[2]footnote

# A multivariate extension of the Erdös-Taylor theorem

Dimitris Lygkonis, Nikos Zygouras  Department of Mathematics, University of
Warwick, Coventry CV4 7AL, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The Erdös-Taylor theorem [ET60] states that if $\mathsf{L}_{N}$ is the local
time at zero, up to time $2N$, of a two-dimensional simple, symmetric random
walk, then $\tfrac{\pi}{\log N}\,\mathsf{L}_{N}$ converges in distribution to
an exponential random variable with parameter one. This can be equivalently
stated in terms of the total collision time of two independent simple random
walks on the plane. More precisely, if
$\mathsf{L}_{N}^{(1,2)}=\sum_{n=1}^{N}\mathds{1}_{\\{S_{n}^{(1)}=S_{n}^{(2)}\\}}$,
then $\tfrac{\pi}{\log N}\,\mathsf{L}^{(1,2)}_{N}$ converges in distribution
to an exponential random variable of parameter one. We prove that for every
$h\geq 3$, the family $\big{\\{}\frac{\pi}{\log
N}\,\mathsf{L}_{N}^{(i,j)}\big{\\}}_{1\leq i<j\leq h}$, of logarithmically
rescaled, two-body collision local times between $h$ independent simple,
symmetric random walks on the plane converges jointly to a vector of
independent exponential random variables with parameter one, thus providing a
multivariate version of the Erdös-Taylor theorem. We also discuss connections
to directed polymers in random environments.

###### Key words and phrases:

planar random walk collisions, Erdös-Taylor theorem, Schrödinger operators
with point interactions, directed polymer in random environment

###### 2010 Mathematics Subject Classification:

82B44, 60G50, 60H15, 82D60, 47D08

###### Contents

  1. 1 Introduction.
  2. 2 Chaos expansions and auxiliary results
    1. 2.1 Chaos expansion for two-body collisions and renewal framework.
    2. 2.2 Chaos expansion for many-body collisions.
    3. 2.3 Functional analytic framework and some auxiliary estimates
  3. 3 Approximation steps and proof of the theorem
    1. 3.1 Reduction to 2-body collisions and finite order chaoses.
    2. 3.2 Diffusive spatial truncation.
    3. 3.3 Scale separation.
    4. 3.4 Rewiring.
    5. 3.5 Final step

##  1\. Introduction.

Let $S^{(1)},\dots,S^{(h)}$ be independent, simple, symmetric random walks on
$\mathbb{Z}^{2}$ starting at the origin. We will use $\mathrm{P}_{x}$ and
$\mathrm{E}_{x}$ to denote the probability and expectation with respect to the
law of the simple random walk when starting from $x\in\mathbb{Z}^{2}$ and we
will omit the subscripts when the walk starts from $0$. For $1\leq i<j\leq h$
we define the collision local time between $S^{(i)}$ and $S^{(j)}$ up to time
$N$ by

$\mathsf{L}_{N}^{(i,j)}:=\sum_{n=1}^{N}\mathds{1}_{\\{S_{n}^{(i)}=S_{n}^{(j)}\\}}\,.$

Notice that given $1\leq i<j\leq h$, $\mathsf{L}_{N}^{(i,j)}$ has the same law
as the number of returns to zero, before time $2N$, for a single simple,
symmetric random walk $S$ on $\mathbb{Z}^{2}$, that is
$\mathsf{L}_{N}^{(i,j)}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathsf{L}_{N}:=\sum_{n=1}^{N}\mathds{1}_{\\{S_{2n}=0\\}}$.
This equality is a consequence of the independence of $S^{(i)}$, $S^{(j)}$ and
the symmetry of the simple random walk. A first moment calculation shows that

$R_{N}:=\mathrm{E}\big{[}\mathsf{L}_{N}\big{]}=\sum_{n=1}^{N}\mathrm{P}(S_{2n}=0)\stackrel{{\scriptstyle
N\to\infty}}{{\approx}}\frac{\log N}{\pi}\,,$ (1.1)

see Section 2 for more details. It was established by Erdös and Taylor, $60$
years ago [ET60], that under normalisation (1.1), $\mathsf{L}_{N}$ satisfies
the following limit theorem.

###### Theorem A ([ET60]).

Let $\mathsf{L}_{N}:=\sum_{n=1}^{N}\mathds{1}_{\\{S_{2n}=0\\}}$ be the local
time at zero, up to time $2N$, of a two-dimensional, simple, symmetric random
walk $(S_{n})_{n\geq 1}$ starting at $0$. Then, as $N\to\infty$,

$\frac{\pi}{\log N}\,\mathsf{L}_{N}\xrightarrow{(d)}Y\,,$

where $Y$ has an exponential distribution with parameter $1$.

Theorem A was recently generalised in [LZ21]. In particular,

###### Theorem B ([LZ21]).

Let $h\in\mathbb{N}$ with $h\geq 2$ and $S^{(1)},...,S^{(h)}$ be $h$
independent two-dimensional, simple random walks starting all at zero. Then,
for $\beta\in(0,1)$, it holds that the total collision time $\sum_{1\leq
i<j\leq h}\mathsf{L}_{N}^{(i,j)}$ satisfies

$\mathrm{E}^{\otimes h}\bigg{[}e^{\frac{\pi\,\beta}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}_{N}^{(i,j)}}\bigg{]}\xrightarrow[N\to\infty]{}\bigg{(}\frac{1}{1-\beta}\bigg{)}^{\frac{h(h-1)}{2}}\,,$

and, consequently,

$\frac{\pi}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}_{N}^{(i,j)}\xrightarrow[N\to\infty]{(d)}\Gamma\big{(}\tfrac{h(h-1)}{2},1\big{)}\,,$

where $\Gamma\big{(}\tfrac{h(h-1)}{2},1\big{)}$ denotes a Gamma variable,
which has a density $\Gamma(h(h-1)/2)^{-1}x^{\tfrac{h(h-1)}{2}-1}e^{-x}$;
$\Gamma(\cdot)$, in the expression of the density, denotes the Gamma function.

Given the fact that a gamma distribution $\Gamma(k,1)$, with parameter $k\geq
1$, arises as the distribution of the sum of $k$ independent random variables
each one distributed according to an exponential random variable with
parameter one (denoted as ${\rm Exp}(1)$), Theorem B raises the question as to
whether the joint distribution of the individual rescaled collision times
$\Big{\\{}\frac{\pi}{\log N}\,\mathsf{L}_{N}^{(i,j)}\Big{\\}}_{1\leq i<j\leq
h}$ converges to that of a family of independent $\text{Exp}(1)$ random
variables. This is what we prove in this work. In particular,

###### Theorem 1.1.

Let $h\in\mathbb{N}$ with $h\geq 2$ and
$\boldsymbol{\beta}:=\\{\beta_{i,j}\\}_{1\leq i<j\leq
h}\in\mathbb{R}^{\frac{h(h-1)}{2}}$ with $\beta_{i,j}<1$ for all $1\leq
i<j\leq h$. Then we have that

$\mathrm{E}^{\otimes h}\bigg{[}e^{\frac{\pi}{\log N}\sum_{1\leq i<j\leq
h}\beta_{i,j}\mathsf{L}_{N}^{(i,j)}}\bigg{]}\xrightarrow[N\to\infty]{}\prod_{1\leq
i<j\leq h}\frac{1}{1-\beta_{i,j}}\,$ (1.2)

and, consequently,

$\Big{\\{}\tfrac{\pi}{\log N}\,\mathsf{L}_{N}^{(i,j)}\Big{\\}}_{1\leq i<j\leq
h}\xrightarrow[N\to\infty]{(d)}\big{\\{}Y^{(i,j)}\big{\\}}_{1\leq i<j\leq
h}\,,$

where $\big{\\{}Y^{(i,j)}\big{\\}}_{1\leq i<j\leq h}$ are independent and
identically distributed random variables following an $\text{Exp}(1)$
distribution.

An intuitive way to understand the convergence of the individual collision
times, or equivalently of the local time of a planar walk, to an exponential
variable is the following. By (1.1), the number of visits to zero of a planar
walk, which starts at zero, is $O(\log N)$ and, thus, much smaller than the
time horizon $2N$. Typically, also, these visits happen within a short time,
much smaller than $2N$, so that every time the random walk is back at zero,
the probability that it will return there again before time $2N$ is not
essentially altered. This results in the local time $\mathsf{L}_{N}$ being
close to a geometric random variable with parameter of order $(\log N)^{-1}$
(as also manifested by (1.1)), which when rescaled suitably converges to an
exponential random variable.

The fact that the joint distribution of $\Big{\\{}\tfrac{\pi}{\log
N}\,\mathsf{L}_{N}^{(i,j)}\Big{\\}}_{1\leq i<j\leq h}$ converges to that of
independent  exponentials is much less apparent as the collision times have
obvious correlations. A way to understand this is, again, through the fact
that collisions happen at time scales much shorter than the time horizon $N$
and, thus, it is not really possible to actually distinguish which pairs
collide when collisions take place. More crucially, the logarithmic scaling,
as indicated via (1.1), introduces a separation of scales between collisions
of different pairs of walks, which is what, essentially, leads to the
asymptotic factorisation of the Laplace transform (1.2). This intuition is
reflected in the two main steps in our proof, which are carried out in
Sections 3.3 and 3.4.

Even though the Erdös-Taylor theorem appeared a long time ago, the
multivariate extension that we establish here appears to be new. In [GS09] it
was shown that the law of $\tfrac{\pi}{\log N}\,\mathsf{L}^{(1,2)}_{N}$,
conditioned on $S^{(1)}$, converges a.s. to that of an ${\rm Exp}(1)$ random
variable. This implies that $\big{\\{}\tfrac{\pi}{\log
N}\,\mathsf{L}^{(1,i)}_{N}\big{\\}}_{1<i\leq h}$ converge to independent
exponentials. However, it does not address the full independence of the family
of all pairwise collisions $\big{\\{}\tfrac{\pi}{\log
N}\,\mathsf{L}^{(i,j)}_{N}\big{\\}}_{1\leq i<j\leq h}$.

In the continuum, phenomena of independence in functionals of planar Brownian
motions have appeared in works around log-scaling laws see [PY86] (where the
term log-scaling laws was introduced) as well as [Y91] and [Kn93]. These works
are mostly concerned with the problem of identifying the limiting distribution
of windings of a planar Brownian motion around a number of points
$z_{1},...,z_{k}$, different than the starting point of the Brownian motion,
or the winding around the origin of the differences $B^{(i)}-B^{(j)}$ between
$k$ independent Brownian motions $B^{(1)},...,B^{(k)}$, starting all from
different points, which are also different than zero. Without getting into
details, we mention that the results of [PY86, Y91, Kn93] establish that the
windings (as well as some other functionals that fall within the class of log-
scaling laws) converge, when logarithmically scaled, to independent Cauchy
variables. [Kn93] outlines a proof that the local times of the differences
$B^{(i)}-B^{(j)},1\leq i<j\leq k$, on the unit circle
$\\{z\in\mathbb{R}^{2}\colon|z|=1\\}$ converge, jointly, to independent
exponentials ${\rm Exp}(1)$, when logarithmically scaled, in a fashion similar
to the scaling of Theorem 1.1. The methods employed in the above works rely
heavily on continuous techniques (Itô calculus, time changes etc.), which do
not have discrete counterparts. In fact, the passage from continuous to
discrete is not straightforward either at a technical level (see e.g. the
discussion on page 41 of [Kn93] and [Kn94]) or at a phenomenological level
(see e.g. discussion on page 736 of [PY86]).

The approach we follow towards Theorem 1.1 starts with expanding the joint
Laplace transform in the form of chaos series, which take the form of Feynman-
type diagrams. To control (and simplify) these diagrams, we start by inputing
a renewal representation as well as a functional analytic framework. The
renewal theoretic framework was originally introduced in [CSZ19a] in the
context of scaling limits of random polymers (we will come back to the
connection with polymers later on) and it captures the stream of collisions
within a single pair of walks. The functional analytic framework can be traced
back to works on spectral theory of delta-Bose gases [DFT94, DR04] and was
also recently used in works on random polymers [GQT21, CSZ21, LZ21]. The core
of this framework is to establish operator norm bounds for the total Green’s
functions of a set of planar random walks conditioned on a subset of them
starting at the same location and on another subset of them ending up at the
same location. Roughly speaking, the significance of these operator estimates
is to control the redistribution of collisions when walks switch pairs. The
operator framework (together with the renewal one) allows to reduce the number
of Feynman-type diagrams that need to be considered. For the reduced Feynman
diagrams one, then, needs to look into the logarithmic structure, which
induces the separation of scales and leads to the fact that, asymptotically,
the structure of the Feynman diagrams becomes that of the product of Feynman
diagrams corresponding to Laplace transforms of single pairs of random walks.

Exponential moments of collision times arise naturally when one looks at
moments of partition functions of the model of directed polymer in a random
environment (DPRE), we refer to [C17] for an introduction to this model. For a
family of i.i.d. variables $\big{(}\omega_{n,x}\colon
n\in\mathbb{N},x\in\mathbb{Z}^{2}\big{)}$ with log-moment generating function
$\lambda(\beta)$, the partition function of the directed polymer measure is
defined as

$\displaystyle
Z_{N,\beta}(x):=\mathrm{E}_{x}\Big{[}\exp\Big{(}\sum_{n=1}^{N}\big{(}\beta\omega_{n,S_{n}}-\lambda(\beta)\big{)}\Big{)}\Big{]},$

where $\mathrm{E}_{x}$ is the expected value with respect to a simple,
symmetric walk starting at $x\in\mathbb{Z}^{2}$. Considering, for simplicity,
$\omega_{n,x}$ to be a standard normal variable, we can easily compute, for
$\beta_{N}:=\beta\sqrt{\tfrac{\pi}{\log N}}$, that

$\displaystyle{\mathbb{E}}\Big{[}\big{(}Z_{N,\beta_{N}}(x)\big{)}^{h}\Big{]}=\mathrm{E}^{\otimes
h}\bigg{[}e^{\frac{\pi\beta^{2}}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}_{N}^{(i,j)}}\bigg{]}.$

A corollary of our Theorem 1.1 is that the limit of the $h$-th moment of the
DPRE partition function converges to $(1-\beta^{2})^{-h(h-1)/2}$; a result
that was previously obtained in [LZ21] combining upper bounds on moments,
established in [LZ21], with results on the distributional convergence of the
partition function established in [CSZ17]. Using Theorem 1.1, we can further
extend this to convergence of mixed moments. More precisely, for
$\beta_{i,N}:=\beta_{i}\sqrt{\tfrac{\pi}{\log N}}$ and $i=1,...,h$, we have
that

$\displaystyle{\mathbb{E}}\big{[}Z_{N,\beta_{1,N}}(x)\cdots
Z_{N,\beta_{h,N}}(x)\big{]}\xrightarrow[N\to\infty]{}\prod_{1\leq i<j\leq
N}\frac{1}{1-\beta_{i}\beta_{j}}.$

Moment estimates are important in establishing fine properties, such as
structure of maxima, of the field of partition functions $\big{\\{}\sqrt{\log
N}\big{(}\log Z_{N,\beta}(x)-{\mathbb{E}}[\log Z_{N,\beta}(x)]\big{)}\colon
x\in\mathbb{Z}^{2}\big{\\}}$, which is known to converge to a log-correlated
gaussian field [CSZ18]. We refer to [CZ21] for more details. We expect that
the independence structure of the collision local times, that we establish
here, to be useful towards these investigations. An interesting problem, in
relation to this (but also of broader interest), is how large can the number
$h$ of random walks be (depending on $N$), before we start seeing correlations
in the limit of the rescaled collisions. The work of Cosco-Zeitouni [CZ21] has
shown that there exists $\beta_{0}\in(0,1)$ such that for all
$\beta\in(0,\beta_{0})$ and $h=h_{N}\in\mathbb{N}$ such that

$\limsup_{N\to\infty}\frac{3\beta}{1-\beta}\frac{1}{\log N}\binom{h}{2}<1\,,$

one has that

$\mathrm{E}^{\otimes h}\bigg{[}e^{\frac{\pi\,\beta}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}_{N}^{(i,j)}}\bigg{]}\leq
c(\beta)\,\Big{(}\frac{1}{1-\beta}\Big{)}^{\binom{h}{2}(1+\varepsilon_{N})}\,,$

with $c(\beta)\in(0,\infty)$ and
$0\leq\varepsilon_{N}=\varepsilon(\beta,N)\downarrow 0$ as $N\to\infty$. This
suggests that the threshold might be $h=h_{N}=O(\sqrt{\log N})$.

