_{1\leq i<j\leq N}V(B_{i},B_{j}),\qquad\mbox{where}\qquad
V(f,g)=\int_{0}^{1}v(|f(s)-g(s)|)\,\mathrm{d}s\,\,\,\mbox{for}\,\,f,g\in\mathcal{C}_{1},$

where we recall from (1.7) that
$B_{j}=(B_{j}(s))_{s\in[0,1]}:=(B((j-1)+s))_{s\in[0,1]}$ is the $j$-th leg of
the Brownian motion $B$. Then $0\leq H_{N}(B)\leq LN$ since $0\leq v\leq L$.
Also recall that $\mu^{{\scriptscriptstyle{({1}})}}_{x,y}$ denotes the
canonical Brownian bridge measure in the time interval $[0,1]$. We write
$\mathbb{P}_{N}$ for the standard Wiener measure on $\mathcal{C}_{N}$. Then
the tilted measure ${\mathbb{Q}}_{\alpha,N}$ on $\mathcal{C}_{N}$ is defined
by

${\mathbb{Q}}_{\alpha,N}(\mathrm{d}B):=\mathrm{e}^{-\alpha
H_{N}(B)}\,\mathbb{P}_{N}(\mathrm{d}B),\qquad
N\in\mathbb{N},\alpha\in[0,\infty).$ (4.2)

Note that ${\mathbb{Q}}_{\alpha,N}$ is not normalized. Its normalized version
is the announced variant of the self-repellent Brownian motion. For $N=1,$
there is no interaction, so we have ${\mathbb{Q}}_{\alpha,1}=\mathbb{P}_{1}$.
The finite dimensional marginals of ${\mathbb{Q}}_{\alpha,N}$ have continuous
Lebesgue densities. In particular, we define

$\Gamma_{\alpha,N}(x):=\frac{{\mathbb{Q}}_{\alpha,N}(B_{N}\in\mathrm{d}x)}{\mathrm{d}x}.$
(4.3)

Then $\Gamma_{k}$ defined in (1.12) for $v$ replaced by $\alpha v$ (and
$\beta=1$) is equal to $\Gamma_{\alpha,k}(0)$. Note that $\mathrm{e}^{-\alpha
LN}\varphi_{N}(x)\leq\Gamma_{\alpha,N}(x)\leq\varphi_{N}(x)$ for any
$N\in\mathbb{N}$ and $x\in\mathbb{R}^{d}$. Now define the Green function for
the self-repellent Brownian motion as follows: for $\lambda>0,$ let

$G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x):=\sum_{N=1}^{\infty}\lambda^{N}\Gamma_{\alpha,N}(x)\qquad\mbox{and
put}\quad\lambda_{\mathrm{cr}}(\alpha):=\sup\big{\\{}\lambda>0\colon\|G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\|_{1}<\infty\big{\\}}.$
(4.4)

Then $\lambda_{\mathrm{cr}}(\alpha)$ is equal to $\lambda_{\rm c}(1)$ defined
in (1.13) for $v$ replaced by $\alpha v$.

Our main result in the present section is:

######  Theorem 4.1.

Assume $d\geq 5$. Then there exists $\alpha_{0}=\alpha_{0}(v,d)>0$ such that
for $\alpha\in(0,\alpha_{0})$

$G_{\alpha,\lambda_{\mathrm{cr}}(\alpha)}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq
5G(x),\qquad x\in\mathbb{R}^{d}.$

In particular,
$\sum_{N=1}^{\infty}\lambda_{\mathrm{cr}}(\alpha)^{N}\Gamma_{\alpha,N}(0)<\infty$
for $\alpha\in(0,\alpha_{0})$.

Theorem 4.1 evidently implies Theorem 1.2. The rest of the article is devoted
to the proof of Theorem 4.1.

For $\lambda<\lambda_{\mathrm{cr}}(\alpha)$, consider

$\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}:=G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\star\varphi,$
(4.5)

so that $\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$ is
evidently integrable, and has bounded and continuous derivatives of all
orders. We first observe the following simple fact:

######  Lemma 4.2.

Fix $\alpha\in(0,\infty)$ such that ${\rm e}^{\alpha L}\leq 2$. Then the
function

$f_{\alpha}(\lambda):=\sup_{x\in\mathbb{R}^{d}}\frac{\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)}{G(x)}$

is continuous and increasing in $\lambda\in[0,\lambda_{\mathrm{cr}}(\alpha)).$

###### Proof.

It is clear that $f_{\alpha}$ is increasing. For the continuity, it suffices
to prove that $f_{\alpha}(\cdot)$ is continuous on
$[\lambda_{0}^{\prime},\lambda_{0}]$ for any
$0<\lambda_{0}^{\prime}<\lambda_{0}<\lambda_{\mathrm{cr}}(\alpha).$

We first prove that there exists $R(\lambda_{0}^{\prime},\lambda_{0})$ such
that for $\lambda\in[\lambda_{0}^{\prime},\lambda_{0}]$

$f_{\alpha}(\lambda)=\sup_{|x|\leq
R(\lambda_{0}^{\prime},\lambda_{0})}\frac{\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)}{G(x)}.$
(4.6)

Since $\mathrm{e}^{-\alpha LN}\varphi_{N}\leq\Gamma_{\alpha,N}$, and since
$\sum_{n}\lambda^{n}\varphi_{n}$ does not converge for $\lambda>1,$ we get
that $\lambda_{\mathrm{cr}}(\alpha)\leq\mathrm{e}^{\alpha L}\leq 2$, by the
choice of $\alpha$. Note that the lower bound
$\lambda_{\mathrm{cr}}(\alpha)\geq 1$ is evident as $H_{N}(B)\geq 0.$ Define
$\lambda_{1}:=(\lambda_{0}+\lambda_{\mathrm{cr}}(\alpha))/2.$ For any
$N\in\mathbb{N}$, using that $\lambda_{\mathrm{cr}}(\alpha)\leq 2$ and
$\Gamma_{\alpha,n}(\cdot)\leq\varphi_{n}(\cdot)$,

$\displaystyle\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)$
$\displaystyle=\sum_{n=1}^{N-1}\lambda^{n}(\Gamma_{\alpha,n}\star\varphi)(x)+\sum_{n=N}^{\infty}\Big{(}\frac{\lambda}{\lambda_{1}}\Big{)}^{n}\lambda_{1}^{n}(\Gamma_{\alpha,n}\star\varphi)(x)$
(4.7)
$\displaystyle\leq\sum_{n=1}^{N-1}2^{n}\varphi_{n+1}(x)+\sum_{n=N}^{\infty}\Big{(}\frac{\lambda_{0}}{\lambda_{1}}\Big{)}^{n}\lambda_{1}^{n}(\Gamma_{\alpha,n}\star\varphi)(x)$
$\displaystyle\leq\sum_{n=1}^{N-1}2^{n}\frac{1}{(2\pi(n+1))^{\frac{d}{2}}}\,\exp[-|x|^{2}/2(n+1)]+\Big{(}\frac{\lambda_{0}}{\lambda_{1}}\Big{)}^{N}\|\overline{G}_{\alpha,\lambda_{1}}^{{\scriptscriptstyle{({\Gamma}})}}\|_{\infty}.$

Remark that by definition of $\lambda_{1}$ we have
$G_{\alpha,\lambda_{1}}^{{\scriptscriptstyle{({\Gamma}})}}\in
L^{1}(\mathbb{R}^{d})$ and therefore
$\|\overline{G}_{\alpha,\lambda_{1}}^{{\scriptscriptstyle{({\Gamma}})}}\|_{\infty}=\|G^{{\scriptscriptstyle{({\Gamma}})}}_{\alpha,\lambda_{1}}\star\varphi\|_{\infty}<\infty$.
By choosing

$N:=\frac{|x|}{2\sqrt{2\log 2}}$

one sees that there are constants $c,C$, depending on $\lambda_{0},$ such that
for $\lambda\leq\lambda_{0}$

$\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq
C\exp[-c|x|].$

As $\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(0)$ is
bounded away from $0$ for $\lambda\geq\lambda_{0}^{\prime}$, and the Green
function $G(x)$ decays like $\mathrm{const}\times|x|^{-d+2}$ as
$|x|\to\infty$, (4.6) follows.

To prove the desired continuity, we appeal to the representation of
$\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$ in the
first identity in (4.7). Then $(\Gamma_{\alpha,n}\star\varphi)(x)$ is
continuous, and therefore uniformly continuous on $B_{R}:=\\{x\colon|x|\leq
R\\}.$ Therefore, for any $N,$ the first part
$\sum_{n=1}^{N-1}\lambda^{n}(\Gamma_{\alpha,n}\star\varphi)(\cdot)$ is
uniformly continuous, also uniformly in
$\lambda\in[\lambda_{0}^{\prime},\lambda_{0}].$ Now since
$\lambda_{0}<\lambda_{\mathrm{cr}}$, by definition of
$\lambda_{\mathrm{cr}}(\alpha)$, the second summand
$\sum_{n=N}^{\infty}\lambda_{0}^{n}(\Gamma_{\alpha,n}\star\varphi)(x)$
converges to $0$ as $N\to\infty$ uniformly in $x$ and
$\lambda\in[\lambda_{0}^{\prime},\lambda_{0}],$ we conclude that
$\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)$ is
uniformly continuous in $x\in B_{R}$ and
$\lambda\in[\lambda_{0}^{\prime},\lambda_{0}].$ The same remains true for
$\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)/G(x)$.
Therefore $\sup_{x\in
B_{R}}\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)/G(x)$
is continuous in $\lambda\in[\lambda_{0}^{\prime},\lambda_{0}]$, as claimed. ∎

The key to proof of Theorem 4.1 is the following.

######  Proposition 4.3.

There exists $\alpha_{0}>0$ such that for $\alpha\leq\alpha_{0},$ there is no
$\lambda<\lambda_{\mathrm{cr}}(\alpha)$ with $f_{\alpha}(\lambda)\in(2,3].$

The proof is based on two main lemmas, namely Lemma 4.4 and Lemma 4.6, which
will be proved in the next two sections. Let us first note that the above
proposition, together with Lemma 4.2, implies Theorem 4.1:

###### Proof of Theorem 4.1 (assuming Proposition 4.3).

We will show Theorem 4.1 with $\alpha_{0}$ replaced by
$\min\\{\alpha_{0},\frac{1}{L}\log 2\\}$, where $\alpha_{0}$ is from
Proposition 4.3.