The structure of the article is as follows: In Section 2 we set the framework
of the chaos expansion, its graphical representations in terms of Feynman-type
diagrams, as well as the renewal and functional analytic frameworks. In
Section 3 we carry out the approximation steps, which lead to our theorem. At
the beginning of Section 3 we also provide an outline of the scheme.

##  2\. Chaos expansions and auxiliary results

In this section we will introduce the framework, within which we work, and
which consists of setting chaos expansions for the joint Laplace transform

$M^{\boldsymbol{\beta}}_{N,h}:=\mathrm{E}^{\otimes h}\bigg{[}e^{\sum_{1\leq
i<j\leq h}\frac{\pi\beta_{i,j}}{\log N}\,\mathsf{L}_{N}^{(i,j)}}\bigg{]}\,,$
(2.1)

for a fixed collection of numbers
$\boldsymbol{\beta}:=\\{\beta_{i,j}\\}_{1\leq i<j\leq
h}\in\mathbb{R}^{\frac{h(h-1)}{2}}$ with $\beta_{i,j}\in(0,1)$ for all $1\leq
i<j\leq h$. We denote by

$\bar{\beta}:=\max_{1\leq i<j\leq h}\beta_{i,j}<1\,,$ (2.2)

and define

$\displaystyle\sigma_{N}^{i,j}:=\sigma_{N}^{i,j}(\beta_{i,j}):=e^{\beta^{i,j}_{N}}-1\qquad\text{with}\qquad\beta^{i,j}_{N}:=\frac{\pi\,\beta_{i,j}}{\log
N}.$ (2.3)

We will use the notation $q_{n}(x):=\mathrm{P}(S_{n}=x)$ for the transition
probability of the simple, symmetric random walk. The expected collision local
time between two independent simple, symmetric random walks will be

$R_{N}:=\mathrm{E}^{\otimes
2}\Big{[}\sum_{n=1}^{N}\mathds{1}_{S_{n}^{(1)}=S_{n}^{(2)}}\Big{]}=\sum_{n=1}^{N}q_{2n}(0)$
(2.4)

and by Proposition 3.2 in [CSZ19a] we have that in the two-dimensional setting

$R_{N}=\frac{\log N}{\pi}+\frac{\alpha}{\pi}+o(1)\,,$ (2.5)

as $N\to\infty$, with $\alpha=\gamma+\log 16-\pi\simeq 0.208$ and
$\gamma\simeq 0.577$ is the Euler constant.

###  2.1. Chaos expansion for two-body collisions and renewal framework.

We start with the Laplace transform of the simple case of two-body collisions
$\mathrm{E}\Big{[}e^{\beta^{i,j}_{N}\,\mathsf{L}^{(i,j)}_{N}}\Big{]}$ and
deduce its chaos expansion as follows:

$\displaystyle\mathrm{E}\Big{[}e^{\beta^{i,j}_{N}\,\mathsf{L}^{(i,j)}_{N}}\Big{]}$
$\displaystyle=\mathrm{E}\bigg{[}e^{\beta^{i,j}_{N}\,\sum_{n=1}^{N}\sum_{x\in\mathbb{Z}^{2}}\mathds{1}_{\\{S_{n}^{(i)}=x\\}}\mathds{1}_{\\{S_{n}^{(j)}=x\\}}}\bigg{]}=\mathrm{E}\bigg{[}\prod_{\begin{subarray}{c}1\leq
n\leq N\\\
x\in\mathbb{Z}^{2}\end{subarray}}\Big{(}1+\Big{(}e^{\beta^{i,j}_{N}}-1\Big{)}\,\mathds{1}_{\\{S_{n}^{(i)}=x\\}}\mathds{1}_{\\{S_{n}^{(j)}=x\\}}\Big{)}\bigg{]}$
$\displaystyle=1+\sum_{k\geq
1}(\sigma_{N}^{i,j})^{k}\sum_{\begin{subarray}{c}1\leq n_{1}<\dots<n_{k}\leq
N\\\
x_{1},...,x_{k}\in\mathbb{Z}^{2}\end{subarray}}\mathrm{E}\bigg{[}\prod_{a=1}^{k}\,\mathds{1}_{\\{S_{n_{a}}^{(i)}=x_{a}\\}}\mathds{1}_{\\{S_{n_{a}}^{(j)}=x_{a}\\}}\bigg{]}$
$\displaystyle=1+\sum_{k\geq
1}(\sigma_{N}^{i,j})^{k}\sum_{\begin{subarray}{c}1\leq n_{1}<\dots<n_{k}\leq
N,\,\\\
x_{1},\dots,x_{k}\in\mathbb{Z}^{2}\end{subarray}}\prod_{a=1}^{k}q^{2}_{n_{a}-n_{a-1}}(x_{a}-x_{a-1})$
(2.6)

where in the last equality we used the Markov property, in the third we
expanded the product and in the second we used the simple fact that

$\displaystyle
e^{\beta_{N}^{i,j}\,\mathds{1}_{\\{S_{n}^{(i)}=S_{n}^{(j)}=x\\}}}$
$\displaystyle=1+\Big{(}e^{\beta_{N}^{i,j}\,\mathds{1}_{\\{S_{n}^{(i)}=S_{n}^{(j)}=x\\}}}-1\Big{)}=1+\Big{(}e^{\beta_{N}^{i,j}}-1\Big{)}\mathds{1}_{\\{S_{n}^{(i)}=S_{n}^{(j)}=x\\}}$
$\displaystyle=1+\sigma_{N}^{i,j}\,\mathds{1}_{\\{S_{n}^{(i)}=S_{n}^{(j)}=x\\}},$

with $\sigma^{i,j}_{N}$ defined in (2.3). We will express (2.1) in terms of
the following quantity $U^{\beta}_{N}(n,x)$, which plays an important role in
our formulation. For $\beta>0$,
$\sigma_{N}:=\sigma_{N}(\beta):=e^{\frac{\pi\beta}{\log N}}-1$ and
$(n,x)\in\mathbb{N}\times\mathbb{Z}^{2}$, we define

$\begin{split}U^{\beta}_{N}(n,x)&:=\sigma_{N}\,q^{2}_{n}(x)\\\ &+\sum_{k\geq
1}\sigma_{N}^{k+1}\sum_{\begin{subarray}{c}0<n_{1}<\dots<n_{k}<n\\\
z_{1},z_{2},\dots,z_{k}\in\mathbb{Z}^{2}\end{subarray}}q^{2}_{n_{1}}(z_{1})\Big{\\{}\prod_{j=2}^{k}q^{2}_{n_{j}-n_{j-1}}(z_{j}-z_{j-1})\Big{\\}}\,q^{2}_{n-n_{k}}(x-z_{k}).\end{split}$
(2.7)

and $U_{N}^{\beta}(n,x):=\mathds{1}_{\\{x=0\\}}$, if $n=0$. Moreover, for
$n\in\mathbb{N}$ we define

$U^{\beta}_{N}(n):=\sum_{x\in\mathbb{Z}^{2}}U^{\beta}_{N}(n,x)\,.$

$U^{\beta}_{N}(n,x)$ represents the Laplace transform of the two-body
collisions, scaled by $\beta$, between a pair of random walks that are
constrained to end at the spacetime point
$(n,x)\in\\{1,\dots,N\\}\times\mathbb{Z}^{2}$, starting from $(0,0)$. In
particular, for any $1\leq i<j\leq h$, we can write (2.1) as

$\displaystyle\mathrm{E}\Big{[}e^{\beta^{i,j}_{N}\,\mathsf{L}^{(i,j)}_{N}}\Big{]}=\sum_{n=0}^{N}\sum_{x\in\mathbb{Z}^{2}}U^{\beta_{i,j}}_{N}(n,x)=\sum_{n=0}^{N}U^{\beta_{i,j}}_{N}(n).$

We will call $U^{\beta}_{N}(n,x)$ a replica and for
$\sigma_{N}(\beta)=e^{\frac{\pi\beta}{\log N}}-1$ we will graphically
represent $\sigma_{N}(\beta)\,U^{\beta}_{N}(n,x)$ as

$\displaystyle\sigma_{N}(\beta)\,U^{\beta}_{N}(b-a,y-x)$
$\displaystyle\equiv\quad\leavevmode\hbox to75.88pt{\vbox
to22.9pt{\pgfpicture\makeatletter\hbox{\hskip 12.62234pt\lower-20.13666pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.56068pt}{0.0pt}\pgfsys@curveto{2.56068pt}{1.41423pt}{1.41423pt}{2.56068pt}{0.0pt}{2.56068pt}\pgfsys@curveto{-1.41423pt}{2.56068pt}{-2.56068pt}{1.41423pt}{-2.56068pt}{0.0pt}\pgfsys@curveto{-2.56068pt}{-1.41423pt}{-1.41423pt}{-2.56068pt}{0.0pt}{-2.56068pt}\pgfsys@curveto{1.41423pt}{-2.56068pt}{2.56068pt}{-1.41423pt}{2.56068pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{51.21461pt}{0.0pt}\pgfsys@moveto{53.7753pt}{0.0pt}\pgfsys@curveto{53.7753pt}{1.41423pt}{52.62885pt}{2.56068pt}{51.21461pt}{2.56068pt}\pgfsys@curveto{49.80038pt}{2.56068pt}{48.65393pt}{1.41423pt}{48.65393pt}{0.0pt}\pgfsys@curveto{48.65393pt}{-1.41423pt}{49.80038pt}{-2.56068pt}{51.21461pt}{-2.56068pt}\pgfsys@curveto{52.62885pt}{-2.56068pt}{53.7753pt}{-1.41423pt}{53.7753pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21461pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{} {}{} {}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {}{}{}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}}
{{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}}
}}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}}
}{{{{}{}{{}} }}{{}} {{}{}{}{}} {{}{}{}{}} }{{{{}{}{{}} }}{{}} {{}{}{}{}}
{{}{}{}{}} }{{{{}{}{{}} }}{{}} {}{}{} }{{{{}{}{{}} }}{{}} {{}} }
\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{0.71132pt}{0.0pt}{1.06697pt}{1.1381pt}{1.77829pt}{1.1381pt}\pgfsys@curveto{2.29327pt}{1.1381pt}{2.73714pt}{0.58272pt}{3.20093pt}{0.0pt}\pgfsys@curveto{3.6647pt}{-0.5827pt}{4.10857pt}{-1.1381pt}{4.62357pt}{-1.1381pt}\pgfsys@curveto{5.13855pt}{-1.1381pt}{5.58241pt}{-0.58272pt}{6.0462pt}{0.0pt}\pgfsys@curveto{6.50998pt}{0.5827pt}{6.95384pt}{1.1381pt}{7.46884pt}{1.1381pt}\pgfsys@curveto{7.98383pt}{1.1381pt}{8.42769pt}{0.58272pt}{8.89148pt}{0.0pt}\pgfsys@curveto{9.35526pt}{-0.5827pt}{9.79912pt}{-1.1381pt}{10.31412pt}{-1.1381pt}\pgfsys@curveto{10.8291pt}{-1.1381pt}{11.27296pt}{-0.58272pt}{11.73676pt}{0.0pt}\pgfsys@curveto{12.20053pt}{0.5827pt}{12.6444pt}{1.1381pt}{13.1594pt}{1.1381pt}\pgfsys@curveto{13.67438pt}{1.1381pt}{14.11824pt}{0.58272pt}{14.58203pt}{0.0pt}\pgfsys@curveto{15.0458pt}{-0.5827pt}{15.48967pt}{-1.1381pt}{16.00467pt}{-1.1381pt}\pgfsys@curveto{16.51965pt}{-1.1381pt}{16.96352pt}{-0.58272pt}{17.4273pt}{0.0pt}\pgfsys@curveto{17.89108pt}{0.5827pt}{18.33495pt}{1.1381pt}{18.84995pt}{1.1381pt}\pgfsys@curveto{19.36493pt}{1.1381pt}{19.80879pt}{0.58272pt}{20.27258pt}{0.0pt}\pgfsys@curveto{20.73636pt}{-0.5827pt}{21.18022pt}{-1.1381pt}{21.69522pt}{-1.1381pt}\pgfsys@curveto{22.2102pt}{-1.1381pt}{22.65407pt}{-0.58272pt}{23.11786pt}{0.0pt}\pgfsys@curveto{23.58163pt}{0.5827pt}{24.0255pt}{1.1381pt}{24.5405pt}{1.1381pt}\pgfsys@curveto{25.05548pt}{1.1381pt}{25.49934pt}{0.58272pt}{25.96313pt}{0.0pt}\pgfsys@curveto{26.42691pt}{-0.5827pt}{26.87077pt}{-1.1381pt}{27.38577pt}{-1.1381pt}\pgfsys@curveto{28.0971pt}{-1.1381pt}{28.45276pt}{0.0pt}{29.16408pt}{0.0pt}\pgfsys@lineto{51.21461pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.28934pt}{-14.80365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.8}{$(a,x)$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{42.50444pt}{-14.80365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.8}{$(b,y)$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$\displaystyle:=\sum_{k\geq 1}\sum_{\begin{subarray}{c}n_{1}<\cdots<n_{k}\\\
x_{1},...,x_{k}\end{subarray}}\leavevmode\hbox to150.45pt{\vbox
to27.47pt{\pgfpicture\makeatletter\hbox{\hskip
11.46118pt\lower-19.63666pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{
}\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.56068pt}{0.0pt}\pgfsys@curveto{2.56068pt}{1.41423pt}{1.41423pt}{2.56068pt}{0.0pt}{2.56068pt}\pgfsys@curveto{-1.41423pt}{2.56068pt}{-2.56068pt}{1.41423pt}{-2.56068pt}{0.0pt}\pgfsys@curveto{-2.56068pt}{-1.41423pt}{-1.41423pt}{-2.56068pt}{0.0pt}{-2.56068pt}\pgfsys@curveto{1.41423pt}{-2.56068pt}{2.56068pt}{-1.41423pt}{2.56068pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@moveto{28.16798pt}{0.0pt}\pgfsys@curveto{28.16798pt}{1.41423pt}{27.02153pt}{2.56068pt}{25.6073pt}{2.56068pt}\pgfsys@curveto{24.19307pt}{2.56068pt}{23.04662pt}{1.41423pt}{23.04662pt}{0.0pt}\pgfsys@curveto{23.04662pt}{-1.41423pt}{24.19307pt}{-2.56068pt}{25.6073pt}{-2.56068pt}\pgfsys@curveto{27.02153pt}{-2.56068pt}{28.16798pt}{-1.41423pt}{28.16798pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{25.6073pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{51.21461pt}{0.0pt}\pgfsys@moveto{53.7753pt}{0.0pt}\pgfsys@curveto{53.7753pt}{1.41423pt}{52.62885pt}{2.56068pt}{51.21461pt}{2.56068pt}\pgfsys@curveto{49.80038pt}{2.56068pt}{48.65393pt}{1.41423pt}{48.65393pt}{0.0pt}\pgfsys@curveto{48.65393pt}{-1.41423pt}{49.80038pt}{-2.56068pt}{51.21461pt}{-2.56068pt}\pgfsys@curveto{52.62885pt}{-2.56068pt}{53.7753pt}{-1.41423pt}{53.7753pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{51.21461pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{102.42923pt}{0.0pt}\pgfsys@moveto{104.98991pt}{0.0pt}\pgfsys@curveto{104.98991pt}{1.41423pt}{103.84346pt}{2.56068pt}{102.42923pt}{2.56068pt}\pgfsys@curveto{101.015pt}{2.56068pt}{99.86855pt}{1.41423pt}{99.86855pt}{0.0pt}\pgfsys@curveto{99.86855pt}{-1.41423pt}{101.015pt}{-2.56068pt}{102.42923pt}{-2.56068pt}\pgfsys@curveto{103.84346pt}{-2.56068pt}{104.98991pt}{-1.41423pt}{104.98991pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{102.42923pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{128.03654pt}{0.0pt}\pgfsys@moveto{130.59723pt}{0.0pt}\pgfsys@curveto{130.59723pt}{1.41423pt}{129.45078pt}{2.56068pt}{128.03654pt}{2.56068pt}\pgfsys@curveto{126.62231pt}{2.56068pt}{125.47586pt}{1.41423pt}{125.47586pt}{0.0pt}\pgfsys@curveto{125.47586pt}{-1.41423pt}{126.62231pt}{-2.56068pt}{128.03654pt}{-2.56068pt}\pgfsys@curveto{129.45078pt}{-2.56068pt}{130.59723pt}{-1.41423pt}{130.59723pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{128.03654pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} {}{{}}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{7.06125pt}{7.06125pt}{18.54605pt}{7.06125pt}{25.6073pt}{0.0pt}\pgfsys@curveto{32.66855pt}{7.06125pt}{44.15337pt}{7.06125pt}{51.21461pt}{0.0pt}\pgfsys@curveto{53.2768pt}{2.06218pt}{54.70006pt}{3.84113pt}{57.61644pt}{3.84113pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{7.06125pt}{-7.06125pt}{18.54605pt}{-7.06125pt}{25.6073pt}{0.0pt}\pgfsys@curveto{32.66855pt}{-7.06125pt}{44.15337pt}{-7.06125pt}{51.21461pt}{0.0pt}\pgfsys@curveto{53.2768pt}{-2.06218pt}{54.70006pt}{-3.84113pt}{57.61644pt}{-3.84113pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{96.0274pt}{3.84113pt}\pgfsys@curveto{98.94377pt}{3.84113pt}{100.36703pt}{2.06218pt}{102.42923pt}{0.0pt}\pgfsys@curveto{109.49048pt}{7.06125pt}{120.9753pt}{7.06125pt}{128.03654pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{} {{}{}}{{}}
{{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{96.0274pt}{-3.84113pt}\pgfsys@curveto{98.94377pt}{-3.84113pt}{100.36703pt}{-2.06218pt}{102.42923pt}{0.0pt}\pgfsys@curveto{109.49048pt}{-7.06125pt}{120.9753pt}{-7.06125pt}{128.03654pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.12817pt}{-14.55365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.7}{$(a,x)$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.26836pt}{-14.55365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.7}{$(n_{1},x_{1})$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{40.87567pt}{-14.55365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.7}{$(n_{2},x_{2})$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{91.88614pt}{-14.55365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.7}{$(n_{k},x_{k})$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{120.41515pt}{-14.55365pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scalebox{0.7}{$(b,y)$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{73.07191pt}{-2.5pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\cdots$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

In the second line we have assigned weights $q_{n^{\prime}-n}(x^{\prime}-x)$
to the solid lines going from $(n,x)$ to $(n^{\prime},x^{\prime})$ and we have
assigned the weight $\sigma_{N}(\beta)=e^{\frac{\pi\beta}{\log N}}-1$ to every
solid dot.

$U^{\beta}_{N}(n)$ and $U^{\beta}_{N}(n,x)$ admit a very useful probabilistic
interpretation in terms of certain renewal processes. More specifically,
consider the family of i.i.d. random variables
$(T^{\scriptscriptstyle(N)}_{i},X^{\scriptscriptstyle(N)}_{i})_{i\geq 1}$ with
law

$\displaystyle\mathrm{P}\Big{(}\,\big{(}T^{\scriptscriptstyle(N)}_{1},X^{\scriptscriptstyle(N)}_{1}\big{)}=(n,x)\Big{)}=\frac{q_{n}^{2}(x)}{R_{N}}\,\mathds{1}_{\\{n\leq
N\\}}\,.$

and $R_{N}$ defined in (2.4). Define the random variables
$\tau^{\scriptscriptstyle(N)}_{k}:=T_{1}^{\scriptscriptstyle(N)}+\dots+T_{k}^{\scriptscriptstyle(N)}$,
$S^{\scriptscriptstyle(N)}_{k}:=X^{\scriptscriptstyle(N)}_{1}+\dots+X^{\scriptscriptstyle(N)}_{k}$,
if $k\geq 1$, and $(\tau_{0},S_{0}):=(0,0)$, if $k=0$. It is not difficult to
see that $U^{\beta}_{N}(n,x)$ and $U^{\beta}_{N}(n)$ can, now, be written as

$U^{\beta}_{N}(n,x)=\sum_{k\geq
0}(\sigma_{N}R_{N})^{k}\,\mathrm{P}\big{(}\tau^{\scriptscriptstyle(N)}_{k}=n,S^{\scriptscriptstyle(N)}_{k}=x\big{)}\quad\text{and}\quad
U^{\beta}_{N}(n)=\sum_{k\geq
0}(\sigma_{N}R_{N})^{k}\,\mathrm{P}\big{(}\tau^{\scriptscriptstyle(N)}_{k}=n\big{)}$
(2.8)

This formalism was developed in [CSZ19a] and is very useful in obtaining sharp
asymptotic estimates. In particular, it was shown in [CSZ19a] that the
rescaled process $\Big{(}\frac{\tau^{(N)}_{\lfloor s\log
N\rfloor}}{N},\frac{S^{(N)}_{\lfloor s\log N\rfloor}}{\sqrt{N}}\Big{)}$
converges in distribution for $N\to\infty$ with the law of the marginal
limiting process for $\tfrac{\tau^{(N)}_{\lfloor s\log N\rfloor}}{N}$ being
the Dickman subordinator, which was defined in [CSZ19a] as a truncated, zero-
stable Lévy process.

An estimate that follows easily from this framework, which is useful for our
purposes here, is the following: for $\beta<1$, it holds

$\displaystyle\limsup_{N\to\infty}\sum_{n=0}^{N}U^{\beta}_{N}(n)$
$\displaystyle=\limsup_{N\to\infty}\sum_{k\geq
0}(\sigma_{N}R_{N})^{k}\,\mathrm{P}\big{(}\tau^{\scriptscriptstyle(N)}_{k}\leq
N\big{)}$ $\displaystyle\leq\limsup_{N\to\infty}\sum_{k\geq
0}(\sigma_{N}R_{N})^{k}$
$\displaystyle=\limsup_{N\to\infty}\frac{1}{1-\sigma_{N}R_{N}}=\frac{1}{1-\beta},$
(2.9)

where we used the fact that

$\displaystyle\sigma_{N}R_{N}=\big{(}e^{\frac{\pi\beta}{\log
N}}-1\big{)}\cdot\Big{(}\frac{\log
N}{\pi}+\frac{\alpha}{\pi}+o(1)\Big{)}\xrightarrow[N\to\infty]{}\beta<1.$
(2.10)

###  2.2. Chaos expansion for many-body collisions.

We now move to the expansion of the Laplace transform of the many-body
collisions $M^{\boldsymbol{\beta}}_{N,h}$. The goal is to obtain an expansion
in the form of products of certain Markovian operators. The desired expression
will be presented in (2.17). This expansion will be instrumental in obtaining
some important estimates in Section 2.3.

The first steps are similar as in the expansion for the two-body collisions,
above. In particular, we have

$\displaystyle\mathrm{E}^{\otimes h}\bigg{[}e^{\sum_{1\leq i<j\leq
h}\beta^{i,j}_{N}\,\mathsf{L}^{(i,j)}_{N}}\bigg{]}$
$\displaystyle=\mathrm{E}\bigg{[}\prod_{1\leq i<j\leq
h}\,\prod_{\begin{subarray}{c}1\leq n\leq N\\\
x\in\mathbb{Z}^{2}\end{subarray}}\Big{(}1+\sigma_{N}^{i,j}\,\mathds{1}_{\\{S_{n}^{(i)}=x\\}}\mathds{1}_{\\{S_{n}^{(j)}=x\\}}\Big{)}\bigg{]}$
(2.11) $\displaystyle=1+\sum_{k\geq
1}\,\,\sum_{\begin{subarray}{c}(i_{a},j_{a},n_{a},x_{a})\in{\mathcal{A}}_{h},\,\,\text{for}\,\,a=1,...,k\\\
\text{distinct}\end{subarray}}\,\,\mathrm{E}\Big{[}\prod_{a=1}^{k}\sigma_{N}^{i_{a},j_{a}}\,\mathds{1}_{\\{S_{n_{a}}^{(i_{a})}=x_{a}\\}}\,\mathds{1}_{\\{S_{n_{a}}^{(j_{a})}=x_{a}\\}}\Big{]}$

where the last sum is over $k$ distinct elements of the set

$\displaystyle{\mathcal{A}}_{h}:=\big{\\{}(i,j,n,x)\in\mathbb{N}^{3}\times\mathbb{Z}^{2}\colon
1\leq i<j\leq h\big{\\}}.$

$\cdots\cdots$$N$ Figure 1.  This is a graphical representation of expansion
(2.11) corresponding to the collisions of four random walks, each starting
from the origin. Each solid line will be marked with the label of the walk
that it corresponds to throughout the diagram. The solid dots, which mark a
collision among a subset $A$ of the random walks, is given a weight
$\prod_{i,j\in A}\sigma_{N}^{i,j}$. Any solid line between points
$(m,x),(n,y)$ is assigned the weight of the simple random walk transition
kernel $q_{m-n}(y-x)$. The hollow dots are assigned weight $1$ and they mark
the places where we simply apply the Chapman-Kolmogorov formula.

The graphical representation of expansion (2.11) is depicted in Figure 1.
There, we have marked with black dots the space-time points $(n,x)$ where some
of the walks collide and have assigned to such each one the weight
$\prod_{1\leq i<j\leq
h}\sigma_{N}^{i,j}\mathds{1}_{\\{S_{n}^{(i)}=S_{n}^{(j)}=x\\}}$.

We now want to write the above expansion as a convolution of Markovian
operators, following the Markov property of the simple random walks. We can
partition the time interval $\\{0,1,...,N\\}$ according to the times when
collisions take place; these are depicted in Figure 1 by vertical lines. In
between two successive times $m,n$, the walks will move from their locations
$(x^{(i)})_{i=1,...,h}$ at time $m$ to their new locations
$(y^{(i)})_{i=1,...,h}$ at time $n$ (some of which might coincide) according
to their transition probabilities, giving a total weight to this transition of
$\prod_{i=1}^{h}q_{n-m}(y^{(i)}-x^{(i)})$. We, now, want to encode in this
product the coincidences that may take place within the sets
$(x^{(i)})_{i=1,...,h}$ and $(y^{(i)})_{i=1,...,h}$. To this end, we consider
partitions $I$ of the set of indices $\\{1,...,h\\}$, which we denote by
$I\vdash\\{1,...,h\\}$. We also denote by $|I|$ the number of parts of $I$.
Given a partition $I\vdash\\{1,\dots,h\\}$, we define an equivalence relation
$\stackrel{{\scriptstyle I}}{{\sim}}$ in $\\{1,\dots,h\\}$ such that
$k\stackrel{{\scriptstyle I}}{{\sim}}\ell$ if and only if $k$ and $\ell$
belong to the same part of partition $I$. Given a vector
$\boldsymbol{y}=(y_{1},\dots,y_{h})\in(\mathbb{Z}^{2})^{h}$ and
$I\vdash\\{1,\dots,h\\}$, we shall use the notation $\boldsymbol{y}\sim I$ to
mean that $y_{k}=y_{\ell}$ for all pairs $k\stackrel{{\scriptstyle
I}}{{\sim}}\ell$. We use the symbol $\circ$ to denote the one-part
partition***the notation $\circ$, with which we denote the one-part partition,
here, should not be confused with the $\circ$ that appears in the figures,
where it just marks places where we apply the Chapman-Kolmogorov., that is,
$\circ:=\\{1,\dots,h\\}$, and $*$ to denote the partition consisting only of
singletons, that is $*:=\bigsqcup_{i=1}^{h}\\{i\\}$. Moreover, given
$I\vdash\\{1,\dots,h\\}$ such that $|I|=h-1$ and
$I=\\{i,j\\}\sqcup\bigsqcup_{k\neq i,j}\\{k\\}$, by slightly abusing notation,
we may identify and denote $I$ by its non-trivial part $\\{i,j\\}$.