Fix $\alpha\in(0,\alpha_{0})$ so small that $\lambda_{\mathrm{cr}}(\alpha)\leq
2$. As $\Gamma_{\alpha,n}\leq\varphi_{n}$ for any $n\in\mathbb{N}$, it is
evident that $f_{\alpha}(\lambda)\leq 1$ for $\lambda\leq 1$. Then Proposition
4.3 and the continuity in $\lambda$ shown in Lemma 4.2 imply that
$f_{\alpha}(\lambda)\leq 2$ for $\lambda<\lambda_{\mathrm{cr}}(\alpha).$ This
means that
$\overline{G}_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq
2G(x)$ for any $x\in\mathbb{R}^{d}.$ As $\Gamma_{\alpha,n}(x)\geq 0$ for all
$x,$ it follows that
$\overline{G}_{\alpha,\lambda_{\mathrm{cr}}(\alpha)}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq
2G(x)$. Observe now that from $H_{n}(B)\leq H_{n+1}(B)$, it follows that

$\Gamma_{\alpha,n}\star\varphi\geq\Gamma_{\alpha,n+1}.$

This implies

$\overline{G}_{\alpha,\lambda_{\mathrm{cr}}(\alpha)}^{{\scriptscriptstyle{({\Gamma}})}}\geq\frac{G_{\alpha,\lambda_{\mathrm{cr}}}^{{\scriptscriptstyle{({\Gamma}})}}-\varphi}{\lambda_{\mathrm{cr}}(\alpha)},$

and therefore, using
$\overline{G}_{\alpha,\lambda_{\mathrm{cr}}(\alpha)}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq
2G(x),$ and $\lambda_{\mathrm{cr}}(\alpha)\leq 2,$ it follows that

$\displaystyle
G_{\alpha,\lambda_{\mathrm{cr}}(\alpha)}^{{\scriptscriptstyle{({\Gamma}})}}\leq\lambda_{\mathrm{cr}}(\alpha)\overline{G}_{\alpha,\lambda_{\mathrm{cr}}(\alpha)}^{{\scriptscriptstyle{({\Gamma}})}}+\varphi$
$\displaystyle\leq 4G+\varphi\leq 5G,$ (4.8)

which implies Theorem 4.1. ∎

###  4.2. Outline of the proof of Proposition 4.3.

We outline the arguments for Proposition 4.3. We have to show that if $\alpha$
is small enough, then the following implication is true for all
$\lambda<\lambda_{\mathrm{cr}}(\alpha)$:

$f_{\alpha}(\lambda)\leq 3\quad\Longrightarrow\quad f_{\alpha}(\lambda)\leq
2.$

So, we assume $f_{\alpha}(\lambda)\leq 3$. For the ease of notation, we drop
the parameter $\alpha$. The proof then splits into three main steps: First, by
the argument leading to (4.8) we see that this leads to

$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\leq\lambda\overline{G}_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}+\varphi\leq
7G(x).$ (4.9)

Next, we build on a modification of the lace expansion technique. This
provides a complicated but rather explicit bound for an inverse of
$I+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$, under convolution, where
$I$ is the identity operator (formally given by the Dirac $\delta$ function).
This inverse is of the form

$(I+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}})^{-1}=I+\lambda\varphi+G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}},\qquad\mbox{where
}G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)=\sum_{N=2}^{\infty}\lambda^{N}\Pi_{N}(x)$

for some functions $\Pi_{N}(x)$ that we will soon describe below. The crucial
point is that under the condition
$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq 7G(x)$ for all
$x\in\mathbb{R}^{d}$, we obtain good decay properties for
$G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x).$ Also
$G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)$ is in a suitable sense small
if $\alpha$ is small.

In the final step, which is quite orthogonal to lace expansion used in the
preceding step, we just show that if
$G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ is small, in a sense that will
be made precise, we can invert
$I+\lambda\varphi+G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ under
convolution using a Neumann series. This then gives an estimate of
$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$ in terms of “smallness
properties” of $G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}},$ and in
particular it gives the estimate
$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq 2G(x)$ if $\alpha$ is
small enough, implying

$\overline{G}_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}=G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\star\varphi\leq
2G(x)\star\varphi\leq 2G.$

This will subsequently complete the proof of Proposition 4.3.

###  4.3. The main lemmas, and proof of Proposition 4.3.

For the purposes mentioned above, let us first define the functions $\Pi_{N}$:
For $1\leq k<m,\ k,m\in\mathbb{N}$, let $\mathcal{G}_{k,m}$ be the set of
simple graphs on the set $\\{k,k+1,\ldots,m\\}.$ We also write
$\mathcal{G}_{m}:=\mathcal{G}_{1,m}$ and set

$\displaystyle U_{\alpha}(f,g)=1-\mathrm{e}^{-\alpha V(f,g)}\qquad
f,g\in{\mathcal{C}}_{1}\qquad\mbox{and}$ $\displaystyle
U_{i,j}^{{\scriptscriptstyle{({\alpha}})}}=U_{\alpha}(B_{i},B_{j})\qquad
i,j\in\mathbb{N}.$ (4.10)

Then with $H_{N}(B)=\sum_{1\leq i<j\leq N}V(B_{i},B_{j})$ we can expand

$\mathrm{e}^{-\alpha H_{N}(B)}=\prod_{1\leq i<j\leq
N}(1-U_{i,j}^{{\scriptscriptstyle{({\alpha}})}})=\sum_{{\mathfrak{g}}\in\mathcal{G}_{N}}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{i,j}^{{\scriptscriptstyle{({\alpha}})}}),$

where we identify the graph ${\mathfrak{g}}$ with its vertex set. For
notational convenience, from now on we will suppress the dependence on
$\alpha$ on the function $\Gamma_{N}=\Gamma_{\alpha,N}$ defined in (4.3), on
the tilted measure ${\mathbb{Q}}_{N}={\mathbb{Q}}_{\alpha,N}$ defined in (4.2)
and on the functions $U=U_{\alpha}$ and
$U_{i,j}=U_{i,j}^{\scriptscriptstyle{({\alpha}})}$ defined in (4.10). Then
${\mathbb{Q}}_{N}$ can be rewritten as

${\mathbb{Q}}_{N}(\mathrm{d}B)=\sum_{{\mathfrak{g}}\in\mathcal{G}_{N}}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{i,j})\,\mathbb{P}_{N}(\mathrm{d}B).$
(4.11)

We call an index $k\in\\{1,\ldots,N\\}$ a _breakpoint_ of a graph
${\mathfrak{g}}\in\mathcal{G}_{N}$ if there is no edge
$(i,j)\in{\mathfrak{g}}$ with $i\leq k<j$. If a graph has no breakpoint, we
call it _irreducible_. Remark that we use the notion in a way that is slightly
different from the standard one in lace expansions: For instance, if
${\mathfrak{g}}$ contains a bond $(k,j)$ with $j>k,$ then $k$ is not a
breakpoint. We write $\mathcal{J}_{N}$ for the set of irreducible graphs in
$\mathcal{G}_{N}$. This makes sense only for $N\geq 2.$ For
${\mathfrak{g}}\in\mathcal{G}_{N}\backslash\mathcal{J}_{N},$ we denote by
$b({\mathfrak{g}})$ the smallest breakpoint of ${\mathfrak{g}}$, and write
$\mathcal{G}_{N}^{{\scriptscriptstyle{({k}})}}$ for the set of graphs in
$\mathcal{G}_{N}\backslash\mathcal{J}_{N}$ for which $k$ is the smallest
breakpoint. For convenience, we set $b({\mathfrak{g}}):=N$ for
${\mathfrak{g}}\in\mathcal{J}_{N},$ and accordingly
$\mathcal{G}_{N}^{{{{(N})}}}:=\mathcal{J}_{N}.$

If $\mathbb{E}_{N}$ denotes expectation with respect to the Wiener measure
$\mathbb{P}_{N}$ on ${\mathcal{C}}_{N}$, we define

$\Pi_{N}^{{\scriptscriptstyle{({\alpha}})}}(x)=\Pi_{N}(x):=\sum_{{\mathfrak{g}}\in\mathcal{J}_{N}}\mathbb{E}_{N}\Big{[}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{i,j}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}x\\}}\Big{]}\Big{/}{\rm
d}x,\qquad N\geq 2,$ (4.12)

and define the Green function of $\Pi$ by

$G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x):=\sum_{N=2}^{\infty}\lambda^{N}\Pi_{N}(x),$
(4.13)

provided the series is absolutely summable. The precise formulation for the
second step sketched above is provided by the following lemma.

######  Lemma 4.4.

Fix $\alpha\in(0,\infty)$ and pick $\lambda>0$ such that

$G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)\leq 7G(x),\qquad
x\in\mathbb{R}^{d}.$ (4.14)

Then the right-hand side of (4.13) is absolutely summable and there is a
$C=C_{\alpha,\lambda}>0$ such that

$\sum_{N=2}^{\infty}\lambda^{N}|\Pi_{N}(x)|\leq C\alpha(1+|x|)^{6-3d},\qquad
x\in\mathbb{R}^{d}.$ (4.15)

In particular

$|G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)|\leq
C\alpha(1+|x|)^{6-3d},\qquad x\in\mathbb{R}^{d}.$ (4.16)

The proof of Lemma 4.4 can be found in Section 5. The reason to consider
$G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ is that
$I+\lambda\varphi+G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ is essentially
an inverse of $I+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$ under
convolution. This is based on the following simple result:

######  Lemma 4.5.

With $\Gamma_{0}:=\delta_{0}$ and $\Gamma_{1}=\varphi$ we have

$\Gamma_{N}=\varphi\star\Gamma_{N-1}+\sum_{k=2}^{N}\Pi_{k}\star\Gamma_{N-k},\qquad
N\geq 2.$ (4.17)

###### Proof.

Using (4.11), we have

$\Gamma_{N}(x)=\mathbb{E}_{N}\Big{[}\sum_{{\mathfrak{g}}\in\mathcal{G}_{N}}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{ij}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in{\rm
d}x\\}}\Big{]}\Big{/}{\rm
d}x=\sum_{k=1}^{N}\mathbb{E}_{N}\Big{[}\sum_{{\mathfrak{g}}\in\mathcal{G}_{N}^{(k)}}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{ij}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in{\rm
d}x\\}}\Big{]}\Big{/}{\rm d}x.$

The summand for $k=N$ restricts the summation over ${\mathfrak{g}}$ to
${\mathfrak{g}}\in\mathcal{J}_{N}$, which gives $\Pi_{N}=\Pi_{N}\star\delta$.
We next consider the summand for $k=1$. We remark that $1$ is a break point of
${\mathfrak{g}}$ if and only if there is no edge in ${\mathfrak{g}}$ starting
in $1$. In the product $\prod_{(i,j)\in{\mathfrak{g}}}(-U_{i,j})$ there is
then no interaction between the first leg $B_{1}$ and any other leg. Using
translation invariance, we get

$\sum_{{\mathfrak{g}}\in\mathcal{G}_{N}^{{\scriptscriptstyle{({1}})}}}\mathbb{E}_{N}\Big{[}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{i,j}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}x\\}}\Big{]}\Big{/}{\rm
d}x=\int\varphi(y)\Gamma_{N-1}(x-y)\,\mathrm{d}y.$

This argument can simply be extended to handle the summation over graphs in
$\mathcal{G}_{N}^{{\scriptscriptstyle{({k}})}}$, for $2\leq k\leq N-1$. A
graph ${\mathfrak{g}}$ is in $\mathcal{G}_{N}^{{\scriptscriptstyle{({k}})}}$
if and only if it is the union of an irreducible graph on $\\{1,\ldots,k\\}$
and an arbitrary graph on $\\{k+1,\ldots,N\\}$. Summing over all the
possibilities, we get for $2\leq k\leq N-1$

$\sum_{{\mathfrak{g}}\in\mathcal{G}_{N}^{{\scriptscriptstyle{({k}})}}}\mathbb{E}_{N}\Big{[}\prod\nolimits_{(i,j)\in{\mathfrak{g}}}(-U_{i,j}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}x\\}}\Big{]}\Big{/}{\rm
d}x=\int\Pi_{k}(y)\Gamma_{N-k}(x-y)\,\mathrm{d}y.$

We remark that for $k=N-1,$ there is no graph on $\\{N\\},$ and we get
$\Pi_{N-1}\star\varphi.$ Summing over $k$ we conclude the proof of the lemma.
∎

###### Remark 2

A consequence of the convolution equation (4.17) is that

$\displaystyle G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$
$\displaystyle=\sum_{N=1}^{\infty}\lambda^{N}\Gamma_{N}=\sum_{N=1}^{\infty}\lambda^{N}\sum_{k=1}^{N}\Pi_{k}\star\Gamma_{N-k}=\lambda\varphi+\bigg{(}\sum_{k=1}^{\infty}\lambda^{k}\Pi_{k}\star\bigg{(}\sum_{N=k}^{\infty}\lambda^{N-k}\Gamma_{N-k}\bigg{)}\bigg{)}$
$\displaystyle=(\lambda\varphi+G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}})\star(I+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}),$

that is,

$(I-\lambda\varphi-
G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}})\star(I+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}})=I.$
(4.18)

As we want to work only with smooth functions, we write this equation as

$-(\lambda\varphi+G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}})\star
G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}-(\lambda\varphi+G_{\lambda}^{\Pi})=0.$
(4.19)

$\diamondsuit$

The precise formulation of the third step for the proof of Proposition 4.3 is
provided by the following lemma.