Given this formalism, we denote the total transition weight of the $h$ walks,
from points $\boldsymbol{x}=(x^{(1)},...,x^{(h)})\in(\mathbb{Z}^{2})^{h}$,
subject to constraints $\boldsymbol{x}\sim I$ at time $m$, to points
$\boldsymbol{y}=(y^{(1)},...,y^{(h)})\in(\mathbb{Z}^{2})^{h}$, subject to
constraints $\boldsymbol{y}\sim J$ at time $n$, by

$Q^{I,J}_{n-m}(\boldsymbol{x},\boldsymbol{y}):=\mathds{1}_{\\{\boldsymbol{x}\sim
I\\}}\,\prod_{i=1}^{h}q_{n-m}(y^{(i)}-x^{(i)})\,\mathds{1}_{\\{\boldsymbol{y}\sim
J\\}}\,.$ (2.12)

We will call this operator the constrained evolution. Furthermore, for a
partition $I\vdash\\{1,\dots,h\\}$ and
$\boldsymbol{\beta}=\\{\beta_{i,j}\\}_{1\leq i<j\leq h}$ we define the mixed
collision weight subject to $I$ as

$\sigma_{N}(I):=\sigma_{N}(I,\\{\beta_{i,j}\\}_{1\leq i<j\leq
h})=\prod_{\begin{subarray}{c}1\leq i<j\leq h,\\\ i\stackrel{{\scriptstyle
I}}{{\sim}}j\end{subarray}}\sigma_{N}^{i,j},$ (2.13)

with $\sigma_{N}^{i,j}$ as defined in (2.3). We can then rewrite (2.11) in the
form

$1+\sum_{r=1}^{\infty}\sum_{\circ:=I_{0},I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\sum_{\begin{subarray}{c}1\leq
n_{1}<\dots<n_{r}\leq N\\\
0:=\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\prod_{i=1}^{r}Q^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\,.$
(2.14)

$\cdots\cdots$$N$ Figure 2.  This is the simplified version of Figure’s 1
graphical representation of the expansion (2.14), where we have grouped
together the blocks of consecutive collisions between the same pair of random
walks. These are now represented by the wiggle lines (replicas) and we call
the evolution in strips that contain only one replica as replica evolution
(although strip seven is the beginning of another wiggle line, we have not
represented it as such since we have not completed the picture beyond that
point). The wiggle lines (replicas) between points $(n,x),(m,y)$,
corresponding to collisions of a single pair of walks $S^{(k)},S^{(\ell)}$,
are assigned weight $U_{N}^{\beta_{k,\ell}}(m-n,y-x)$. A solid line between
points $(m,x),(n,y)$ is assigned the weight of the simple random walk
transition kernel $q_{m-n}(y-x)$.

We want to make one more simplification in this representation, which,
however, contains an important structural feature. This is to group together
consecutive constrained evolution operators
$\sigma_{N}(I_{i})Q^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})$
for which $I_{i-1}=I_{i}$. An example in Figure 1 is the sequence of
evolutions in the first three strips and another one is the group of
evolutions in strips five and six. Such groupings can be captured by the
following definition: For a partition $I\vdash\\{1,\dots,h\\}$ of the form
$I=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$ and
$\boldsymbol{x}=(x^{(1)},\dots,x^{(h)})$,
$\boldsymbol{y}=(y^{(1)},\dots,y^{(h)})\in(\mathbb{Z}^{2})^{h}$, we define the
replica evolution as

$\mathsf{U}^{I}_{n}(\boldsymbol{x},\boldsymbol{y}):=\mathds{1}_{\\{\boldsymbol{x},\boldsymbol{y}\sim
I\\}}\cdot U^{\beta_{k,\ell}}_{N}(n,y^{(k)}-x^{(k)})\cdot\prod_{i\neq
k,\ell}q_{n}(y^{(i)}-x^{(i)})\,,$ (2.15)

with $U^{\beta}_{N}(n,y^{(k)}-x^{(k)})$ defined in (2.7). We name this replica
evolution since in the time interval $[0,n]$ we see a stream of collisions
between only two of the random walks. The simplified version of expansion
(2.14) (and Figure 1) is presented in Figure 2.

In order to re-express (2.14) with the reduction of the replica evolution
(2.15), we need to introduce one more formalism, which is

$\displaystyle
P^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y}):=\begin{dcases}\sum_{\begin{subarray}{c}m_{1}\geq
1,\,m_{2}\geq 0\colon\,m_{1}+m_{2}=n\,,\\\
\boldsymbol{z}\in(\mathbb{Z}^{2})^{h}\end{subarray}}Q^{I;J}_{m_{1}}(\boldsymbol{x},\boldsymbol{z})\cdot\mathsf{U}^{J}_{m_{2}}(\boldsymbol{z},\boldsymbol{y})\,,&\text{
if }|J|=h-1\,,\\\ \qquad\qquad\qquad
Q_{n}^{I;J}(\boldsymbol{x},\boldsymbol{y})\,,&\text{ if
}|J|<h-1\,,\end{dcases}$ (2.16)

where we recall that $|J|$ is the number of parts of $J$ and so $|J|=h-1$
means that $J$ has the form $\\{k,\ell\\}\bigsqcup\sqcup_{i\neq
k,\ell}\\{i\\}$, corresponding to a pairwise collision, while $|J|<h-1$ means
that there are multiple collisions (the latter would correspond to the end of
the eighth strip in Figure 1). In other words, the operator $P_{n}^{I;J}$
groups together the replica evolutions with its preceding constrained
evolution.

We, finally, arrive to the desired expression for the Laplace transform of the
many-body collisions:

$M^{\boldsymbol{\beta}}_{N,h}=1+\sum_{r=1}^{\infty}\sum_{\circ:=I_{0},I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\sum_{\begin{subarray}{c}1\leq
n_{1}<\dots<n_{r}\leq N\,,\\\
0:=\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\prod_{i=1}^{r}P^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\,.$
(2.17)

###  2.3. Functional analytic framework and some auxiliary estimates

Let us start with some, fairly easy, bounds on operators $Q$ and $\mathsf{U}$
(with the estimate on the latter being an upgrade of estimate (2.1)).

###### Lemma 2.1.

Let the operators $Q^{I;J}_{n},\mathsf{U}^{J}_{n}$ be defined in (2.12) and
(2.15), respectively. For all partitions $I\neq J$ with $|J|=h-1$,
$\bar{\beta}<1$ defined in (2.2) and $\sigma_{N}(I)$ defined in (2.13), we
have the bounds

$\sum_{0\leq n\leq
N,\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})\leq\frac{1}{1-{\bar{\beta}}\,^{\prime}}\,\quad\text{and}\quad\sigma_{N}(J)\cdot\Bigg{(}\sum_{1\leq
n\leq
N,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})\Bigg{)}\leq\bar{\beta}\,^{\prime}\,,$
(2.18)

for all large enough $N$ and a $\bar{\beta}\,^{\prime}\in(\bar{\beta},1)$ .

  * Proof.

We start by proving the first bound in (2.18). By definition (2.15) we have
that

$\begin{split}\sum_{n\geq
0,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})&:=\sum_{n\geq
0,\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}\mathds{1}_{\\{\boldsymbol{x},\boldsymbol{y}\sim
J\\}}\cdot U^{\beta_{k,\ell}}_{N}(n,y^{(k)}-x^{(k)})\cdot\prod_{j\neq
k,\ell}q_{n}(y^{(j)}-x^{(j)})\\\ &=\sum_{n\geq
0}U^{\beta_{k,\ell}}_{N}(n)\,,\end{split}$

by using that $\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)=1$ to sum all the kernels
$q_{n}(y^{(j)}-x^{(j)})$ for $j\neq k,\ell$ and
$\sum_{z\in\mathbb{Z}^{2}}U^{\beta}_{N}(n,z)=U^{\beta}_{N}(n)$. Moreover, by
definition (2.7) and (2.3), since $\beta_{k,\ell}\leq\bar{\beta}$, we have

$\sum_{n\geq 0}U^{\beta_{k,\ell}}_{N}(n)\leq\sum_{n\geq
0}U^{\bar{\beta}}_{N}(n)=\sum_{k\geq
0}(\sigma_{N}(\bar{\beta})R_{N})^{k}\,\mathrm{P}(\tau^{\scriptscriptstyle(N)}_{k}=n)\,,$

and by (2.10) we have that for any $\bar{\beta}\,^{\prime}\in(\bar{\beta},1)$
and all $N$ large enough

$\sum_{n\geq 0}U^{\beta_{k,\ell}}_{N}(n)\leq\sum_{k\geq
0}(\bar{\beta}\,^{\prime})^{k}\,\mathrm{P}(\tau^{\scriptscriptstyle(N)}_{k}=n)\leq\sum_{k\geq
0}(\bar{\beta}\,^{\prime})^{k}=\frac{1}{1-\bar{\beta}\,^{\prime}}\,.$

Therefore,

$\sum_{n\geq
0,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})\leq(1-\bar{\beta}\,^{\prime})^{-1}\,.$

For the second bound in (2.18) we recall from (2.12) that when
$J=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$, then

$Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y}):=\bigg{(}\mathds{1}_{\\{\boldsymbol{x}\sim
I\\}}\,\prod_{j\neq k,\ell}q_{n}(y^{(j)}-x^{(j)})\bigg{)}\cdot
q_{n}(y^{(k)}-x^{(k)})\cdot q_{n}(y^{(k)}-x^{(\ell)})\,,$

since $\boldsymbol{y}\sim J$ means that $y_{k}=y_{\ell}$. Therefore,
$\sigma_{N}(J)=\sigma_{N}(\beta_{i,j})\leq\sigma_{N}(\bar{\beta})$. We, now,
use that $\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)=1$ in order to sum the kernels
$q_{n}(y^{(j)}-x^{(j)}),\,j\neq k,\ell$, while we also have by Cauchy-Schwarz
that

$\sigma_{N}(J)\cdot\bigg{(}\sum_{1\leq n\leq
N,\,y_{k}\in\mathbb{Z}^{2}}q_{n}(y^{(k)}-x^{(k)})\cdot
q_{n}(y^{(k)}-x^{(\ell)})\bigg{)}\leq\sigma_{N}(\bar{\beta})\cdot\Big{(}\sum_{n=1}^{N}q_{2n}(0)\Big{)}\leq\bar{\beta}\,^{\prime}\,,$

by (2.10), for all $N$ large enough, thus establishing the second bound in
(2.18). ∎

Next, in Proposition 2.2, we are going to recall some norm estimates from
[LZ21] on the Laplace transform of operators $P^{I;J}_{n}$, defined (2.16).
For this, we need to set up the functional analytic framework. We start by
defining
$(\mathbb{Z}^{2})^{h}_{I}:=\\{\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}:\boldsymbol{y}\sim
I\\}$ and, for $q\in(1,\infty)$, the $\ell^{q}((\mathbb{Z}^{2})^{h}_{I})$
space of functions $f:(\mathbb{Z}^{2})^{h}_{I}\to\mathbb{R}$ which have finite
norm

$\left\lVert
f\right\rVert_{\ell^{q}((\mathbb{Z}^{2})^{h}_{I})}:=\Bigg{(}\sum_{\boldsymbol{y}\in((\mathbb{Z}^{2})^{h}_{I})}\big{|}f(\boldsymbol{y})\big{|}^{q}\Bigg{)}^{\frac{1}{q}}\,.$

For $q\in(1,\infty)$ and for an integral operator
$\mathsf{T}:\ell^{q}\big{(}(\mathbb{Z}^{2})^{h}_{J}\big{)}\rightarrow\ell^{q}\big{(}(\mathbb{Z}^{2})^{h}_{I}\big{)}$,
one can define the pairing

$\langle
f,\mathsf{T}g\rangle:=\sum_{\boldsymbol{x}\,\in(\mathbb{Z}^{2})^{h}_{I},\boldsymbol{y}\,\in(\mathbb{Z}^{2})^{h}_{J}}f(\boldsymbol{x})\mathsf{T}(\boldsymbol{x},\boldsymbol{y})g(\boldsymbol{y})\,.$
(2.19)

The operator norm will be given by

$\left\lVert\mathsf{T}\right\rVert_{\ell^{q}\to\ell^{q}}:=\sup_{\left\lVert
g\right\rVert_{\ell^{q}_{J}}\leq
1}\left\lVert\mathsf{T}g\right\rVert_{\ell^{q}_{I}}=\sup_{\left\lVert
f\right\rVert_{\ell^{p}_{I}}\leq 1,\,\left\lVert
g\right\rVert_{\ell^{q}_{J}}\leq 1}\langle f,\mathsf{T}g\rangle\,,$ (2.20)

for $p,q\in(1,\infty)$ conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$.

We introduce the weighted Laplace transforms of operators $Q^{I,J}$ and
$\mathsf{U}^{J}$. In particular, let $w(x)$ be any continuous function in
$L^{\infty}(\mathbb{R}^{2})\cap L^{1}(\mathbb{R}^{2})$ such that $\log w(x)$
is Lipschitz (one can think of $w(x)=e^{-|x|}$) and define
$w_{N}(x):=w(x/\sqrt{N})$. Also, for a function
$g\colon\mathbb{R}^{2}\to\mathbb{R}$ we define the tensor product $g^{\otimes
h}(x_{1},...,x_{h})=g(x_{1})\cdots g(x_{h})$, The weighted Laplace transforms
are now defined as

$\begin{split}&\widehat{\mathsf{Q}}^{I;J}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y}):=\bigg{(}\sum_{n\geq
1}^{N}e^{-\lambda\frac{n}{N}}Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})\bigg{)}\cdot\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes h}_{N}(\boldsymbol{y})}\,,\\\
&\widehat{\mathsf{U}}^{J}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y}):=\bigg{(}\sum_{n\geq
0}^{N}e^{-\lambda\frac{n}{N}}\,\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})\bigg{)}\cdot\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes h}_{N}(\boldsymbol{y})}\,.\end{split}$
(2.21)

The passage to a Laplace transform will help to estimate convolutions
involving operators $Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})$ and
$\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})$ and the introduction of
the weight comes handy in improving integrability when these operators are
applied to functions which are not in $\ell^{1}((\mathbb{Z}^{2})^{h})$. We
will see this in Lemma 2.3 below.

We also define the Laplace transform operator of the combined evolution
(2.16):

$\widehat{\mathsf{P}}^{I;J}_{N,\lambda}=\begin{dcases}\widehat{\mathsf{Q}}^{I;J}_{N,\lambda}\,,&\text{
if }|J|<h-1\\\
\widehat{\mathsf{Q}}^{I;J}_{N,\lambda}\,\widehat{\mathsf{U}}^{J}_{N,\lambda}\,,&\text{
if }|J|=h-1\,.\end{dcases}$ (2.22)

For our purposes, it will be sufficient to take $\lambda=0$ and consider
operators $\widehat{\mathsf{Q}}^{I;J}_{N,0},\widehat{\mathsf{U}}^{J}_{N,0}$
and $\widehat{\mathsf{P}}^{I;J}_{N,0}$.

Using the above formalism we summarise in the next proposition some key
estimates of [LZ21] (see Propositions 3.2-3.4 and the proof of Theorem 1.3
therein), which are refinements of estimates in [CSZ21] (Section 6) and
[DFT94] (Section 3).

###### Proposition 2.2.

Consider the operators $\widehat{\mathsf{Q}}^{I;J}_{N,0}$ and
$\widehat{\mathsf{P}}^{I;J}_{N,0}$ defined in (2.21) and (2.22) with
$\lambda=0$ and a weight function $w\in L^{\infty}(\mathbb{R}^{2})\cap
L^{1}(\mathbb{R}^{2})$ such that $\log w(x)$ is Lipschitz. Then there exists a
constant $C=C(h,\bar{\beta},w)\in(0,\infty)$ (recall $\bar{\beta}$ from (2.2))
such that for all $p,q\in(1,\infty)$ with $\frac{1}{p}+\frac{1}{q}=1$ and all
partitions $I,J\vdash\\{1,\dots,h\\}$, such that $I\neq J$ and $|I|,|J|\leq
h-1$, we have that

$\left\lVert\widehat{\mathsf{P}}_{N,0}^{I;J}\right\rVert_{\ell^{q}\to\ell^{q}}\leq
C\,p\,q\,.$ (2.23)

Moreover, if $g\in\ell^{q}(\mathbb{Z}^{2})$,

$\left\lVert\widehat{\mathsf{Q}}^{*;I}_{N,0}\,g^{\otimes
h}\right\rVert_{\ell^{p}}\leq C\,q\,N^{\frac{1}{q}}\left\lVert
g\right\rVert^{h}_{\ell^{p}}\,,$ (2.24)

for $g^{\otimes h}(x_{1},...,x_{h}):=g(x_{1})\cdots g(x_{h})$.

Let us now present the following lemma, which demonstrates how the above
functional analytic framework will be used. This lemma will be useful in the
first approximation, that we will perform in the next Section, in showing that
contributions from multiple, i.e. three or more, collisions are negligible.

###### Lemma 2.3.

Let $H_{r,N}$ be the $r^{th}$ term in the expansion (2.17), that is,

$H_{r,N}:=\sum_{\circ:=I_{0},I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\sum_{\begin{subarray}{c}1\leq
n_{1}<\dots<n_{r}\leq N,\\\
0:=\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\prod_{i=1}^{r}P^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\,,$
(2.25)

and $H^{\mathsf{(multi)}}_{r,N}$ be the corresponding term with the additional
constraint that there is at least one multiple collision (i.e. at some point,
three or more walks meet), that is,

$H^{\mathsf{(multi)}}_{r,N}:=\sum_{\circ:=I_{0},I_{1},\dots,I_{r}}\Bigg{(}\prod_{i=1}^{r}\sigma_{N}(I_{i})\Bigg{)}\mathds{1}_{\\{\exists\,1\leq
j\leq r\,:\,|I_{j}|<h-1\\}}\sum_{\begin{subarray}{c}1\leq
n_{1}<\dots<n_{r}\leq N\\\
0:=\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\prod_{i=1}^{r}P^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\,.$

Then the following bounds hold:

$\displaystyle H_{r,N}\leq\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{r}N^{\frac{h+1}{q}}\qquad\text{and}\qquad
H^{\mathsf{(multi)}}_{r,N}\leq\frac{r}{\log N}\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{r}N^{\frac{h+1}{q}}.$ (2.26)

for any $p,q\in(1,\infty)$ with $\tfrac{1}{p}+\tfrac{1}{q}=1$ and a constant
$C$ that depends on $h$ and $\bar{\beta}$ but is independent of $N,r,p,q$.

  * Proof.

We start by considering $w(x)=e^{-|x|},w_{N}(x):=w(\tfrac{x}{\sqrt{N}})$ and
$w_{N}^{\otimes h}(x_{1},...,x_{h})=\prod_{i=1}^{h}w_{N}(x_{i})$ and by
including in the expression (2.25) the term

$\frac{1}{w_{N}^{\otimes
h}(\boldsymbol{x}_{0})}\Big{(}\prod_{i=1}^{r}\frac{w_{N}^{\otimes
h}(\boldsymbol{x}_{i-1})}{w_{N}^{\otimes
h}(\boldsymbol{x}_{i})}\Big{)}w_{N}^{\otimes h}(\boldsymbol{x}_{r})=1,$

thus rewriting $H_{r,N}$ as

$\begin{split}H_{r,N}&=\sum_{\circ:=I_{0},I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\\\
&\times\sum_{\begin{subarray}{c}1\leq n_{1}<\dots<n_{r}\leq N\\\
0:=\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\frac{1}{w_{N}^{\otimes
h}(\boldsymbol{x}_{0})}\prod_{i=1}^{r}P^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\frac{w_{N}^{\otimes
h}(\boldsymbol{x}_{i-1})}{w_{N}^{\otimes h}(\boldsymbol{x}_{i})}\cdot
w_{N}^{\otimes h}(\boldsymbol{x}_{r})\,.\end{split}$

We can extend the summation on $\boldsymbol{x}_{0}$ from
$\boldsymbol{x}_{0}=0$ to $\boldsymbol{x}_{0}\in\mathbb{Z}^{2}$ by introducing
a delta function $\delta_{0}^{\otimes h}$ at zero. Then

$\begin{split}H_{r,N}&=\sum_{*:=I_{0},I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\\\
&\times\sum_{\begin{subarray}{c}1\leq n_{1}<\dots<n_{r}\leq N\\\
\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\frac{\delta_{0}^{\otimes
h}(\boldsymbol{x}_{0})}{w_{N}^{\otimes
h}(\boldsymbol{x}_{0})}\prod_{i=1}^{r}P^{I_{i-1};I_{i}}_{n_{i}-n_{i-1}}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\frac{w_{N}^{\otimes
h}(\boldsymbol{x}_{i-1})}{w_{N}^{\otimes h}(\boldsymbol{x}_{i})}\cdot
w_{N}^{\otimes h}(\boldsymbol{x}_{r})\,.\end{split}$

We can, now, bound the last expression by extending the temporal range of
summations from $1\leq n_{1}<\dots<n_{r}\leq N$ to
$n_{i}-n_{i-1}\in\\{1,\dots,N\\}$ for all $i=1,...,r$. Recalling the
definition of the Laplace transforms of the operators (2.21), (2.22), we,
thus, obtain the upper bound

$H_{r,N}\leq\sum_{*:=I_{0},I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\sum_{\boldsymbol{x}_{0},\boldsymbol{x}_{1},\dots,\,\boldsymbol{x}_{r}\in(\mathbb{Z}^{2})^{h}}\frac{\delta_{0}^{\otimes
h}(\boldsymbol{x}_{0})}{w_{N}^{\otimes
h}(\boldsymbol{x}_{0})}\prod_{i=1}^{r}\widehat{\mathsf{P}}^{I_{i-1};I_{i}}_{N,0}(\boldsymbol{x}_{i-1},\boldsymbol{x}_{i})\cdot
w_{N}^{\otimes h}(\boldsymbol{x}_{r})\,,$

which we can write in the more compact and useful notation, using the brackets
(2.19), as

$H_{r,N}\leq\sum_{I_{1},\dots,I_{r}}\Big{\langle}\,\frac{\delta_{0}^{\otimes
h}}{w_{N}^{\otimes
h}},\widehat{\mathsf{Q}}^{*;I_{1}}_{N,0}\widehat{\mathsf{P}}^{I_{1};I_{2}}_{N,0}\cdots\widehat{\mathsf{P}}^{I_{r-1};I_{r}}_{N,0}\,w^{\otimes
h}_{N}\Big{\rangle}\prod_{i=1}^{r}\sigma_{N}(I_{i}).$

We note, here, that in the right-hand side we set the $I_{0}$ partition to be
equal to $I_{0}=\\{1\\}\sqcup\cdots\sqcup\\{h\\}$. The delta function
$\delta_{0}^{\otimes h}(\boldsymbol{x}_{0})$ will force all points of
$\boldsymbol{x}_{0}$ to coincide at zero, thus, forcing $I_{0}$ to be equal to
the partition $\circ=\\{1,...,h\\}$ but, at the stage of operators, we do not
yet need to enforce this constraint. At this stage we can proceed with the
estimate using the operator norms (2.20) as

$H_{r,N}\leq\sum_{I_{1},\dots,I_{r}}\left\lVert\widehat{\mathsf{Q}}^{*,I_{1}}_{N,0}\frac{\delta^{\otimes
h}_{0}}{w^{\otimes
h}_{N}}\right\rVert_{\ell^{p}}\,\prod_{i=2}^{r}\,\left\lVert\widehat{\mathsf{P}}_{N,0}^{I_{i-1};I_{i}}\right\rVert_{\ell^{q}\to\ell^{q}}\,\left\lVert
w_{N}^{\otimes
h}\right\rVert_{\ell^{q}}\,\,\cdot\,\prod_{i=1}^{r}\sigma_{N}(I_{i}),$ (2.27)

By (2.24) of Proposition 2.2 we have that

$\left\lVert\widehat{\mathsf{Q}}^{*,I_{1}}_{N,0}\frac{\delta^{\otimes
h}_{0}}{w^{\otimes h}_{N}}\right\rVert_{\ell^{p}}\leq
C\,q\,N^{\frac{1}{q}}\left\lVert\frac{\delta_{0}}{w_{N}}\right\rVert^{h}_{\ell^{p}}=C\,q\,N^{\frac{1}{q}},$

and by (2.23) we have that for all $1\leq i\leq r-1$,

$\left\lVert\widehat{\mathsf{P}}_{N,0}^{I_{i-1};I_{i}}\right\rVert_{\ell^{q}\to\ell^{q}}\leq
C\,p\,q\,.$

Inserting these estimates in (2.27) we deduce that

$\displaystyle H_{r,N}$
$\displaystyle\leq(C\,p\,q)^{r}N^{\frac{1}{q}}\left\lVert\frac{\delta_{0}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert
w_{N}\right\rVert_{\ell^{q}}^{h}\sum_{I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})$
$\displaystyle=(C\,p\,q)^{r}N^{\frac{1}{q}}\left\lVert
w_{N}\right\rVert_{\ell^{q}}^{h}\sum_{I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\,,$
(2.28)

for a constant $C=C(h,\bar{\beta})\in(0,\infty)$, not depending on $p,q,r,N$.
We now notice that for any partition $I\vdash\\{1,...,h\\}$, it holds that
$\sigma_{N}(I)\leq C/\log N$ (recall definitions (2.13) and (2.3)) and that,
by Riemann summation, $N^{-h/q}\left\lVert w_{N}\right\rVert_{\ell^{q}}^{h}$
is bounded uniformly in $N$. Therefore, applying these on (Proof.) we arrive
at the bound

$\displaystyle H_{r,N}\leq\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{r}N^{\frac{h+1}{q}}$

which is the first claimed estimate in (2.26). For the second estimate in
(2.26) we follow the same steps until we arrive at the bound

$H^{\mathsf{(multi)}}_{r,N}\leq(C\,p\,q)^{r}N^{\frac{1}{q}}\left\lVert
w_{N}\right\rVert_{\ell^{q}}^{h}\sum_{I_{1},\dots,I_{r}}\prod_{i=1}^{r}\sigma_{N}(I_{i})\mathds{1}_{\\{\exists\,1\leq
j\leq r:|I_{j}|<h-1\\}}.$

Then we notice that for a partition $I\vdash\\{1,...,h\\}$ with $|I|<h-1$ it
will hold that $\sigma_{N}(I)\leq C(\log N)^{-2}$ (recall definitions (2.13)
and (2.3)). This fact, together with the fact that there are $r$ possible
choices among the partitions $I_{1},...,I_{r}$ that can be chosen so that
$|I_{j}|<h-1$, leads to the second bound in (2.26). ∎

##  3\. Approximation steps and proof of the theorem

In this section we prove Theorem 1.1 through a series of approximations on the
chaos expansion (2.11), (2.17). The first step, in Section 3.1, is to
establish that the series in the chaos expansion (2.17) can be truncated up to
a finite order and that the main contribution comes from diagrams where, at
any fixed time, we only have at most two walks colliding. The second step,
Section 3.2, is to show that the main contribution to the expansion and to
diagrams like in Figure 3, comes when all jumps between marked dots (see
Figure 3) happen within diffusive scale. The third step, in Section 3.3,
captures the important feature of scale separation. This is intrinsic to the
two-dimensionality and can be seen as the main feature that leads to the
asymptotic independence of the collision times. With reference to Figure 3,
this says that the time between two consecutive replicas, say $a_{4}-b_{3}$ in
Figure 3 must be much larger than the time between the previous replicas, say
$b_{3}-b_{2}$. This would then lead to the next step in Section 3.4, see also
Figure 4, which is that we can rewire the links so that the solid lines
connect only replicas between the same pairs of walks. The final step, which
is performed in Section 3.5 is to reverse all the above approximations within
the rewired diagrams, to which we arrived in the previous step. The summation,
then, of all rewired diagrams leads, in the limit, to the right hand of (1.2),
thus completing the proof of the theorem.

###  3.1. Reduction to 2-body collisions and finite order chaoses.

In this step, we use the functional analytic framework and estimates of the
previous section to show that for each $r\geq 1$, $H_{r,N}$ decays
exponentially in $r$, uniformly in $N\in\mathbb{N}$ and that it is
concentrated on configurations which contain only two-body collisions between
the $h$ random walks.

###### Proposition 3.1.

There exist constants $\mathsf{a}\in(0,1)$ and
$\bar{C}=C(h,\bar{\beta},\mathsf{a})\in(0,\infty)$ and such that for all
$r\geq 1$,

$\sup_{N\in\mathbb{N}}H_{r,N}\leq\bar{C}\,\mathsf{a}^{r}\,,\qquad\text{and}\qquad
H^{\mathsf{(multi)}}_{r,N}\leq\frac{\bar{C}}{\log N}\,r\,\mathsf{a}^{r}\,.$
(3.1)

  * Proof.