######  Lemma 4.6.

Assume that $\alpha\in(0,\infty)$ is such that
$G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ satisfies (4.16) for
$\lambda<\lambda_{\mathrm{cr}}(\alpha),$ where $C=C_{\alpha}$ does not depend
on $\lambda$. If $\alpha$ is small enough, then there exists a unique
continuous, bounded, rotational symmetric function $F$ (depending on $\alpha$)
that satisfies

$-(\lambda\varphi+G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Pi}})}})\star
F+F=\lambda\varphi+G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ (4.20)

and

$F(x)\leq 2G(x),\qquad x\in\mathbb{R}^{d}.$ (4.21)

The proof of Lemma 4.6 can be found in Section 6.

###### Proof of Proposition 4.3 (assuming Lemma 4.4 and Lemma 4.6).

If $f_{\alpha}(\lambda)\leq 3,$ then we have already observed in (4.14) that
$G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\leq 7G$. By Lemma 4.4,
$G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ is finite and satisfies
$\|G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Pi}})}}\|\leq C\alpha$ for some
$C=C_{\alpha}$ that does not depend on $\lambda$. By Lemma 4.6, if $\alpha$ is
small enough, there is, for any $\lambda<\lambda_{\rm cr}(\alpha)$, a unique
$F$ (depending on $\lambda$) that satisfies (4.20). By (4.19), we have
$F=G_{\alpha,\lambda}^{{\scriptscriptstyle{({\Gamma}})}},$ and then by (4.21),
$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\leq 2G.$ This implies that
$f_{\alpha}(\lambda)\leq 2$ and finishes the proof. ∎

##  5\. Proof of Lemma 4.4: lace expansion

In the standard version of lace expansions, one considers sets of paths that
visit a single point on the lattice possibly several times. This is not the
case in our situation, since the steps of our random walk have a Lebesgue
density. What we do instead is to consider continuous-time paths that on time
intervals of order one can be close together. The closeness is measured by the
following functions $u_{n,\alpha},\ n\geq 2,$ depending on $2n$ locations:

$u_{n}(\mathbf{x})=u_{n,\alpha}(\mathbf{x}):=\int_{\mathcal{C}_{1}^{n}}\prod_{i=1}^{n-1}U(f_{i},f_{i+1})\prod_{i=1}^{n}\mu_{x_{2i-1},x_{2i}}^{{\scriptscriptstyle{({1}})}}(\mathrm{d}f_{i})\qquad\mbox{for}\,\,\mathbf{x}\in(\mathbb{R}^{d})^{2n},$
(5.1)

with $U=U_{\alpha}$ defined in (4.10). In words, this is (up to normalization)
the expectation over a vector of $n$ independent Brownian bridges from
$x_{2i-1}$ to $x_{2i}$ for $i=1,\dots,n$ with interactions between each two
subsequent ones of them. Clearly, $u_{n,\alpha}$ is invariant under shifting
the components of $\mathbf{x}=(x_{1},\ldots,x_{2n})$ by a single vector
$y\in\mathbb{R}^{d}$. Also, $U(f,g)$ is close to $0$ unless $f$ and $g$ are
close together. Therefore, $\ u_{n,\alpha}(\mathbf{x})\approx 0$ unless the
components of $\mathbf{x}$ are close together. The quantitative estimates are
expressed in Lemma 7.1 in Section 7.

Again for notational simplicity, we will usually drop the parameter $\alpha$
appearing in $u_{n,\alpha}$ (as we did for the terms $\Gamma_{\alpha,N}$,
${\mathbb{Q}}_{\alpha,N}$, $U_{\alpha}$ and
$U_{ij}^{{\scriptscriptstyle{({\alpha}})}}$), but the reader should be aware
that the parameter is present in essentially every expression.

###  5.1. The set of laces $\mathbf{{\mathcal{L}}_{\it N}}$ and their
characterization.

Recall from Section 4.3 that $\mathcal{G}_{N}$ denotes the set of simple
graphs on $\\{1,\dots,N\\}$ and $\mathcal{J}_{N}$ denotes the set of
irreducible graphs (i.e., graphs with no breakpoint) in $\mathcal{G}_{N}$ for
$N\geq 2$. We also recall the function $\Pi_{N}$ defined in (4.12). For the
estimation of $\Pi_{N}$, we are going to introduce now a decomposition (called
a lace expansion) of $\mathcal{J}_{N}$ according to the minimal number of
edges the graph has so that it is still irreducible. Any corresponding minimal
subgraph is called a lace, i.e., a subgraph that is not irreducible if any
edge is removed. Every non-irreducible graph may have many laces, but we are
going to construct now for any ${\mathfrak{g}}\in{\mathcal{J}}_{N}$ a
particular lace $\mathrm{lace}({\mathfrak{g}})$ that will be crucial in our
expansion of $\Pi_{N}$. Indeed, the starting point of our lace expansion is

$\sum_{{\mathfrak{g}}\in{\mathcal{J}}_{N}}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{ij})=\sum_{\ell\in{\mathcal{L}}_{N}}\sum_{{\mathfrak{g}}\in{\mathcal{J}}_{N}\colon\mathrm{lace}({\mathfrak{g}})=\ell}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{ij}),$
(5.2)

where ${\mathcal{L}}_{N}$ denotes the set of laces.

Evidently, if the graph on $\\{1,\ldots,N\\}$ contains the edge $(1,N)$ then
it is irreducible. We denote the set of graphs that contain this particular
edge by $\mathcal{J}_{N}^{{\scriptscriptstyle{({1}})}}$.

For other irreducible graphs, the construction of its lace is more
complicated. If
${\mathfrak{g}}\in\mathcal{J}_{N}\setminus\mathcal{J}_{N}^{{\scriptscriptstyle{({1}})}},$
we define its lace, denoted by $\mathrm{lace}({\mathfrak{g}})$, as a subgraph
of ${\mathfrak{g}}$ in the following way: Let ${\mathfrak{g}}$ contain an edge
$(1,j)$ (as otherwise it is not irreducible). We take the largest such $j$
(which is not $N$, as otherwise we would have
${\mathfrak{g}}\in\mathcal{J}_{N}^{{\scriptscriptstyle{({1}})}}$), and denote
it by $j_{1}.$ This edge $(1,j_{1})$ is the first member of
$\mathrm{lace}({\mathfrak{g}})$. For the next, we know that there is an edge
$(i^{\prime},j^{\prime})$ with $i^{\prime}\leq j<j^{\prime},$ as otherwise
${\mathfrak{g}}$ would not be connected. We take the largest possible
$j^{\prime}$ and afterwards the smallest possible $i^{\prime}$ to this
$j^{\prime}$. We denote this edge by $(i_{2},j_{2})\in{\mathfrak{g}}$. If
$j_{2}=N$ we are finished with the construction of
$\mathrm{lace}({\mathfrak{g}})$. Otherwise, we go on in this way. Finally, we
end up with a collection

$\ell=\big{\\{}(i_{1}=1,j_{1}),(i_{2},j_{2}),\ldots,(i_{k},j_{k}=N)\big{\\}}.$
(5.3)

This is the uniquely defined $\mathrm{lace}({\mathfrak{g}})$. We write
$\mathcal{J}_{N}^{{\scriptscriptstyle{({k}})}}$ for the set of irreducible
graphs whose lace has $k$ edges. We write $\mathcal{L}_{N}$ for the set of
laces, i.e., irreducible graphs that are no longer irreducible when leaving
out any edge.

######  Lemma 5.1 (Characterization of laces).

Let $\ell\in\mathcal{L}_{N}$ and ${\mathfrak{g}}\in\mathcal{J}_{N}.$ Then
$\mathrm{lace}({\mathfrak{g}})=\ell$ if and only if all the edges
$(i,j)\in{\mathfrak{g}}\backslash\ell$ have the property that

$\mathrm{lace}(\ell\cup\\{(i,j)\\})=\ell.$

###### Proof.

If $\mathrm{lace}({\mathfrak{g}})=\ell$ then evidently all edges
$(i,j)\in{\mathfrak{g}}\backslash\ell$ satisfy
$\mathrm{lace}(\ell\cup\\{(i,j)\\})=\ell.$

For the other direction, assume $\ell\in\mathcal{L}_{N}$ satisfies
$\ell\subset{\mathfrak{g}},$ and $\mathrm{lace}(\ell\cup\\{(i,j)\\})=\ell$ for
all $(i,j)\in{\mathfrak{g}}\backslash\ell.$ We want to prove that
$\mathrm{lace}({\mathfrak{g}})=\ell.$ (we remind the reader that
${\mathfrak{g}}$ may have many subsets which are laces). Write $\ell$ in the
form (5.3). If $(1,j_{1})\notin\mathrm{lace}({\mathfrak{g}}),$ then
${\mathfrak{g}}$ would have to contain an edge $(1,k)$ with $k>j_{1},$ and
then $\mathrm{lace}(\ell\cup\\{(1,k)\\})\neq\ell.$ So
$(1,j_{1})\in\mathrm{lace}({\mathfrak{g}}).$ Next, if
$(i_{2},j_{2})\notin\mathrm{lace}({\mathfrak{g}}),$ we would have an edge
$(i^{\prime},j^{\prime})$ with either $j^{\prime}>j_{2},$ or
($j^{\prime}=j_{2}$ and $i^{\prime}<i_{2}$). In both cases
$\mathrm{lace}(\ell\cup\\{(i^{\prime},j^{\prime})\\})\neq\ell,$ and we
conclude $(i_{2},j_{2})\in\mathrm{lace}({\mathfrak{g}}).$ In this way it
follows that all edges in $\ell$ are members of
$\mathrm{lace}({\mathfrak{g}}),$ i.e.,
$\ell\subset\mathrm{lace}({\mathfrak{g}}).$ But then, if
$\mathrm{lace}({\mathfrak{g}})$ would contain an edge $\notin\ell,$ it would
not be a lace, by the minimality property of laces. ∎

We write for $\ell\in\mathcal{L}_{N}$

$\mathcal{C}_{N}(\ell):=\big{\\{}(i,j)\notin\ell\colon\mathrm{lace}(\ell\cup\\{(i,j)\\})=\ell\big{\\}},$

and call each bond in $\mathcal{C}_{N}(\ell)$ compatible with $\ell$. In this
terminology, Lemma 5.1 can therefore be rephrased by saying

$\mathrm{lace}({\mathfrak{g}})=\ell\qquad\Longleftrightarrow\qquad{\mathfrak{g}}\backslash\ell\subset\mathcal{C}_{N}(\ell).$
(5.4)

In particular, (5.4) implies that for a fixed lace $\ell\in\mathcal{L}_{N},$

$\displaystyle\sum_{{\mathfrak{g}}\colon\mathrm{lace}({\mathfrak{g}})=\ell}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{ij})$
$\displaystyle=$
$\displaystyle\bigg{[}\prod\nolimits_{(i,j)\in\ell}(-U_{ij})\bigg{]}\sum_{{\mathfrak{g}}\subset\mathcal{C}_{N}(\ell)}\prod_{(i,j)\in{\mathfrak{g}}}(-U_{ij})$
$\displaystyle=$
$\displaystyle\bigg{[}\prod\nolimits_{(i,j)\in\ell}(-U_{ij})\bigg{]}\prod_{(i,j)\in\mathcal{C}_{N}(\ell)}(1-U_{ij}).$

Now define

$\Pi_{N}^{{\scriptscriptstyle{({k}})}}(x):=\sum_{{\mathfrak{g}}\in\mathcal{J}_{N}^{{\scriptscriptstyle{({k}})}}}\mathbb{E}_{N}\Big{[}\prod\nolimits_{(i,j)\in{\mathfrak{g}}}(-U_{ij}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}x\\}}\Big{]}\Big{/}{\rm
d}x,\qquad\mbox{so
that}\,\,\Pi_{N}=\sum_{k=1}^{\infty}\Pi_{N}^{{\scriptscriptstyle{({k}})}}.$
(5.5)

Since $\Pi_{N}^{{\scriptscriptstyle{({k}})}}=0$ for large enough $k,$ the
above sum is finite, and

$\sum_{N=1}^{\infty}\lambda^{N}|\Pi_{N}(x)|\leq\sum_{k=1}^{\infty}\sum_{N=1}^{\infty}\lambda^{N}|\Pi_{N}^{{\scriptscriptstyle{({k}})}}(x)|.$

We are going to estimate the summands on the right-hand side individually for
every $k$. To control the higher order terms, we first characterise the
structure of a lace. The simple proof of the following lemma is left to the
reader.

######  Lemma 5.2.

Any lace

$\ell=\big{\\{}(i_{1},j_{1}),(i_{2},j_{2}),\ldots,(i_{n},j_{n})\big{\\}}$

with $n\geq 3$ edges, where $i_{1}=1,\ j_{n}=N$ and $i_{k}<j_{k}$ for all $k,$
satisfies

$1=i_{1}<i_{2}\leq j_{1}<i_{3}\leq j_{2}<\leq\cdots<i_{n-1}\leq
j_{n-2}<i_{n}\leq j_{n-1}<j_{n}=N.$ (5.6)

###### Remark 3

The statement of the above lemma is in disagreement with the corresponding
ones for the standard used lace expansions. In our case, there is the
possibility to have $i_{k+1}=j_{k},$ but there are no other possibilities for
equality. To see the point: In the usually used expansions, one can have for a
lace with $3$ edges (starting with $i_{1}=1$)

$1=i_{1}<i_{2}<j_{1}=i_{3}<j_{2}<j_{3}=N.$

This is not possible in our case, as $\\{(i_{1},j_{1}),(i_{3},j_{3})\\}$ would
be irreducible, and therefore it would be the lace in this situation. For
essentially a similar reasoning, $i_{k+1}=j_{k}$ is not possible in the
standard versions, but it is indeed the case according to our definition.
$\diamondsuit$

###  5.2. Types of laces and their classification.

We need to make a finer disctinction of laces, on base of Lemma 5.2. We say
that two neighboring edges $(i_{k},j_{k}),\ (i_{k+1},j_{k+1})$ in the lace
touch if $j_{k}=i_{k+1}$. Then we say that $(i_{k+1},j_{k+1})$ touches
$(i_{k},j_{k})$ from the right. Otherwise, i.e., when $i_{k+1}<j_{k},$ we say
that the edges intersect.

According to Lemma 5.2, we classify the laces on $\\{1,\ldots,N\\}$ of length
$n$ into different types, dictated by (5.6). Given a lace $\ell$ with $n$
edges, we order the set of edges increasingly according to the left end,
writing $e_{1},\ldots,e_{n}$ for the edges. We split the set of edges into
disjoint chains

$\chi_{j}=(e_{n_{j-1}+1},\ldots,e_{n_{j}}),\quad j=1,\dots,k,\qquad\mbox{with
}0=n_{0}<n_{1}<n_{2}<\dots<n_{k}=n.$ (5.7)

Within a chain, and if a chain has at least two elements, $e_{i+1}$ touches
$e_{i}$ from the right, but $e_{n_{j}+1}$ intersects $e_{n_{j}},$
$j=1,\ldots,k-1.$ We write

$\tau:=(n_{1},n_{2}-n_{1},\ldots,n_{k}-n_{k-1})$

and call it the type of the lace. In case $\tau=(1,\ldots,1),$ there are no
touching edges, and if $\tau=(n),$ the lace is a chain of $n$ touching edges.
The set of types is denoted by $\mathfrak{T}_{n}.$ We write
$\mathcal{L}_{N}^{{\scriptscriptstyle{({n,\tau}})}}$ for the set of laces of
type $\tau$ on $\\{1,\ldots,N\\}$ with $n$ edges. We split
$\Pi_{N}^{{\scriptscriptstyle{({n}})}}$ defined in (5.5) according to the
different types by writing

$\Pi_{N}^{{\scriptscriptstyle{({n}})}}=\sum_{\tau\in\mathfrak{T}_{n}}\Pi_{N}^{{\scriptscriptstyle{({n,\tau}})}},$

where $\Pi_{N}^{{\scriptscriptstyle{({n,\tau}})}}$ is the sum over graphs
whose lace have $n$ edges and are of type $\tau$.

Given a lace $\ell$ with $n$ edges, the set of endpoints of its edges can be
written as increasing sequence:

$\mathbf{s}(\ell):=(s_{i})_{i=1,\dots,m},\qquad\mbox{where
}1=s_{1}<s_{2}<\cdots<s_{m-1}<s_{m}=N,$ (5.8)

and $m$ depends on $n$ and the type of the lace. If all neighboring edges
intersect, then $m=2n.$ For every touching of neighbor edges, $m$ is reduced
by one.2††2For instance, if the type is $(2,3,1,2),$ then $n=8$ and $m=12$.

Any $e\in\ell$ is written $e=(s_{\tau_{\star}(e)},s_{\tau^{\star}(e)})$ with
$\tau_{\star}(e)<\tau^{\star}(e)$. Then
$\ell=((s_{\tau_{\star}(e)},s_{\tau^{\star}(e)}))_{e\in\ell}$. In an abuse of
notation, we write $\ell$ also as a sequence

$((\tau_{\star}(i),\tau^{\star}(i)))_{i=1,\ldots,n}$ (5.9)

where $i$ stands for the $i$-th edge, these being ordered increasingly.3††3For
instance if a lace of $n=5$ edges has type type $(1,3,1)$ then the sequence of
pairs $((\tau_{\star}(i),\tau^{\star}(i)))_{i=1}^{5}$ is
$((1,3),(2,4),(4,5),(5,7),(6,8))$.

We define $\ell_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}$ to be the
unique lace of type $\tau$ that has endpoints $\mathbf{s}(\ell)=\mathbf{s}$
with given $\mathbf{s}=(s_{i})_{i=1,\dots,m}$ as in (5.8). We will also write
$\mathbf{t=}(t_{1},\ldots,t_{m})$ where $t_{i}:=s_{i}-s_{i-1}$ and $s_{0}=0$.
We will use $\mathbf{t}$ and $\mathbf{s}$ according to what is more
convenient, but they are tied in the described way.

###  5.3. Proof of Lemma 4.4.

With the notation set up in the previous section, we want to estimate

$\Pi_{N}^{{\scriptscriptstyle{({n,\tau}})}}(y)=\sum_{\ell\in\mathcal{L}_{N}^{{\scriptscriptstyle{({n,\tau}})}}}\mathbb{E}_{N}\bigg{[}\bigg{(}\prod_{(i,j)\in\ell}(-U_{ij})\prod_{(i,j)\in\mathcal{C}_{N}(\ell)}(1-U_{ij})\bigg{)}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}y\\}}\bigg{]}\Big{/}{\rm
d}y.$

Recalling that $\mathbf{t}=(t_{1},\dots,t_{m})$ with $t_{i}=s_{i}-s_{i-1}$ and
hence $\sum_{i=1}^{m}t_{i}=N$, we see that

$\lambda^{N}\Pi_{N}^{(n,\tau)}(y)=\sum_{\mathbf{t}}\Big{[}\prod\nolimits_{i=1}^{m}\lambda^{t_{i}}\Big{]}\Pi_{N}^{(n,\tau,\mathbf{s})}(y),$
(5.10)

where

$\Pi_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y):=\sum_{\ell\in\mathcal{L}_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}}\mathbb{E}_{N}\bigg{[}\prod_{(i,j)\in\ell}(-U_{ij})\prod_{(i,j)\in\mathcal{C}_{N}(\ell)}(1-U_{ij}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}y\\}}\bigg{]}\Big{/}{\rm
d}y.$

We start with estimating

$\displaystyle\big{|}\Pi_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y)\big{|}$
$\displaystyle\leq\sum_{\ell\in\mathcal{L}_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}}\mathbb{E}_{N}\Bigg{[}\bigg{(}\prod\nolimits_{(i,j)\in\ell}U_{ij}\bigg{)}\prod_{j=1}^{m-1}\prod_{s_{j}<u<v\leq
s_{j+1}-1}(1-U_{uv}){\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{B_{N}\in\mathrm{d}y\\}}\Bigg{]}\Big{/}{\rm
d}y$ (5.11)
$\displaystyle=:g_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y).$

We are now going to represent
$\Pi_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y)$ in terms of an
integration over $2m-2$ variables $x_{2},\ldots,x_{2m-1}\in\mathbb{R}^{d}$
representing the positions of the Brownian motion at times $s_{1}=1,\
s_{2}-1,\ s_{2},\ldots,\ s_{m}-1=N-1.$ Thus, $s_{1}-1=0$ and also, $B_{0}=0$
and at time $s_{m}=N,$ the Brownian motion is fixed to be at $y$. We define

$\mathbb{P}_{N,\tau,\mathbf{s}}^{{\scriptscriptstyle{({\mathbf{x},y}})}}(\cdot):=\mathbb{P}_{N}\Big{(}\cdot\big{|}B_{s_{i}-1}=x_{2i-1},B_{s_{i}}=x_{2i}\,\,\mbox{for}\,\,i=1,\ldots,m\Big{)}$

and integrate over the density of the event in the conditioning. By the time-
structure, the Markov inequality can be used at the times $s_{i}-1,s_{i}$ for
$i=1,\dots,m$, which decomposes the path into pieces of time length 1 and
$s_{i}-1-s_{i-1}$, with $s_{0}=0$. Recall that $\varphi_{t}$ is the Gaussian
density with variance $t$. Then