We use the estimates in (2.26) and make the choice
$q=q_{N}:=\frac{\mathsf{a}}{C_{1}}\log N$ with $\mathsf{a}\in(0,1)$ and a
constant $C_{1}$ such that $\frac{Cpq}{\log N}<\mathsf{a}$ (recall that
$\tfrac{1}{p}+\tfrac{1}{q}=1$). Moreover, this choice of $q$ implies that

$N^{\frac{h+1}{q}}=e^{\frac{h+1}{q}\log
N}=e^{\frac{C_{1}\,(h+1)}{\mathsf{a}}}.$

Therefore, choosing $\bar{C}=\,e^{\frac{C_{1}\,(h+1)}{\mathsf{a}}}$ implies
the first estimate in (3.1).

The second estimate follows from the same procedure and the same choice of
$q=q_{N}:=\frac{\mathsf{a}}{C_{1}}\log N$ in the second bound of (2.26). ∎

###### Proposition 3.2.

If $M^{\boldsymbol{\beta}}_{N,h}$ is the joint Laplace transform of the
collision local times $\Big{\\{}\tfrac{\pi}{\log
N}\,\mathsf{L}_{N}^{(i,j)}\Big{\\}}_{1\leq i<j\leq h}$, (2.1) and $H_{r,N}$ is
the $r^{th}$ term in its chaos expansion (2.25), then for any $\varepsilon>0$
there exists $K=K_{\varepsilon}$ such that

$\Big{|}M^{\boldsymbol{\beta}}_{N,h}-\sum_{r=0}^{K}H_{r,N}\Big{|}\leq\varepsilon\,,$

uniformly for all $N\in\mathbb{N}$.

  * Proof.

By Proposition, 3.1, $H_{r,N}$ decay exponentially in $r$, uniformly in
$N\in\mathbb{N}$ and therefore

$\limsup_{K\to\infty}\,\Big{(}\sup_{N\geq 1}\,\sum_{r>K}H_{r,N}\Big{)}=0\,,$

which means that we can truncate the expansion of
$M^{\boldsymbol{\beta}}_{N,h}$ to a finite number of terms $K$ depending only
on $\varepsilon$. ∎

By Proposition 3.1 we can focus on only two-body collisions, since higher
order collisions bear a negligible contribution as $N\to\infty$. Let us
introduce some notation to conveniently describe the expansion of $H_{r,N}$,
after the reduction to only two-body collisions, which we will use in the
sequel. Given $r\geq 1$ we will denote by
$a_{i},b_{i}\in\mathbb{N}\cup\\{0\\}$, $a_{i}\leq b_{i}$, $i=1,\dots,r$ the
times where replicas start and end respectively, see (2.15) and Figure 2,
where replicas are represented by wiggle lines. Thus, $a_{i}$ will be the time
marking the beginning of the $i^{th}$ wiggle line and $b_{i}$ the time marking
its end. Note that, $a_{1}=0$. Moreover, we use the notation
$\vec{\boldsymbol{x}}=(\boldsymbol{x}_{1},\boldsymbol{x}_{2},\dots,\boldsymbol{x}_{r})\in(\mathbb{Z}^{2})^{hr}$
to denote the starting points of the $r$ replicas and
$\vec{\boldsymbol{y}}=(\boldsymbol{y}_{1},\dots,\boldsymbol{y}_{r})\in(\mathbb{Z}^{2})^{hr}$
the corresponding ending points. Again, notice that
$\boldsymbol{x}_{1}=\boldsymbol{0}$. We then define the set

$\mathsf{C}_{r,N}:=\Big{\\{}(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\Big{|}\,0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq
N,\,\vec{\boldsymbol{x}},\vec{\boldsymbol{y}}\in(\mathbb{Z}^{2})^{hr}\,,\boldsymbol{x}_{1}=\boldsymbol{0}\Big{\\}}\,.$
(3.2)

We also define a set of finite sequences of partitions

$\mathcal{I}^{(2)}=\bigcup_{r=0}^{\infty}\bigg{\\{}(I_{1},\dots,I_{r}):I_{j}\neq
I_{j+1}\text{ and }|I_{j}|=h-1\,,\forall j\in\\{1,\dots,r\\}\bigg{\\}}\,.$

Using the notational conventions outlined above we can write
$H_{r,N}=H^{(2)}_{r,N}+H^{\mathsf{(multi)}}_{r,N}$ with

$\displaystyle H^{(2)}_{r,N}$
$\displaystyle:=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}_{r,N}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\sigma_{N}(I_{i}).$
(3.3)

In the next sections will focus on $H^{(2)}_{r,N}$, which by Proposition 3.1
contains the main contributions.

###  3.2. Diffusive spatial truncation.

In this step we show that we can introduce diffusive spatial truncations in
all the kernels appearing in (3.3) which originate from the diffusive
behaviour of the simple random walk in $\mathbb{Z}^{2}$. For a vector
$\boldsymbol{x}=(x^{(1)},\dots,x^{(h)})\in(\mathbb{Z}^{2})^{h}$, we shall use
the notation

$\left\lVert\boldsymbol{x}\right\rVert_{\infty}=\max_{1\leq j\leq
h}|x^{(j)}|\,,$

where $|\cdot|$ denotes the usual Euclidean norm on $\mathbb{R}^{2}$. For each
$r\in\mathbb{N}$, define $H^{\mathsf{(diff)}}_{r,N}$ to be the sum in (3.3)
where $\mathsf{C}_{r,N}$ is replaced by

$\begin{split}\mathsf{C}^{\mathsf{(diff)}}_{r,N,R}:=\mathsf{C}_{r,N}\cap\Big{\\{}(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}}):\,&\left\lVert\boldsymbol{y}_{i}-\boldsymbol{x}_{i}\right\rVert_{\infty}\leq
R\sqrt{b_{i}-a_{i}}\\\ &\text{ and
}\left\lVert\boldsymbol{x}_{i}-\boldsymbol{y}_{i-1}\right\rVert_{\infty}\leq
R\sqrt{a_{i}-b_{i-1}}\text{ for all }1\leq i\leq r\Big{\\}}\,\end{split}$
(3.4)

and similarly we define

$H^{\mathsf{(superdiff)}}_{r,N,R}=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(superdiff)}}_{r,N,R}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i}),$
(3.5)

where

$\begin{split}\mathsf{C}^{\mathsf{(superdiff)}}_{r,N,R}:=\mathsf{C}_{r,N}\cap\Big{\\{}(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}}):\,\exists\,1\leq
i\leq
r:\,&\left\lVert\boldsymbol{y}_{i}-\boldsymbol{x}_{i}\right\rVert_{\infty}>R\sqrt{b_{i}-a_{i}}\\\
&\text{ or
}\left\lVert\boldsymbol{x}_{i}-\boldsymbol{y}_{i-1}\right\rVert_{\infty}>R\sqrt{a_{i}-b_{i-1}}\,\Big{\\}}\,.\end{split}$

Note that then we have that

$H^{(2)}_{r,N}=H^{\mathsf{(diff)}}_{r,N,R}+H^{\mathsf{(superdiff)}}_{r,N,R}\,.$

We have the following Proposition.

###### Proposition 3.3.

For all $r\geq 1$ we have that

$\lim_{R\to\infty}\sup_{N\in\mathbb{N}}H^{\mathsf{(superdiff)}}_{r,N,R}=0\,.$
(3.6)

  * Proof.

We use the bounds established in Lemma 3.4, below, and (2.18) to show (3.6).
We can use a union bound for (3.5) to obtain that

$\begin{split}&H^{\mathsf{(superdiff)}}_{r,N,R}=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(superdiff)}}_{r,N,R}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\\\
&\leq\sum_{j=1}^{r}\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
0:=\boldsymbol{x}_{1},\boldsymbol{y}_{1},\dots,\boldsymbol{x}_{r},\boldsymbol{y}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\bigg{(}\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{y}_{j}-\boldsymbol{x}_{j}\right\rVert_{\infty}>R\sqrt{b_{j}-a_{j}}\big{\\}}}+\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{x}_{j}-\boldsymbol{y}_{j-1}\right\rVert_{\infty}>R\sqrt{a_{j}-b_{j-1}}\big{\\}}}\bigg{)}\\\
&\qquad\times\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\,.\end{split}$
(3.7)

We split the sum on the last two lines of (3.7) according to the two indicator
functions that appear therein. By repeated successive application of the
bounds from (2.18) for $j<i\leq r$ and then by using (3.10), which reads as

$\sum_{\boldsymbol{y}_{j}\in(\mathbb{Z}^{2})^{h},\,a_{j}\leq b_{j}\leq
N}\mathsf{U}^{I_{j}}_{b_{j}-a_{j}}(\boldsymbol{x}_{j},\boldsymbol{y}_{j})\,\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{y}_{j}-\boldsymbol{x}_{j}\right\rVert_{\infty}>R\sqrt{b_{j}-a_{j}}\big{\\}}}\leq
e^{-\kappa R}\,,$

we deduce that

$\begin{split}&\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
0:=\boldsymbol{x}_{1},\boldsymbol{y}_{1},\dots,\boldsymbol{x}_{r},\boldsymbol{y}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{y}_{j}-\boldsymbol{x}_{j}\right\rVert_{\infty}>R\sqrt{b_{j}-a_{j}}\big{\\}}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\\\
&\leq e^{-\kappa
R}\bigg{(}\frac{\bar{\beta}^{\prime}}{1-\bar{\beta}^{\prime}}\bigg{)}^{r-j}\\\
&\qquad\times\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<b_{j-1}<a_{j}\leq N,\\\
0:=\boldsymbol{x}_{1},\boldsymbol{y}_{1},\dots,\boldsymbol{x}_{j}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{j-1}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\\\
&\qquad\qquad\times
Q^{I_{j-1};I_{j}}_{a_{j}-b_{j-1}}(\boldsymbol{y}_{j-1},\boldsymbol{x}_{j})\,\sigma_{N}(I_{j})\,.\end{split}$
(3.8)

We then continue the summation using the bounds from (2.18), to obtain that
the right-hand side of the inequality in (3.8) is bounded by

$e^{-\kappa
R}\bigg{(}\frac{\bar{\beta}^{\prime}}{1-\bar{\beta}^{\prime}}\bigg{)}^{r-j}\,\bigg{(}\frac{\bar{\beta}^{\prime}}{1-\bar{\beta}^{\prime}}\bigg{)}^{j-1}=e^{-\kappa
R}\bigg{(}\frac{\bar{\beta}^{\prime}}{1-\bar{\beta}^{\prime}}\bigg{)}^{r-1}\,.$

Similarly, for the sum involving the second indicator function in (3.7) we
obtain by using (2.18) and (3.9) of Lemma 3.4 that

$\displaystyle\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N\,,\\\
0:=\boldsymbol{x}_{1},\boldsymbol{y}_{1},\dots,\boldsymbol{x}_{r},\boldsymbol{y}_{r}\in(\mathbb{Z}^{2})^{h}\end{subarray}}$
$\displaystyle\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{x}_{j}-\boldsymbol{y}_{j-1}\right\rVert_{\infty}>R\sqrt{a_{j}-b_{j-1}}\big{\\}}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})$
$\displaystyle\times\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})$
$\displaystyle\leq e^{-\kappa
R^{2}}\,\bigg{(}\frac{1}{1-\bar{\beta}^{\prime}}\bigg{)}^{r}\,(\bar{\beta}^{\prime})^{r-1}\,.$

Therefore, the right-hand side of the inequality in (3.7) is bounded by

$\sum_{j=1}^{r}\sum_{(I_{1},...,I_{r})\,\in{\mathcal{I}}^{(2)}}\,e^{-\kappa
R}\Bigg{(}\bigg{(}\frac{\bar{\beta}^{\prime}}{1-\bar{\beta}^{\prime}}\bigg{)}^{r-1}+\bigg{(}\frac{1}{1-\bar{\beta}^{\prime}}\bigg{)}^{r}\,(\bar{\beta}^{\prime})^{r-1}\Bigg{)}\leq
e^{-\kappa
R}\Bigg{(}2r\cdot\frac{(\bar{\beta}^{\prime})^{r-1}}{(1-\bar{\beta}^{\prime})^{r}}\cdot\binom{h}{2}^{r}\,\Bigg{)}\,,$

where the $\binom{h}{2}^{r}$ factor comes from the fact that there are at most
$\binom{h}{2}^{r}$ choices for the sequence
$(I_{1},\dots,I_{r})\in{\mathcal{I}}^{(2)}$. Thus, recalling (3.7) we get that

$\sup_{N\in\mathbb{N}}H^{\mathsf{(superdiff)}}_{r,N,R}\leq e^{-\kappa
R}\Bigg{(}2r\cdot\frac{(\bar{\beta}^{\prime})^{r-1}}{(1-\bar{\beta}^{\prime})^{r}}\cdot\binom{h}{2}^{r}\,\Bigg{)}\xrightarrow{R\to\infty}0.$

∎

###### Lemma 3.4.

Let $I,J\vdash\\{1,...,h\\}$ such that $|I|=|J|=h-1$ and $I\neq J$. For large
enough $R\in(0,\infty)$ and uniformly in
$\boldsymbol{x}\in(\mathbb{Z}^{2})_{I}^{h}$ we have that for a constant
$\kappa=\kappa(h,\bar{\beta})\in(0,\infty)$,

$\sigma_{N}(J)\cdot\Bigg{(}\sum_{1\leq n\leq
N,\,\boldsymbol{y}\,\in(\mathbb{Z}^{2})^{h}}Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})\cdot\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{x}-\boldsymbol{y}\right\rVert_{\infty}>R\sqrt{n}\big{\\}}}\Bigg{)}\leq
e^{-\kappa R^{2}}$ (3.9)

and

$\sum_{1\leq n\leq
N,\,\boldsymbol{y}\,\in(\mathbb{Z}^{2})^{h}}\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})\cdot\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{x}-\boldsymbol{y}\right\rVert_{\infty}>R\sqrt{n}\big{\\}}}\leq
e^{-\kappa R}\,.$ (3.10)

  * Proof.

We start with the proof of (3.9). Since $|J|=h-1$, let us assume without loss
of generality that $J=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$. In
this case, $Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})$ contains $h-2$ random
walk jumps with free endpoints $y^{(j)}$, $j\neq k,\ell$, that is

$\prod_{j\neq k,\ell}q_{n}(y^{(j)}-x^{(j)})\,.$

Moreover, $J$ imposes the constraint that $y^{(k)}=y^{(\ell)}$, which appears
in $Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})$ through the product of
transition kernels

$q_{n}(y^{(k)}-x^{(k)})\cdot q_{n}(y^{(k)}-x^{(\ell)})\,,$

recall (2.12). The constraint
$\left\lVert\boldsymbol{x}-\boldsymbol{y}\right\rVert_{\infty}>R\sqrt{n}$
implies that there exists $1\leq j\leq h$ such that
$|x^{(j)}-y^{(j)}|>R\,\sqrt{n}$. We distinguish two cases:

    1. (1)

There exists $j\neq k,\ell$ such that $|x^{(j)}-y^{(j)}|>R\,\sqrt{n}$, or

    2. (2)

$|x^{(j)}-y^{(j)}|>R\,\sqrt{n}$ for $j=k$ or $j=\ell$.