$\displaystyle g_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{2m-2}}\mathrm{d}\mathbf{x\
}\mathbb{E}_{N,\tau,\mathbf{s}}^{{\scriptscriptstyle{({\mathbf{x,}y}})}}\Big{[}\prod\nolimits_{(i,j)\in\ell_{N}^{(n,\tau,\mathbf{t})}}U_{ij}\Big{]}$
(5.12)
$\displaystyle\quad\times\prod_{j=1}^{m}\mathbb{E}_{N,\tau,\mathbf{s}}^{{\scriptscriptstyle{({\mathbf{x,}y}})}}\Big{[}\prod\nolimits_{s_{j-1}<u<v\leq
s_{j}-1}(1-U_{ij})\Big{]}$
$\displaystyle\quad\times\prod_{j=1}^{m}\varphi_{1}(x_{2j}-x_{2j-1})\prod_{j=2}^{m}\varphi_{s_{j}-1-s_{j-1}}(x_{2j-1}-x_{2(j-1)}).$

Here we followed the convention that $x_{1}:=0,\ x_{2m}=y$ and the integration
is only over $x_{2},\ldots,x_{2m-1}$ and used that under the conditional law
$\mathbb{P}_{N,\tau,\mathbf{s}}^{{\scriptscriptstyle{({\mathbf{x},y}})}}$, the
first product and the factors $\prod_{j=1}^{m-1}$ appearing in the definition
of $g_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}$ on the right hand
side of (5.11) are independent. Also, we have made the following convention:
it is not excluded that $s_{j}=s_{j-1}+1$ in which case, the conditional law
would not be defined, except when $x_{2j}=x_{2j-1}.$ This is taken care of by
interpreting $\varphi_{0}$ as the Dirac function.

Let us now handle factors appearing on the right hand side of (5.12). First we
observe that

$\displaystyle\mathbb{E}_{N,\tau,\mathbf{t}}^{(\mathbf{x,}y)}\Big{[}\prod\nolimits_{s_{j-1}<u<v\leq
s_{j}-1}(1-U_{ij})\Big{]}\varphi_{s_{j}-1-s_{j-1}}(x_{2j}-x_{2j-1})=\Gamma_{s_{j}-s_{j-1}-1}(x_{2j}-x_{2j-1}),$
(5.13)

if $s_{j}\geq s_{j-1}+2.$ In case $s_{j}\geq s_{j-1}+1,$ the product is over
the empty set and therefore equal to $1,$ and so we have to interpret
$\Gamma_{0}$ as $\delta,$ too. Actually, also for $s_{j}=s_{j-1}+2,$ the
product is empty, in which case we arrive at $\Gamma_{1}=\varphi_{1}.$

We next express the factor involving the $U_{ij}$ appearing on the right-hand
side of (5.12) in terms of the function $u_{n}$ defined in (5.1). This factor
depends on the type $\tau=(v_{1},\ldots,v_{k})\in\mathbb{N}^{k}$ of the lace.
If $\chi_{1},\ldots,\chi_{k}$ are the chains (recall (5.7)) then the terms
$\prod\nolimits_{(i,j)\in\chi_{r}}U_{ij},$ $r=1,\ldots,k$ are independent
under the conditional measure
$\mathbb{P}_{N,\tau,\mathbf{t}}^{(\mathbf{x,}y)}.$ Let a chain $\chi$ contain
$b$ edges, labelled $(s_{i_{1}},s_{i_{2}}),\
(s_{i_{2}},s_{i_{3}}),\ldots,(s_{i_{b}},s_{i_{b+1}})$. Then
$i_{k+1}-i_{k}=1,2$ or $3,$ depending whether the chain is in relation to the
others. In any case, if $b\geq 3,$ then $i_{k+1}-i_{k}=1$ for
$k=2,\ldots,b-1$. Then by recalling the definition of $u_{n}(\cdot)$ from
(5.1),

$\mathbb{E}_{N,\tau,\mathbf{t}}^{(\mathbf{x,}y)}\Big{[}\prod\nolimits_{(i,j)\in\chi}U_{ij}\Big{]}\prod_{k=1}^{b+1}\varphi_{1}(x_{i_{k}}-x_{i_{k}-1})=u_{b+1}(x_{i_{1}-1},x_{i_{1}},x_{i_{2}-1},x_{i_{2}},\ldots,x_{i_{b+1}-1},x_{i_{b+1}}).$
(5.14)

Now we can apply the crucial estimate from Lemma 7.1 and obtain

$u_{b+1}(x_{i_{1}-1},x_{i_{1}},\ldots,x_{i_{b+1}-1},x_{i_{b+1}})\leq
C^{b+1}\alpha^{b}\exp\Big{[}-\frac{1}{8}\sum\nolimits_{k=1}^{b+1}|x_{i_{k}-1}-x_{i_{k}}|^{2}-A\sum_{k=1}^{b}|x_{i_{k+1}}-x_{i_{k}}|^{2}\Big{]}.$
(5.15)

Combining (5.12)-(5.14) we therefore get

$\displaystyle g_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y)$
$\displaystyle\leq C^{n}\alpha^{n}\int{\rm d}\mathbf{x\
}\prod_{j=1}^{m-1}\Gamma_{s_{j+1}-1-s_{j}}(x_{2j+1}-x_{2j})$ (5.16)
$\displaystyle\quad\times\exp\Big{[}-\frac{1}{8}\sum\nolimits_{j=1}^{m-1}|x_{2j}-x_{2j-1}|^{2}-A\sum\nolimits_{i=1}^{n}|x_{2\tau^{\star}(i)}-x_{2\tau_{\star}(i)}|^{2}\Big{]},$

where in the last display we used the notation $\tau^{\star}(i)$ and
$\tau_{\star}(i)$ defined in (5.9).

We are heading now towards a proof of Lemma 4.4. In the following, we use
$C>0$ as a generic constant that may change its value from appearance to
appearance and does not depend on $n$ nor on $m$ or $\tau$ and so on, but may
depend only on $d$, $\alpha$ and $\lambda$, which we fix now such that (4.14)
holds.

From the preceding, we get, for any type $\tau$ and any ${\mathbf{t}}$,
recalling that $N=\sum_{i=1}^{m}t_{i}=\sum_{i=1}^{m}(s_{i}-s_{i-1})$,

$\displaystyle\sum_{N}\lambda^{N}\sum_{\ell\in\mathcal{L}_{N}^{{\scriptscriptstyle{({n,\tau}})}}}g_{N}^{{\scriptscriptstyle{({n,\tau,\mathbf{s}}})}}(y)$
$\displaystyle\leq C^{n}\alpha^{n}\lambda^{m-1}\int{\rm d}\mathbf{x\
}\prod_{j=1}^{m-1}\sum_{k=0}^{\infty}\lambda^{k}\Gamma_{k}(x_{2j+1}-x_{2j})$
(5.17)
$\displaystyle\quad\times\exp\Big{[}-\frac{1}{8}\sum\nolimits_{j=1}^{m}|x_{2j}-x_{2j-1}|^{2}\Big{]}\exp\Big{[}-A\sum\nolimits_{i=1}^{n}|x_{2\tau^{\star}(i)}-x_{2\tau_{\star}(i)}|^{2}\Big{]}$
$\displaystyle\leq C^{n}\alpha^{n}\int{\rm
d}\mathbf{x}\prod_{j=2}^{m}\Big{[}\delta(x_{2j-1}-x_{2j-2})+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x_{2j-1}-x_{2j-2})\Big{]}$
$\displaystyle\quad\times\exp\Big{[}-\frac{1}{8}\sum\nolimits_{j=1}^{m}|x_{2j}-x_{2j-1}|^{2}\Big{]}\exp\Big{[}-A\sum\nolimits_{i=1}^{n}|x_{2\tau^{\star}(i)}-x_{2\tau_{\star}(i)}|^{2}\Big{]},$

where we remind on $m\leq 2n.$ We integrate now over all the $x_{2j-1}$
(except of course $x_{1}$ which is $0$ by convention). Indeed, using the
assumption $G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}\leq 7G$ from (4.14)
of Lemma 4.4, we get

$\displaystyle\int{\rm
d}x_{2j-1}\bigg{[}\delta(x_{2j-1}-x_{2j-2})+G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x_{2j-1}-x_{2j-2})\bigg{]}\exp\Big{[}-\frac{1}{8}|x_{2j}-x_{2j-1}|^{2}\Big{]}\leq
Cf_{d}(|x_{2j}-x_{2j-2}|),$ (5.18)

where we abbreviated

$f_{d}(x):=(1+|x|)^{-d+2}.$ (5.19)

Combining the last two estimates we obtain

$\sum_{N}\lambda^{N}\sum_{\ell\in\mathcal{L}_{N}^{\left(n,\tau\right)}}g_{N}^{\left(n,\tau,\mathbf{s}\right)}\left(y\right)\leq
C^{n}\beta^{n}\left(\varphi_{1}\ast X_{n,\tau}\right)\left(y\right),$

where for $n\geq 2$, with renaming $z_{i}:=x_{2i},$ we have written

$\displaystyle X_{n,\tau}\left(x\right)$ $\displaystyle:=\int{\rm
d}z_{2}\cdots{\rm d}z_{m-1}$
$\displaystyle\qquad\times\exp\bigg{[}-A\sum\nolimits_{i=1}^{n-1}\left|z_{\tau^{\ast}\left(i\right)}-z_{\tau_{\ast}\left(i\right)}\right|^{2}\bigg{]}\exp\bigg{[}-A\left|z_{\tau_{\ast}\left(n\right)}-x\right|^{2}\bigg{]}$
$\displaystyle\qquad\times
f_{d}\left(x-z_{m-1}\right)\prod_{j=2}^{m-1}f_{d}\left(z_{j}-z_{j-1}\right),$

where $z_{1}:=0.$ We also remark that $m$ in the expression above is a
function of $\tau$ that satisfies $m\left(\tau\right)\leq 2n.$ Also $m=2$ for
$n=1,$ and $m\geq 3$ for $n\geq 2.$

We will prove by induction on $n$ that for some constant $C,$ depending only
on $A$, and $d$

$\sup_{\tau\in\mathfrak{T}_{n}}X_{n,\tau}\left(x\right)\leq
C^{n}f_{d}\left(x\right),\qquad n\in\mathbb{N}.$ (5.21)

We remark that from this, it also follows that

$\sup_{\tau\in\mathfrak{T}_{n}}\left(\varphi_{1}\ast
X_{n,\tau}\right)\left(x\right)\leq C^{n}f_{d}\left(x\right),\qquad
n\in\mathbb{N},$

with a slightly changed constant $C$. As the number of types of laces with $n$
edges is bounded by $2^{n},$ this will subsequently prove Lemma 4.4.

We will now prove (5.21) by induction. For $n=1$, there is only one type, and
one simply has

$X_{1}\left(x\right)=\exp\big{[}-A\left|x\right|^{2}\big{]}f_{d}\left(x\right)$
(5.22)

so (5.21) is evident for $n=1.$

Let $n=2.$ Then there are the two types $\left(1,1\right)$ and $(2).$ For
$\tau=\left(2\right)$

$\displaystyle X_{2}\left(x\right)$ $\displaystyle=$ $\displaystyle\int{\rm
d}z\exp\big{[}-A\left|z\right|^{2}\big{]}f_{d}\left(z\right)\exp\big{[}-A\left|x-z\right|^{2}\big{]}f_{d}\left(x-z\right)$
$\displaystyle\leq$ $\displaystyle\int{\rm
d}z\exp\big{[}-A\left|z\right|^{2}-A\left|x-z\right|^{2}\big{]},$

which has Gaussian tails, and so we have only to take care of
$\tau=\left(1,1\right).$ There we have $m=4,$ and

$\displaystyle X_{2}\left(x\right)$ $\displaystyle=$ $\displaystyle\int{\rm
d}z_{1}\ {\rm
d}z_{2}\,\,\exp\big{[}-A\big{(}\left|z_{2}\right|^{2}+\left|x-z_{1}\right|^{2}\big{)}\big{]}\,\,f_{d}\left(z_{1}\right)f_{d}\left(z_{2}-z_{1}\right)f_{d}\left(x-z_{2}\right),$

which, according to Lemma 7.2, is bounded by
$C\left(1+\left|y\right|\right)^{-3d+6}.$

We assume now $n\geq 3,$ and that (5.21) holds up to $n-1$ instead of $n$. We
consider first the case where the type $\tau$ starts with
$\left(1,1,\ldots\right).$ Then the expression for $X_{n,\tau}$ starts with

$\displaystyle\int{\rm d}\mathbf{z}\
\exp\big{[}-A\big{(}\left|z_{2}\right|^{2}+\left|z_{4}-z_{1}\right|^{2}+\left|z_{5}-z_{3}\right|^{2}+\cdots\big{)}\big{]}$
$\displaystyle\qquad\times
f_{d}\left(z_{1}\right)f_{d}\left(z_{2}-z_{1}\right)f_{d}\left(z_{3}-z_{2}\right)f_{d}\left(z_{4}-z_{3}\right)f_{d}\left(z_{5}-z_{4}\right)\cdot\cdots$

or $\left|z_{5}-z_{3}\right|^{2}$ replaced by $\left|z_{6}-z_{3}\right|^{2}$
if $n>3$ and the third entry in $\tau$ is $\neq 1.$ Then

$\displaystyle\int{\rm d}z_{1}{\rm
d}z_{4}\,f_{d}\left(z_{1}\right)f_{d}\left(z_{2}-z_{1}\right)f_{d}\left(z_{3}-z_{2}\right)f_{d}\left(z_{4}-z_{3}\right)f_{d}\left(z_{5}-z_{4}\right)\exp\left[-A\left|z_{4}-z_{1}\right|^{2}\right]$
$\displaystyle\leq$ $\displaystyle\sqrt{\int
f_{d}^{2}\left(z_{1}\right)f_{d}^{2}\left(z_{4}-z_{3}\right)\exp\left[-A\left|z_{4}-z_{1}\right|^{2}\right]\,{\rm
d}z_{1}{\rm d}z_{4}}$ $\displaystyle\times\sqrt{\int
f_{d}^{2}\left(z_{2}-z_{1}\right)f_{d}^{2}\left(z_{5}-z_{4}\right)\exp\left[-A\left|z_{4}-z_{1}\right|^{2}\right]\,{\rm
d}z_{1}{\rm d}z_{4}}$ $\displaystyle\leq$ $\displaystyle C\sqrt{\int
f_{d}^{2}\left(z_{1}\right)f_{d}^{2}\left(z_{1}-z_{3}\right)\,{\rm
d}z_{1}}\sqrt{\int
f_{d}^{2}\left(z_{2}-z_{1}\right)f_{d}^{2}\left(z_{5}-z_{1}\right)\,{\rm
d}z_{1}}$ $\displaystyle\leq$ $\displaystyle
Cf_{d}\left(z_{3}\right)f_{d}\left(z_{5}-z_{2}\right).$

To justify the last step in the above bound, we note that
$f_{d}^{2}(x)=(1+|x|)^{4-2d}$ is integrable in $d\geq 5$. Then the first
integral of the fourth line above is a convolution of $f_{d}^{2}$ with itself,
and we can take Fourier transform of this convolution, use a uniform bound on
the Fourier coefficient of $f_{d}^{2}$ and then take inverse Fourier transform
to obtain the desired bound.

Also, if $n>3$ and the third entry in $\tau$ is $\neq 1,$ $z_{5}$ is replaced
by $z_{6}.$ Implementing this inequality then leads to

$X_{n,\tau}\leq CX_{n-1,\tau^{\prime}}$ (5.23)

with $\tau^{\prime}$ obtained from $\tau$ by replacing
$\left(1,1,t_{3},\ldots\right)$ by $\left(1,t_{3},\ldots\right).$

We next consider the case where $\tau$ is $\left(1,t_{2},t_{3},\ldots\right)$
with $t_{2}\neq 1.$ Here the expression for $X_{n,\tau}$ starts with

$\displaystyle X_{n,\tau}\left(y\right)$ $\displaystyle=$
$\displaystyle\int{\rm d}\mathbf{z}\
\exp\bigg{[}-A\big{(}\left|z_{2}\right|^{2}+\left|z_{1}-z_{3}\right|^{2}+\left|z_{5}-z_{3}\right|^{2}+\cdots\big{)}\bigg{]}$
$\displaystyle\qquad\times
f_{d}\left(z_{1}\right)f_{d}\left(z_{2}-z_{1}\right)f_{d}\left(z_{3}-z_{2}\right)f_{d}\left(z_{4}-z_{3}\right)\cdot\cdots$
$\displaystyle\leq$ $\displaystyle\int{\rm d}\mathbf{z}\
\exp\bigg{[}-A\big{(}\left|z_{2}\right|^{2}+\left|z_{1}-z_{3}\right|^{2}+\left|z_{5}-z_{3}\right|^{2}+\cdots\big{)}\bigg{]}$
$\displaystyle\qquad\times
f_{d}\left(z_{1}\right)f_{d}\left(z_{2}-z_{1}\right)f_{d}\left(z_{4}-z_{3}\right)\cdot\cdots$
$\displaystyle\leq$ $\displaystyle C\int{\rm d}z_{2}\cdots\
\exp\bigg{[}-A\big{(}\left|z_{3}-z_{1}\right|^{2}+\left|z_{6}-z_{4}\right|^{2}+\cdots\big{)}\bigg{]}$
$\displaystyle\qquad\times
f_{d}\left(z_{3}\right)f_{d}\left(z_{3}-z_{2}\right)f_{d}\left(z_{4}-z_{3}\right)\cdot\cdots$

That is, we have (5.23) with $\tau^{\prime}=\left(1,t_{3},\ldots\right).$

Finally, we have to handle $\tau=\left(k,t_{2},\ldots\right)$ with $k\geq 2.$
We first look at the case $\tau=(k)$ in which case $k=n$ if
$\tau\in\mathfrak{T}_{n}.$ Then

$\displaystyle X_{n,\tau}\left(x\right)$ $\displaystyle=$
$\displaystyle\int{\rm
d}\mathbf{z}\exp\bigg{[}-A\big{(}\left|z_{1}\right|^{2}+\left|z_{2}-z_{1}\right|^{2}+\cdots+\left|x-z_{n-1}\right|\big{)}\bigg{]}$
$\displaystyle\qquad\times
f_{d}\left(z_{1}\right)f_{d}\left(z_{2}-z_{1}\right)\cdot\cdots\cdot
f_{d}\left(x-z_{n-1}\right)$ $\displaystyle\leq$ $\displaystyle\int{\rm
d}\mathbf{z}\exp\bigg{[}-A\big{(}\left|z_{1}\right|^{2}+\left|z_{2}-z_{1}\right|^{2}+\cdots+\left|x-z_{n-1}\right|\big{)}\bigg{]}$
$\displaystyle\leq$ $\displaystyle
c^{n}\exp\bigg{[}-\frac{A}{n}\left|x\right|^{2}\bigg{]}$

for a constant $c>0$ (depending on $A,d$ only). It is easily checked that
$\exp\big{[}-\frac{A}{n}\left|x\right|^{2}\big{]}\leq\left(c^{\prime}\right)^{n}f_{d}\left(x\right)$
for some constant $c^{\prime}=c^{\prime}\left(A\right)>0.$ (actually,
$n^{\gamma}f_{d}\left(x\right)$ for some $\gamma>0$ is already an upper
bound). So (5.21) is satisfied for $X_{n,(n)}$ for all $n,$ if $C$ is chosen
appropriately. We remark that in this case we do not use induction, and
estimate the expressions for every $n$ directly. Therefore, we have outside of
any inductive assumptions

$X_{n,\left(n\right)}\leq C_{1}^{n}f_{d},\qquad n\in\mathbb{N},$

for some constant $C_{1}>0$ depending on $A$ and $d$ only.

Let now $\tau=\left(k,\tau^{\prime}\right)$ with $k\geq 2.$ Take the types
$\left(k-1\right)\in\mathfrak{T}_{k-1}$ and
$\left(1,\tau^{\prime}\right)\in\mathfrak{T}_{n-k+1}.$ Observe that

$X_{n,\left(k,\tau^{\prime}\right)}=X_{k-1,\left(k-1\right)}\ast
X_{n-k+1,\left(1,\tau^{\prime}\right)}.$

As $k\geq 2,$ we can use the induction assumption for the second factor and
obtain

$\displaystyle X_{n,\left(k,\tau^{\prime}\right)}$ $\displaystyle\leq$
$\displaystyle C_{1}^{k-1}C^{n-k+1}\left(f_{d}\ast f_{d}\right)\leq\gamma
C_{1}^{k-1}C^{n-k+1}f_{d},$

where we use $f_{d}\ast f_{d}\leq\gamma f_{d}$ for some constant
$\gamma=\gamma_{d}>0$ (because $3d-6>d$). Therefore, if $\gamma C_{1}\leq C$,
we have

$X_{n,\left(k,\tau^{\prime}\right)}\leq C^{n}f_{d},$

finishing the induction argument for (5.21). Therefore, Lemma 4.4 is proved.

##  6\. Proof of Lemma 4.6: deconvolution

In this section we prove Lemma 4.6. For this, we introduce a crucial norm on
the set of all continous and bounded functions $\mathbb{R}^{d}\to\mathbb{R}$
and derive some properties of the Green’s function with respect to this norm,
and in Section 6.2 we finish the proof.