In both cases, we can use $\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)=1$ to sum the
kernels $q_{n}(y^{(j)}-x^{(j)})$, $j\neq k,\ell$ to which we do not impose any
super-diffusive constraints. By symmetry and translation invariance we can
upper bound the left-hand side of (3.9) by

$\begin{split}\sigma_{N}(J)\cdot\Bigg{(}(h-2)\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q_{n}(z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}&\cdot\Big{\\{}\sup_{u\,\in\mathbb{Z}^{2}}\sum_{\,z\in\mathbb{Z}^{2}}q_{n}(z)q_{n}(z+u)\Big{\\}}\\\
+2\,\sup_{u\,\in\mathbb{Z}^{2}}\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}&q_{n}(z)q_{n}(z+u)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\Bigg{)}\,.\end{split}$
(3.11)

Looking at the first summand in (3.11) we have by Cauchy-Schwarz that

$\begin{split}\sup_{u\in\mathbb{Z}^{2}}\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)\,q_{n}(z+u)\leq\Big{(}\sum_{z\in\mathbb{Z}^{2}}q_{n}^{2}(z)\Big{)}^{1/2}\cdot\sup_{u\in\mathbb{Z}^{2}}\Big{(}\sum_{z\in\mathbb{Z}^{2}}q^{2}_{n}(z+u)\Big{)}^{1/2}\leq
q_{2n}(0)\,,\end{split}$ (3.12)

since $\sum_{z\in\mathbb{Z}^{2}}q^{2}_{n}(z)=q_{2n}(0)$. Let us recall the
deviation estimate for the simple random walk, which can be found in [LL10],
that is

$\mathrm{P}\Big{(}\max_{0\leq k\leq n}|S_{k}|>R\sqrt{n}\Big{)}\leq
e^{-cR^{2}}\,,$ (3.13)

for a constant $c\in(0,\infty)$ and all $R\in(0,\infty)$. By using bound
(3.12) and subsequently (3.13) on the first summand of (3.11) we get that

$\begin{split}&\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q_{n}(z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\cdot\Big{\\{}\sup_{u\,\in\mathbb{Z}^{2}}\sum_{\,z\in\mathbb{Z}^{2}}q_{n}(z)q_{n}(z+u)\Big{\\}}\\\
\leq&\sum_{1\leq n\leq N,\,z\in\mathbb{Z}^{2}}q_{2n}(0)\cdot
q_{n}(z)\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\\\
\leq&\,e^{-cR^{2}}R_{N}\,.\end{split}$

We recall from (2.4) and (2.5) that
$R_{N}=\sum_{n=1}^{N}q_{2n}(0)\stackrel{{\scriptstyle
N\to\infty}}{{\approx}}\frac{\log N}{\pi}$, therefore,

$\begin{split}&\sigma_{N}(J)\cdot\Bigg{(}(h-2)\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q_{n}(z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\cdot\Big{\\{}\sup_{u\,\in\mathbb{Z}^{2}}\sum_{\,z\in\mathbb{Z}^{2}}q_{n}(z)q_{n}(z+u)\Big{\\}}\Bigg{)}\\\
\leq&\,(h-2)\,\sigma_{N}(J)\,R_{N}\,e^{-cR^{2}}\\\
\leq&\,(h-2)\,\bar{\beta}^{\prime}\,\,e^{-cR^{2}}\,,\end{split}$ (3.14)

for some $\bar{\beta}^{\prime}\in(\bar{\beta},1)$. The second summand in the
parenthesis in (3.11) can be bounded via Cauchy-Schwarz by

$\Bigg{(}\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q^{2}_{n}(z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\Bigg{)}^{\frac{1}{2}}\cdot\Bigg{(}\sum_{1\leq
n\leq N,\,z\in\mathbb{Z}^{2}}q^{2}_{n}(z+u)\Bigg{)}^{\frac{1}{2}}\,.$ (3.15)

For the first term in (3.15), using that
$\sup_{z\in\mathbb{Z}^{2}}q_{n}(z)\leq\frac{C}{n}$ we get

$\begin{split}\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q^{2}_{n}(z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\leq\,C\sum_{1\leq
n\leq
N,\,z\in\mathbb{Z}^{2}}\frac{q_{n}(z)}{n}\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\leq\,C\,e^{-cR^{2}}\,\log
N\end{split}$ (3.16)

For the second term in (3.11), we have that for all $u\in\mathbb{Z}^{2}$

$\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q^{2}_{n}(z+u)=\sum^{N}_{n=1}q_{2n}(0)\stackrel{{\scriptstyle
N\to\infty}}{{\approx}}\frac{\log N}{\pi}\,.$

Thus, by (3.15) together with (3.16) we conclude that for the second summand
in (3.11) we have

$\sigma_{N}(J)\cdot\Bigg{(}2\,\sup_{u\,\in\mathbb{Z}^{2}}\sum_{1\leq n\leq
N,\,z\in\mathbb{Z}^{2}}q_{n}(z)q_{n}(z+u)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\Bigg{)}\leq
C\,e^{-\frac{cR^{2}}{2}}\,.$

Therefore, recalling (3.14) we deduce that there exists a constant
$\kappa(h,\bar{\beta})\in(0,\infty)$ such that

$\sigma_{N}(J)\cdot\Bigg{(}\sum_{1\leq n\leq
N,\,\boldsymbol{y}\,\in(\mathbb{Z}^{2})^{h}}Q^{I;J}_{n}(\boldsymbol{x},\boldsymbol{y})\cdot\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{x}-\boldsymbol{y}\right\rVert_{\infty}>R\sqrt{n}\big{\\}}}\Bigg{)}\leq
e^{-\kappa R^{2}}\,.$

We move to the proof of (3.10). Similar to the proof of (3.9), we can bound
the left-hand side of (3.10) by

$(h-1)\sum_{1\leq n\leq
N,\,w,\,z\in\mathbb{Z}^{2}}U_{N}^{\bar{\beta}}(n,w)\cdot
q_{n}(z)\mathds{1}_{\\{|z|>R\sqrt{n}\\}}+\sum_{1\leq n\leq
N,z\in\mathbb{Z}^{2}}U^{\bar{\beta}}_{N}(n,z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\,.$
(3.17)

For the first summand in (3.17), by (3.13) we have that

$\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\leq
e^{-cR^{2}}\,,$

and $\sum_{1\leq n\leq
N,\,w\in\mathbb{Z}^{2}}U_{N}^{\bar{\beta}}(n,w)\leq\frac{1}{1-\bar{\beta}^{\prime}}$,
therefore

$(h-1)\sum_{1\leq n\leq
N,\,w,\,z\in\mathbb{Z}^{2}}U_{N}^{\bar{\beta}}(n,w)\cdot
q_{n}(z)\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\leq\frac{h-1}{1-\bar{\beta}^{\prime}}\,e^{-cR^{2}}\,.$
(3.18)

For the second summand, we use the renewal representation of
$U^{\bar{\beta}}_{N}(\cdot,\cdot)$ introduced in (2.8). In particular, we have
that

$\sum_{1\leq n\leq
N,z\in\mathbb{Z}^{2}}U^{\bar{\beta}}_{N}(n,z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}=\sum_{k\geq
0}(\sigma_{N}(\bar{\beta})R_{N})^{k}\sum_{n=0}^{N}\mathrm{P}\Big{(}\big{|}S^{\scriptscriptstyle(N)}_{k}\big{|}>R\sqrt{n},\tau^{\scriptscriptstyle(N)}_{k}=n\Big{)}\,.$
(3.19)

Then, by conditioning on the times $(T^{\scriptscriptstyle(N)}_{i})_{1\leq
i\leq k}$ for which
$\tau_{k}^{\scriptscriptstyle(N)}=T^{\scriptscriptstyle(N)}_{1}+\dots+T^{\scriptscriptstyle(N)}_{k}$
we have that

$\begin{split}\mathrm{P}\Big{(}\big{|}S^{\scriptscriptstyle(N)}_{k}\big{|}>&R\sqrt{n},\,\tau^{\scriptscriptstyle(N)}_{k}=n\Big{)}\\\
&=\sum_{n_{1}+\dots+n_{k}=n}\mathrm{P}\Big{(}\big{|}S^{\scriptscriptstyle(N)}_{k}\big{|}>R\sqrt{n}\,\Big{|}\cap_{i=1}^{k}\big{\\{}T^{\scriptscriptstyle(N)}_{i}=n_{i}\big{\\}}\Big{)}\,\prod_{i=1}^{k}\mathrm{P}\big{(}T^{\scriptscriptstyle(N)}_{i}=n_{i}\big{)}\,.\end{split}$
(3.20)

Note that when we condition on $\cap_{i=1}^{k}\big{\\{}T^{(\scriptscriptstyle
N)}_{i}=n_{i}\big{\\}}$, $S_{k}^{\scriptscriptstyle(N)}$ is a sum of $k$
independent random variables $(\xi_{i})_{1\leq i\leq k}$ taking values in
$\mathbb{Z}^{2}$, with law

$\mathrm{P}(\xi_{i}=x)=\frac{q^{2}_{n_{i}}(x)}{q_{2n_{i}}(0)}\,.$

The proof of Proposition 3.4 in [LZ21] showed that there exists a constant
$C\in(0,\infty)$ such that for all $\lambda\geq 0$

$\mathrm{E}\Big{[}e^{\lambda|\sum_{i=1}^{k}\xi_{i}|}\Big{]}\leq
2e^{4C\lambda^{2}n}\,.$ (3.21)

Therefore, by (3.21) with $\lambda=\frac{1}{\sqrt{n}}$ and Markov’s inequality
we obtain that

$\mathrm{P}\Big{(}\big{|}S^{\scriptscriptstyle(N)}_{k}\big{|}>R\sqrt{n}\,\Big{|}\cap_{i=1}^{k}\\{T^{\scriptscriptstyle(N)}_{i}=n_{i}\\}\Big{)}\leq
2e^{4C-R}\,.$

Thus, looking back at (3.20) we have that for all $k\geq 0$,

$\mathrm{P}\Big{(}\big{|}S^{\scriptscriptstyle(N)}_{k}\big{|}>R\sqrt{n},\,\tau^{\scriptscriptstyle(N)}_{k}=n\Big{)}\leq
2e^{4C-R}\,\,\mathrm{P}(\tau^{\scriptscriptstyle(N)}_{k}=n)\,,$

therefore, plugging the last inequality into (3.19), we get that

$\sum_{1\leq n\leq
N,z\in\mathbb{Z}^{2}}U^{\bar{\beta}}_{N}(n,z)\cdot\mathds{1}_{\\{|z|>R\sqrt{n}\\}}\leq
2e^{4C-R}\sum_{n=1}^{N}U^{\bar{\beta}}_{N}(n)\leq\frac{2e^{4C-R}}{1-\bar{\beta}^{\prime}}\,,$
(3.22)

therefore by (3.18) and (3.22) we have that there exists a constant
$\kappa(h,\bar{\beta})\in(0,\infty)$ such that

$\sum_{1\leq n\leq
N,\,\boldsymbol{y}\,\in(\mathbb{Z}^{2})^{h}}\mathsf{U}^{J}_{n}(\boldsymbol{x},\boldsymbol{y})\cdot\mathds{1}_{\big{\\{}\left\lVert\boldsymbol{x}-\boldsymbol{y}\right\rVert_{\infty}>R\sqrt{n}\big{\\}}}\leq
e^{-\kappa R}\,,$

for large enough $R\in(0,\infty)$, thus concluding the proof of (3.10). ∎

###  3.3. Scale separation.

In this step we show that given $r\in\mathbb{N}$, $r\geq 2$, the main
contribution to $H^{\mathsf{(diff)}}_{r,N}$ comes from configurations where
$a_{i+1}-b_{i}>M(b_{i}-b_{i-1})$ for all $1\leq i\leq r$ and large $M$, as
$N\to\infty$. Recall from (3.4) that

$H^{\mathsf{(diff)}}_{r,N,R}=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(diff)}}_{r,N,R}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\cdot\sigma_{N}(I_{i})\,.$

Define the set

$\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}:=\mathsf{C}^{\mathsf{(diff)}}_{r,N,R}\cap\Big{\\{}(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}}):\,a_{i+1}-b_{i}>M(b_{i}-b_{i-1})\text{
for all }1\leq i\leq r-1\,\Big{\\}}\,,$ (3.23)

with the convention $b_{0}:=0$ and accordingly define

$H^{\mathsf{(main)}}_{r,N,R,M}:=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\,.$
(3.24)

We then have the following approximation proposition:

###### Proposition 3.5.

For all fixed $r\in\mathbb{N}$, $r\geq 2$ and $M\in(0,\infty)$,

$\lim_{N\to\infty}\sup_{R\,\in(0,\infty)}\Big{|}H^{\mathsf{(diff)}}_{r,N,R}-H^{\mathsf{(main)}}_{r,N,R,M}\Big{|}=0\,.$
(3.25)

  * Proof.