###  6.1. A convolution algebra.

Consider the Banach space $\mathbf{B}$ of bounded continuous functions
$f\colon\mathbb{R}^{d}\to\mathbb{R}$ that satisfy

$\|f\|:=\max\Big{\\{}\|f\|_{1},\|f\|_{\infty},\sup_{x\in\mathbb{R}^{d}}|x|^{d}|f(x)|\Big{\\}}<\infty.$
(6.1)

######  Lemma 6.1.

If $f,g\in\mathbf{B}$ then also $f\star g\in\mathbf{B}.$ Furthermore

$\|f\star g\|\leq 2^{d+1}\|f\|\|g\|,\qquad f,g\in\mathbf{B}.$

###### Proof.

Clearly $\|f\star g\|_{1}=\|f\|_{1}\|g\|_{1}\leq\|f\|\|g\|,$ and $\|f\star
g\|_{\infty}\leq\|f\|_{1}\|g\|_{\infty}\leq\|f\|\|g\|.$ Concerning the third
part in the definition of $\|\cdot\|,$ let $x\in\mathbb{R}^{d}.$ Then

$\displaystyle|(f\star g)(x)|$ $\displaystyle=\Big{|}\int f(y)g(x-y)\,{\rm
d}y\Big{|}\leq\int|f(y)|\,|g(x-y)|\,{\rm d}y$
$\displaystyle\leq\int_{y\colon|y|>|x-y|}|f(y)|\,|g(x-y)|\,{\rm
d}y+\int_{y\colon|y|\leq|x-y|}|f(y)|\,|g(x-y)|\,{\rm d}y.$

Clearly,

$\big{\\{}y\colon|x-y|<|y|\big{\\}}\subset\big{\\{}y\colon|y|>|x|/2\big{\\}}\qquad\mbox{and}\qquad\big{\\{}y\colon|x-y|\geq|y|\big{\\}}\subset\big{\\{}y\colon|x-y|\geq|x|/2\big{\\}}.$

Therefore,

$\displaystyle|x|^{d}|(f\star g)(x)|$ $\displaystyle\leq|x|^{d}\int_{y\colon
2^{d}|y|^{d}>|x|^{d}}|f(y)|\,|g(x-y)|\,{\rm d}y+|x|^{d}\int_{y\colon
2^{d}|x-y|^{d}\geq|x|^{d}}|f(y)|\,|g(x-y)|\,{\rm d}y$ $\displaystyle\leq
2^{d}\|g\|_{1}\sup_{y\in\mathbb{R}^{d}}|y|^{d}|f(y)|+2^{d}\|f\|_{1}\sup_{y\in\mathbb{R}^{d}}|y|^{d}|g(y)|$
$\displaystyle\leq 2^{d+1}\|f\|\,\|g\|,$

which proves the claim. ∎

We remind the reader on (4.1): $G(x)=\sum_{n=1}^{\infty}\varphi_{n}(x)$ is the
Green function for the random walk with standard normal increments.

###### Remark 4

  1. a)

With the $C=C_{\alpha,\lambda}$ from (4.16) , we obtain from there that
$\|G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\|\leq C\alpha$.

  2. b)

There is $C\in(0,\infty)$ (only depending on $d$) such that, for any $f\in{\bf
B}$ satisfying $\|f\|\leq 1,$ it holds $|f(x)|\leq CG(x)$ for any
$x\in\mathbb{R}^{d}$. (Indeed, use for large $|x|$ that $G(x)\asymp|x|^{2-d}$
and that $|f(x)|\leq|x|^{-d}$ and for small $|x|$ that $G(x)$ is bounded away
from zero.)

$\diamondsuit$

######  Lemma 6.2.

Assume that $\rho\colon\mathbb{R}^{d}\to\mathbb{R}$ is a rotationally
symmetric and continuous function satisfying $|\rho(x)|\leq(1+|x|)^{-d-4}$ for
any $x\in\mathbb{R}^{d}$ and $\int\rho(x)\,{\rm d}x=0$, then $\|\rho\star
G\|<\infty$.

###### Proof.

Note that $G,\rho$ are both bounded and continuous, and $\rho$ is integrable.
Therefore $\rho\star G$ is continuous and bounded and integrable. It remains
to show that $|x|^{d}|(\rho\star G)(x)|$ is bounded in $x$. It suffices to
consider $|x|\geq 1$, which we assume for the rest of the proof.

Evidently

$(\rho\star G)(x)=\int_{|y|\geq 1}G(y)\rho(x-y)\,{\rm
d}y+O(|x|^{-d-4})\qquad\mbox{as }|x|\to\infty.$ (6.2)

In the following we will be using $C$ as a generic positive constant that does
not depend on $x$ nor on $y$ and may change its value from appearance to
appearance. According to Lemma 7.3, there is a function
$h\colon\mathbb{R}^{d}\to[0,\infty)$ such that

$\displaystyle G(y)=a_{d}|y|^{-d+2}+h(y)\qquad\mbox{and }|h(y)|\leq
C(1+|y|)^{-d-2},\qquad|y|\geq 1.$ (6.3)

Using our assumptions on $\rho$ and on $h$, we see that

$\displaystyle\int_{|y|\geq 1}h(y)\rho(x-y)\,{\rm d}y$
$\displaystyle=\int\limits_{1\leq|y|\leq|x|/2}h(y)\rho(x-y)\,{\rm
d}y+\int\limits_{|y|>|x|/2}h(y)\rho(x-y)\,{\rm d}y$ (6.4)
$\displaystyle\leq(1+|x|)^{-d-4}\|h\|_{1}+C(1+|x|)^{-d-2}\|\rho\|_{1}$
$\displaystyle\leq C(1+|x|)^{-d-2}.$

This shows that $|x|^{d}h\star\rho(x)$ is bounded in $x$. It remains to show
that $\int\rho(y)|x-y|^{-d+2}\,{\rm d}y\leq O(|x|^{-d})$ as $|x|\to\infty$. We
split

$\displaystyle\int\rho(y)|x-y|^{-d+2}\,{\rm d}y$
$\displaystyle=\int\limits_{|y|\leq|x|/2}\rho(y)|x-y|^{-d+2}\,{\rm
d}y+\int\limits_{|y|>|x|/2}\rho(y)|x-y|^{-d+2}\,{\rm d}y=:I_{1}(x)+I_{2}(x).$

In the first summand $I_{1}$, we Taylor expand $|x-y|^{-d+2}$ around $x:$

$\displaystyle|x-y|^{-d+2}$
$\displaystyle=|x|^{-d+2}+(2-d)\sum_{i=1}^{d}y_{i}x_{i}|x|^{-d}$
$\displaystyle+(2-d)\sum_{i,j=1}^{d}y_{i}y_{j}\Big{(}-d\cdot
x_{i}x_{j}|x|^{-d-2}+\delta_{ij}|x|^{-d}\Big{)}+O(|y|^{3}|x|^{-d-1}).$

We split $I_{1}$ accordingly into $I_{1}=I_{11}+I_{12}+I_{13}+I_{14}.$ Now for
$I_{11}$ we invoke our assumption $\int\rho(x)\,{\rm d}x=0$ which leads to

$\displaystyle|I_{11}(x)|$
$\displaystyle=|x|^{-d+2}\Big{|}\int_{|y|\geq|x|/2}\rho(y)\,{\rm
d}y\Big{|}\leq|x|^{-d+2}\int_{|y|\geq|x|/2}(1+|y|)^{-d-4}\,{\rm d}y$
$\displaystyle\leq C|x|^{-d}\int_{|y|\geq|x|/2}(1+|y|)^{-d-2}\,{\rm d}y\leq
C|x|^{-d}.$

We will now use the rotational symmetry of $\rho$, which means that
$\rho(y)=\overline{\rho}(|y|)$ for some
$\overline{\rho}:\mathbb{R}_{+}\to\mathbb{R}$. But

$\int y_{i}\overline{\rho}(|y|){\rm d}y=0,\qquad\int
y_{i}y_{j}\overline{\rho}(|y|)=\,\,i\neq j$

and

$\int y_{i}^{2}\overline{\rho}(|y|){\rm d}y=\int
y_{j}^{2}\overline{\rho}(|y|){\rm d}y$

follow by the invariance of Lebesgue measure under rotation and reflection.

Using these observations, we have $I_{12}=0$ and similarly we have the same
statement for the off-diagonal terms $i\neq j$ in $I_{13}(x).$ Now for the
diagonal terms in $I_{13}$ we use that $\int y_{i}^{2}\rho(y)\,{\rm d}y$ does
not depend on $i$ and noting that

$\sum_{i=1}^{d}[-d\cdot x_{i}^{2}|x|^{-d-2}+|x|^{-d}]=0$

Therefore $I_{13}(x)=0$. As for $I_{14},$ we have, by our upper bound on
$\rho$,

$|I_{14}(x)|\leq\int_{|y|\leq|x|/2}|\rho(y)|\,|y|^{3}\,{\rm
d}y\,|x|^{-d-1}=O(|x|^{-d}).$

Summarizing, $|x|^{d}|I_{1}(x)|$ is bounded. Estimating $I_{2}$ is simpler.
Indeed, using the decay of $\rho$ imposed in our assumption of the lemma, one
easily checks that $|I_{2}(x)|\leq C(1+|x|)^{-d-2}$, which is not larger as
$O(|x|^{-}d)$ as well. This shows that $|x|^{d}\rho\star G(x)$ is also bounded
and proves the lemma. ∎

###### Remark 5

Since for the function $\rho$ in Lemma 6.2 $\|\rho\|$ is finite, also
$\|\rho\star G+\rho\|$ is finite. $\diamondsuit$

######  Lemma 6.3.

For $\mu\leq 1$ define

$G_{\mu}(x):=\sum_{n=1}^{\infty}\mu^{n}\varphi_{n}(x),\qquad
x\in\mathbb{R}^{d}.$

If $\rho$ satisfies the conditions of Lemma 6.2, then there is a
$C\in(0,\infty)$ (depending only on $\rho$ and $d$) such that

$\|\rho\star G_{\mu}\|\leq C|\mu|,\qquad\mu\in[-1,1].$

###### Proof.

Let us first verify that $G_{\mu}\in\mathbf{B}$ for any $\mu\in(-1,1)$. We
will be using $C$ as a generic positive constant that does not depend on $x$
nor on $n$ and may change its value from appearance to appearance. Since
$\sup_{x}\varphi_{n}(x)=\varphi_{n}(0)\leq C$, we have
$\|G_{\mu}\|_{\infty}\leq\frac{C|\mu|}{1-|\mu|}$. Since the integral of
$\varphi_{n}$ is one, $\|G_{\mu}\|_{1}\leq\frac{C|\mu|}{1-|\mu|}$.
Furthermore,

$\sup_{x\in\mathbb{R}^{d}}|x|^{d}G_{\mu}(x)\leq\sum_{n=1}^{\infty}|\mu|^{n}\frac{1}{(2\pi
n)^{d/2}}\,\sup_{x\in\mathbb{R}^{d}}|x|^{d}\mathrm{e}^{-|x|^{2}/2n}=d^{d/2}\mathrm{e}^{-d/2}\sum_{n=1}^{\infty}|\mu|^{n}\frac{1}{(2\pi
n)^{d/2}}n^{d/2}\leq\frac{C|\mu|}{1-|\mu|}.$

Hence, $\|G_{\mu}\|\leq\frac{C|\mu|}{1-|\mu|}$. An elementary calculation
shows that

$\rho\star G_{\mu}=\mu(\rho\star G)-(1-\mu)\rho\star G\star
G_{\mu},\qquad\mu\in(-1,1].$

Therefore, for any $\mu\in[-\frac{1}{2},1]$,

$\|\rho\star G_{\mu}\|\leq|\mu|\|\rho\star G\|+(1-\mu)2^{d+1}\|\rho\star
G\|\,\|G_{\mu}\|\leq C|\mu|\|\rho\star G\|$

by applying Lemma 6.1. Now Lemma 6.2 implies the assertion. ∎

###  6.2. Proof of Lemma 4.6.

We would like to apply Lemma 6.2 and Lemma 6.3 to $\rho=-\lambda\varphi-
G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$. However, for applying Lemma 6.2
we need that $\int\rho(x)\,{\rm d}x=0$. Therefore, we add a suitable multiple
of $\varphi,$ by defining

$\displaystyle\rho:=-G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}+(\mu-\lambda)\varphi,\qquad\mbox{where
}\mu=\int\big{[}G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)+\lambda\varphi(x)\big{]}\,{\rm
d}x=\int G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)\,{\rm d}x+\lambda.$
(6.5)

Then $\rho$ is continuous and bounded and rotationally symmetric with
$\int\rho(x)\,{\rm d}x=0$ and satisfies the requested upper bound by (4.16).
We remark that from (4.18), we get

$\Big{(}1-\lambda-\int G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)\,{\rm
d}x\Big{)}\Big{(}1+\int
G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}(x)\,{\rm d}x\Big{)}=1,$

and as the second factor on the left-hand side is $>0,$ we get $\mu\leq 1$.
Furthermore, as $|\int G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}(x)\,{\rm
d}x|=O(\alpha)$ and $\lambda\geq 0,$ we have $\mu\geq-1/2$ for small enough
$\alpha$. Hence, $-\frac{1}{2}\leq\mu\leq 1$ and we can therefore apply Lemma
6.3.

Again, we use $C$ to denote a generic constant that may depend only on $d$ and
$v$ and may change its value from appearance to appearance. Note that
$|\mu|\leq C\alpha$. From Lemma 6.3 we have that $\|\rho\star G_{\mu}\|\leq
C\alpha$. Hence, we have also that $\|\rho\star G_{\mu}+\rho\|\leq C\alpha$.
Defining

$\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}=\lambda\varphi+G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$

one readily checks that

$-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}+G_{\mu}-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\star
G_{\mu}=\rho\star G_{\mu}+\rho.$

Therefore,
$\|-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}+G_{\mu}-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\star
G_{\mu}\|\leq C\alpha$. Therefore, if $\alpha$ is small enough, the Neumann
series

$\displaystyle
Q:=\sum_{n=1}^{\infty}(-1)^{n}(-\widetilde{G}_{\lambda}^{\Pi}+G_{\mu}-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\star
G_{\mu})^{\star n}.$

converges in $\mathbf{B}$, and satisfies $\|Q\|\leq C\alpha$. If we define

$S:=Q+G_{\mu}+Q\star G_{\mu}.$ (6.6)

then one checks that

$S-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\star
S=0.$ (6.7)

Formally this means that if $\delta_{0}$ is the Dirac function at $0,$ one has

$(\delta_{0}+S)\star(\delta_{0}-\widetilde{G}_{\lambda}^{\Pi})=\delta_{0}$

meaning that $\delta_{0}+S$ is the inverse of
$\delta_{0}-G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ under convolution.
The following lemma will complete the argument:

######  Lemma 6.4.

Under condition (4.16), if $\alpha>0$ is small enough, then
$S=G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$.

###### Proof of Lemma 6.4:.

$S$ and $G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}$ are in
$L^{2}(\mathbb{R}^{d})$ which then also contains
$G_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\star S$. The Fourier transform of
(6.7) is given by

$\widehat{S}\,\,-\,\,\widehat{\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}}\,\,-\,\,\widehat{S}\,\,\widehat{\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}}=0,\qquad\mbox{or}\qquad\big{(}1+\widehat{S}\big{)}\big{(}1-\widehat{\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}}\big{)}=1.$

Note that $G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$ satisfies the same
equation
$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}-\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}\star
G_{\lambda}^{\Gamma}=0,$ so we get

$\big{(}1+\widehat{G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}}\big{)}\big{(}1-\widehat{\widetilde{G}_{\lambda}^{{\scriptscriptstyle{({\Pi}})}}}\big{)}=1.$

This implies
$\widehat{S}=\widehat{G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}}$, so
that by Fourier inversion formula, we obtain
$S=G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}$, as required. ∎

###### Completing the proof of Lemma 4.6:.

By Lemma 6.4 and (6.6) we obtain

$G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}=S=Q+G_{\mu}+Q\star G_{\mu}.$

Recall that $G_{\mu}\leq G$. Also, from the arguments preceding Lemma 6.4 we
have that $\|Q+Q\star G_{\mu}\|\leq C\alpha$, so that for small enough
$\alpha$, by Remark 4 b) we conclude that

$\displaystyle G_{\lambda}^{{\scriptscriptstyle{({\Gamma}})}}=S$
$\displaystyle=Q+G_{\mu}+Q\star G_{\mu}\leq Q+G+Q\star G_{\mu}\leq 2G,$

as required. ∎

##  7\. Technical estimates

In the proofs of Lemma 4.4 and Lemma 4.6 we have used some technical
estimates, which we state and prove in this section. Recall that $\varphi$ is
the standard Gaussian density and that the interaction function
$v\colon[0,\infty)\to[0,\infty)$ is continuous and satisfies $v\leq L$ and
$v(r)=0$ for $r>R$ for some parameters $L,R\in(0,\infty)$.

######  Lemma 7.1 (Gaussian decay of $u_{n,\alpha}$).

Recall the function $u_{n}=u_{n,\alpha}$ defined in (5.1). Then there exist
constants $C,A\in(0,\infty)$, depending only on $d,L,R$, such that, for any
$n$ and
$\mathbf{x}=(x_{1},x_{2},\ldots,x_{2n-1},x_{2n})\in(\mathbb{R}^{d})^{2n}$,

${u}_{n}(\mathbf{x})\leq
C^{n}\alpha^{n-1}\exp\Big{[}-\frac{1}{8}\sum\nolimits_{i=1}^{n}|x_{2i}-x_{2i-1}|^{2}-A\sum\nolimits_{i=1}^{n-1}|x_{2i}-x_{2i+2}|^{2}\Big{]}.$

###### Proof.

Let us first assume that $n=2m$ is even (the case of odd $n$ is similar, we
drop it in the following), and identify $u_{2m}(\mathbf{x})$ in terms of
integrals over standard Brownian bridges
$\mathrm{BB}^{{\scriptscriptstyle{({i}})}}$ on $[0,1]$ that start and
terminate at zero. Indeed, if we define
$\overline{x}_{t}^{{\scriptscriptstyle{({i}})}}:=(1-t)x_{2i-1}+tx_{2i}$, then
$(\overline{x}_{t}^{{\scriptscriptstyle{({i}})}}+\mathrm{BB}^{{\scriptscriptstyle{({i}})}}(t))_{t\in[0,1]}$
are independent Brownian bridges from $x_{2i-1}$ to $x_{2i}$, $i=1,\dots,m$.
That is, they have the same distribution as
$\mu^{{\scriptscriptstyle{({1}})}}_{x_{2i-1},x_{2i}}/\varphi(x_{2i}-x_{2i-1})$.
Hence, we can write

$u_{2m}(\mathbf{x})=\prod_{i=1}^{2m}\varphi(x_{2i}-x_{2i-1})\mathbb{E}\bigg{[}\prod_{i=1}^{2m-1}(1-\exp[-\alpha\varpi_{i}])\bigg{]},\\\
$ (7.1)

where

$\varpi_{i}:=\int_{0}^{1}{\rm d}t\,\
v\Big{(}\big{|}\overline{x}_{t}^{{\scriptscriptstyle{({i}})}}+\mathrm{BB}^{{\scriptscriptstyle{({i}})}}(t)-\overline{x}_{t}^{{\scriptscriptstyle{({i+1}})}}-\mathrm{BB}^{{\scriptscriptstyle{({i+1}})}}(t)\big{|}\Big{)}\qquad\mbox{and
}\overline{x}_{t}^{{\scriptscriptstyle{({i}})}}=(1-t)x_{2i-1}+tx_{2i}.$

We will later upper estimate the expectation on the right-hand side of (7.1)
against products of expectations over Brownian brides, using the
Cauchy–Schwarz inequality; hence it will be critical and almost sufficient to
handle the case $n=2$. Hence, first prove the assertion for the case $n=2$. We
will use the well known fact (proved with the help of the reflection
principle) that the distribution of $\sup_{t\in[0,1]}|\mathrm{BB}(t)|$ has
sub-gaussian tails, i.e., there exist constants $c_{1},c_{2}>0$, depending
only on $d,$ such that

$\mathbb{P}\Big{(}\sup_{t\in[0,1]}|\mathrm{BB}(t)|>u\Big{)}\leq
c_{1}\exp[-c_{2}u^{2}],\qquad u\in(0,\infty).$ (7.2)

We first consider the set

$W:=\Big{\\{}\mathbf{x}\in\mathbb{R}^{4}\colon\max_{i=1,2}|x_{2i}-x_{2i-1}|>\frac{1}{4}|x_{4}-x_{2}|\Big{\\}}.$

If $|x_{2}-x_{1}|>|x_{4}-x_{2}|/4,$ we have

$\exp\Big{[}-\frac{1}{2}|x_{2}-x_{1}|^{2}\Big{]}\leq\exp\Big{[}-\frac{1}{4}|x_{2}-x_{1}|^{2}-\frac{1}{64}|x_{4}-x_{2}|^{2}\Big{]},$
(7.3)

and similarly when $|x_{4}-x_{3}|>|x_{4}-x_{2}|/4.$ Therefore, for
$\mathbf{x}\in W,$ we have, also using that $v\leq L$ and that $1-{\rm
e}^{-\alpha L}\leq\alpha L$,

${u}_{2}(\mathbf{x})\leq\frac{L\alpha}{(2\pi)^{d}}\exp\Big{[}-\frac{1}{4}|x_{2}-x_{1}|^{2}-\frac{1}{4}|x_{4}-x_{3}|^{2}-\frac{1}{64}|x_{4}-x_{2}|^{2}\Big{]},$

which shows the claim for $n=2$ on $W$.

On $W^{\rm c},$ and on the event

$\Big{\\{}\sup_{t\in[0,1]}|\mathrm{BB}^{{\scriptscriptstyle{({i}})}}(t)|\leq\frac{1}{2}|x_{4}-x_{2}|-2R\Big{\\}}$

one has $\varpi_{1}=0.$ Therefore, on $W^{\rm c},$

$\displaystyle{u}_{2}(\mathbf{x})$
$\displaystyle\leq\frac{L\alpha}{(2\pi)^{d}}\exp\Big{[}-\frac{1}{2}|x_{2}-x_{1}|^{2}-\frac{1}{2}|x_{4}-x_{3}|^{2}\Big{]}2\mathbb{P}\Big{(}\sup_{t\in[0,1]}|\mathrm{BB}(t)|>\frac{1}{2}|x_{4}-x_{2}|-2R\Big{)}$
(7.4) $\displaystyle\leq
C\alpha\exp\Big{[}-\frac{1}{2}|x_{2}-x_{1}|^{2}-\frac{1}{2}|x_{4}-x_{3}|^{2}-A|x_{4}-x_{2}|^{2}\Big{]},$

using (7.2), with some constants $C,A>0,$ depending only on $d,L,R.$ This