Fix $M>0$. Let us begin by showing (3.25) for the simplest case which is
$r=2$. We have

$\begin{split}H^{\mathsf{(diff)}}_{2,N,R}&-H^{\mathsf{(main)}}_{2,N,R,M}\\\
\leq&\sum_{(I_{1},I_{2})\in\,\mathcal{I}^{(2)}}\,\,\sum_{\begin{subarray}{c}0\leq
b_{1}<a_{2}\leq N,\,a_{2}-b_{1}\leq Mb_{1},\\\
\boldsymbol{y}_{1},\boldsymbol{x}_{2},\boldsymbol{y}_{2}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})Q_{a_{2}-b_{1}}^{I_{1};I_{2}}(\boldsymbol{y}_{1},\boldsymbol{x}_{2})\,\sigma_{N}(I_{2})\mathsf{U}^{I_{2}}_{b_{2}-a_{2}}(\boldsymbol{x}_{2},\boldsymbol{y}_{2}).\end{split}$

We can bound $\sigma_{N}(I_{2})$ by $\frac{\pi\bar{\beta}\,^{\prime}}{\log
N}$, for some $\bar{\beta}\,^{\prime}\in(\bar{\beta},1)$ and use (2.18) to
bound the last replica, i.e. the sum over $(b_{2},\boldsymbol{y}_{2})$, thus
getting

$H^{\mathsf{(diff)}}_{2,N,R}-H^{\mathsf{(main)}}_{2,N,R,M}\leq\frac{\pi\bar{\beta}\,^{\prime}(1-\bar{\beta}\,^{\prime})^{-1}}{\log
N}\sum_{(I_{1},I_{2})\in\,\mathcal{I}^{(2)}}\,\sum_{\begin{subarray}{c}0\leq
b_{1}<a_{2}\leq N,\,a_{2}-b_{1}\leq Mb_{1}\,,\\\
\boldsymbol{y}_{1},\boldsymbol{x}_{2},\boldsymbol{y}_{2}\in(\mathbb{Z}^{2})^{h}\end{subarray}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})Q_{a_{2}-b_{1}}^{I_{1};I_{2}}(\boldsymbol{y}_{1},\boldsymbol{x}_{2})\,.$
(3.26)

Notice that at this stage we can sum out the spatial endpoints of the free
kernels in (3.26) and bound the coupling strength $\beta_{k,\ell}$ of any
replica $\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})$ with
$I_{1}=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$ by $\bar{\beta}$ to
obtain

$\displaystyle H^{\mathsf{(diff)}}_{2,N,R}$
$\displaystyle-H^{\mathsf{(main)}}_{2,N,R,M}$
$\displaystyle\leq\frac{\pi\bar{\beta}\,^{\prime}(1-\bar{\beta}\,^{\prime})^{-1}}{\log
N}\sum_{(I_{1},I_{2})\in\,\mathcal{I}^{(2)}}\,\sum_{\begin{subarray}{c}0\leq
b_{1}<a_{2}\leq N,\,a_{2}-b_{1}\leq Mb_{1}\,,\\\
y_{1},x_{2}\in\mathbb{Z}^{2}\end{subarray}}U^{\bar{\beta}}_{N}(b_{1},y_{1})q_{a_{2}-b_{1}}(x_{2}-y_{1})\,q_{a_{2}}(x_{2})$
$\displaystyle\leq\frac{\pi\bar{\beta}\,^{\prime}(1-\bar{\beta}\,^{\prime})^{-1}}{\log
N}\binom{h}{2}^{2}\sum_{\begin{subarray}{c}0\leq b_{1}<a_{2}\leq
N,\,a_{2}-b_{1}\leq Mb_{1}\,,\\\
y_{1},x_{2}\in\mathbb{Z}^{2}\end{subarray}}U^{\bar{\beta}}_{N}(b_{1},y_{1})q_{a_{2}-b_{1}}(x_{2}-y_{1})\,q_{a_{2}}(x_{2})\,.$
(3.27)

For the last inequality we also have used that the number of possible
partitions $(I_{1},I_{2})\in{\mathcal{I}}^{(2)}$ is bounded by
$\binom{h}{2}^{2}$. For every fixed value of $b_{1}$ in (Proof.), we use
Cauchy-Schwarz for the sum over
$(a_{2},x_{2})\in(0,(1+M)b_{1}]\times\mathbb{Z}^{2}$ in (Proof.) to obtain
that

$\begin{split}\sum_{b_{1}<a_{2}\leq(1+M)b_{1},\,x_{2}\in\mathbb{Z}^{2}}&q_{a_{2}-b_{1}}(x_{2}-y_{1})\,q_{a_{2}}(x_{2})\\\
\leq&\Bigg{(}\sum_{0<a_{2}\leq(1+M)b_{1},\,x_{2}\in\mathbb{Z}^{2}}q^{2}_{a_{2}-b_{1}}(x_{2}-y_{1})\Bigg{)}^{\frac{1}{2}}\Bigg{(}\sum_{b_{1}<a_{2}\leq(1+M)b_{1},\,x_{2}\in\mathbb{Z}^{2}}q^{2}_{a_{2}}(x_{2})\Bigg{)}^{\frac{1}{2}}\\\
=&\Bigg{(}\sum_{0<a_{2}\leq(1+M)b_{1}}q_{2(a_{2}-b_{1})}(0)\Bigg{)}^{\frac{1}{2}}\Bigg{(}\sum_{b_{1}<a_{2}\leq(1+M)b_{1}}q_{2a_{2}}(0)\Bigg{)}^{\frac{1}{2}}\,.\end{split}$
(3.28)

We can bound the leftmost parenthesis in the last line of (3.28) by
$R^{1/2}_{N}=\Big{(}\sum_{n=1}^{N}q_{2n}(0)\Big{)}^{1/2}=O(\sqrt{\log N})$.
For the other term we have

$\sum_{b_{1}<a_{2}\leq(1+M)b_{1}}q_{2a_{2}}(0)\leq
c\,\sum_{b_{1}<a_{2}\leq(1+M)b_{1}}\tfrac{1}{a_{2}}\leq c\log(1+M)\,.$ (3.29)

Therefore, using (3.28) and (3.29) along with $\sum_{0\leq b_{1}\leq
N,\,y_{1}\in\mathbb{Z}^{2}}U^{\bar{\beta}}_{N}(b_{1},y_{1})\leq(1-\bar{\beta}\,^{\prime})^{-1}$
in (Proof.) we obtain that

$H^{\mathsf{(diff)}}_{2,N,R}-H^{\mathsf{(main)}}_{2,N,R,M}\leq
C\pi\bar{\beta}\,^{\prime}(1-\bar{\beta}\,^{\prime})^{-2}\sqrt{\frac{\log(1+M)}{\log
N}}\xrightarrow{N\to\infty}0\,.$

Let us show how this argument can be extended to work for general
$r\in\mathbb{N}$. The key observation is that for every fresh collision
between two random walks, that is $I_{i+1}=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq
k,\ell}\\{j\\}$, happening at time $0<a_{i+1}\leq N$, we have $I_{i}\neq
I_{i+1}$, therefore one of the two colliding walks with labels $k,\ell$ has to
have travelled freely, for time at least $a_{i+1}-b_{i-1}$ from its previous
collision. More precisely, every term in the expansion of
$H^{\mathsf{(diff)}}_{r,N,R}-H^{\mathsf{(main)}}_{r,N,R,M}$ contains for every
$1\leq i\leq r-1$ a product of the form

$q_{a_{i+1}-b_{i}}(x_{i+1}-y_{i})\cdot
q_{a_{i+1}-b_{i-1}}(x_{i+1}-y_{i-1})\,,$

see Figure 3. Recall from (3.2) and (3.24) that we have the expansion

$\begin{split}&H^{\mathsf{(diff)}}_{r,N,R}-H^{\mathsf{(main)}}_{r,N,R,M}\\\
&=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(diff)}}_{r,N,R}\smallsetminus\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\,,\end{split}$
(3.30)

where by definition (3.23) we have that

$\mathsf{C}^{\mathsf{(diff)}}_{r,N,R}\smallsetminus\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}=\mathsf{C}^{\mathsf{(diff)}}_{r,N,R}\,\cap\,\bigcup_{i=1}^{r-1}\Big{\\{}(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}}):\,a_{i+1}-b_{i}\leq
M(b_{i}-b_{i-1})\,\Big{\\}}\,.$ (3.31)

The strategy we are going to follow is to start the summation of (3.30) from
the end until we find the index $1\leq i\leq r-1$ for which the sum over
$a_{i+1}$ is restricted to $\big{(}b_{i},b_{i}+M(b_{i}-b_{i-1})\big{]}$, in
agreement with (3.31), using (2.18) to bound the contribution of the sums over
$b_{j},a_{j+1}$ and the corresponding spatial points for $i<j\leq r-1$. Next,
notice that we can bound the contribution of the sum over
$a_{i+1}\in\big{(}b_{i},b_{i}+M(b_{i}-b_{i-1})\big{]}$ and
$x_{i+1}\in\mathbb{Z}^{2}$, using a change of variables, by a factor of

$\frac{C}{\log N}\Bigg{(}\sup_{1\leq t\leq N,\,u\in\mathbb{Z}^{2}}\sum_{1\leq
n\leq Mt,\,z\in\mathbb{Z}^{2}}q_{n}(z)q_{n+t}(z+u)\Bigg{)}\leq
C\sqrt{\frac{\log(1+M)}{\log N}}\,,$

using Cauchy-Schwarz as in (3.28) and (3.29). The remaining sums over
$b_{j},a_{j-1}$, $1\leq j\leq i$ can be bounded again via (2.18). Therefore,
taking into account that by (3.31) there are $r-1$ choices for the index $i$
such that the sum over $a_{i+1}$ is restricted to
$\big{(}b_{i},b_{i}+M(b_{i}-b_{i-1})\big{]}$, we can give an upper bound to
$H^{\mathsf{(diff)}}_{r,N,R}-H^{\mathsf{(main)}}_{r,N,R,M}$ as follows:

$H^{\mathsf{(diff)}}_{r,N,R}-H^{\mathsf{(main)}}_{r,N,R,M}\leq
C(r-1)\,\binom{h}{2}^{r}\,\Big{(}\frac{\bar{\beta}\,^{\prime}}{1-\bar{\beta}\,^{\prime}}\Big{)}^{r}\sqrt{\frac{\log(1+M)}{\log
N}}\xrightarrow{N\to\infty}0\,,$

where we also used that the number of distinct sequences
$(I_{1},\dots,I_{r})\in{\mathcal{I}}^{(2)}$ is bounded by $\binom{h}{2}^{r}$.
∎

$0:=a_{1}$$b_{1}$$a_{2}$$b_{2}$$a_{3}$$b_{3}$$\\{2,3\\}$$\\{1,2\\}$$\\{2,3\\}$$a_{4}$$b_{4}$$\\{3,4\\}$
Figure 3. A diagramatic representation of a configuration of collisions
between $4$ random walks in $H^{(2)}_{4,N}$ with $I_{1}=\\{2,3\\}$,
$I_{2}=\\{1,2\\}$, $I_{3}=\\{3,4\\}$ and $I_{4}=\\{2,3\\}$. Wiggly lines
represent replica evolution, see (2.15).

###  3.4. Rewiring.

Recall the expansion of $H^{\mathsf{(main)}}_{r,N,R,M}$,

$H^{\mathsf{(main)}}_{r,N,R,M}:=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\,.$
(3.32)

We also remind the reader that we may identify a partition
$I=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$ with its non-trivial
part $\\{k,\ell\\}$. Moreover, if
$\big{(}\\{i_{1},j_{1}\\},\cdots,\\{i_{r},j_{r}\\}\big{)}\in{\mathcal{I}}^{(2)}$
we will use the notation

$\mathsf{p}(m):=\max\big{\\{}k<m:\,\\{i_{m},j_{m}\\}=\\{i_{k},j_{k}\\}\big{\\}}\,,$

with the convention that $\mathsf{p}(m)=0$ if
$\\{i_{k},j_{k}\\}\neq\\{i_{m},j_{m}\\}$ for all $1\leq k<m$. Given this
definition, the time $b_{\mathsf{p}(m)}$ represents the last time walks
${i_{m},j_{m}}$ collided before their new collision at time $a_{m}$. Note that
since we always have $\\{i_{k},j_{k}\\}\neq\\{i_{k+1},j_{k+1}\\}$ by
construction, $\mathsf{p}(m)<m-1$.

Consider a sequence of partitions
$\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{m},j_{m}\\}\big{)}\in{\mathcal{I}}^{(2)}$
and let $m\in\\{2,\dots,r\\}$. The goal of this step will be to show that we
can make the replacement of kernels

$q_{a_{m}-b_{m-1}}\Big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\Big{)}\cdot
q_{a_{m}-b_{m-1}}\Big{(}x^{(j_{m})}_{m}-y^{(j_{m})}_{m-1}\Big{)}\quad\longleftrightarrow\quad
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\Big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\Big{)}\,.$
(3.33)

by inducing an error which is negligible when $M\to\infty$. We iterate this
procedure for all partitions $I_{1},\dots,I_{r}$. We call the procedure
described above rewiring, see Figures 3 and 4. The first step towards the full
rewiring is to show the following lemma which quantifies the error of a single
replacement (3.33).

###### Lemma 3.6.

Let $r\geq 2$ fixed and $m\in\\{2,\dots,r\\}$ with $I_{m}=\\{i_{m},j_{m}\\}$.
Then, for every fixed $R\in(0,\infty)$ and uniformly in
$(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}$
and all sequences of partitions $(I_{1},\dots,I_{r})\in{\mathcal{I}}^{(2)}$,

$q_{a_{m}-b_{m-1}}\Big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\Big{)}\cdot
q_{a_{m}-b_{m-1}}\Big{(}x^{(j_{m})}_{m}-y^{(j_{m})}_{m-1}\Big{)}=q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\Big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\Big{)}\cdot
e^{o_{M}(1)}\,,$ (3.34)

where $o_{M}(1)$ denotes a quantity such that $\lim_{M\to\infty}o_{M}(1)=0$.

  * Proof.

We will show that

$\begin{split}q_{a_{m}-b_{m-1}}\Big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\Big{)}&=q_{a_{m}-b_{\mathsf{p}(m)}}\Big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\Big{)}\cdot
e^{o_{M}(1)}\end{split}$ (3.35)

and by symmetry we will get (3.34). To this end, we invoke the local limit
theorem for simple random walks, which we recall from [LL10]. In particular,
by Theorem 2.3.11 [LL10], we have that there exists $\varrho>0$ such that for
all $n\geq 0$ and $x\in\mathbb{Z}^{2}$ with $|x|<\varrho\,n$,

$q_{n}(x)=g_{\frac{n}{2}}(x)\cdot
e^{O\Big{(}\frac{1}{n}+\frac{|x|^{4}}{n^{3}}\Big{)}}\cdot
2\cdot\mathds{1}_{\big{\\{}(n,x)\in\mathbb{Z}^{3}_{\mathsf{even}}\big{\\}}}\,,$
(3.36)

where $g_{t}(x)=\frac{e^{-\frac{|x|^{2}}{2t}}}{2\pi t}$ denotes the
$2$-dimensional heat kernel and

$\mathbb{Z}^{3}_{\mathsf{even}}:=\\{(n,x)\in\mathbb{Z}\times\mathbb{Z}^{2}:\,n+x_{1}+x_{2}=0\,\,(\text{mod}\,\,2)\\}\,.$

The last constraint in $\eqref{llt}$ is a consequence of the periodicity of
the simple random walk. Let us proceed with the proof of Lemma 3.6.

First, we derive some inequalities which are going to be useful for the
approximations using the local limit theorem. We claim that

$a_{m}-b_{m-1}>(M-1)\,\big{(}b_{m-1}-b_{\mathsf{p}(m)}\big{)}\,.$ (3.37)

The proof of the latter is done by a finite recursion. In particular, notice
that since the inequality $a_{k+1}-b_{k}>M(b_{k}-b_{k-1})$ holds for all
$1\leq k\leq r-1$, we have

$a_{m}-b_{m-1}>M(b_{m-1}-b_{m-2})=M(b_{m-1}-a_{m-1})+M(a_{m-1}-b_{m-2})\,.$
(3.38)

We can then write

$\begin{split}M(a_{m-1}-b_{m-2})&=(M-1)(a_{m-1}-b_{m-2})+(a_{m-1}-b_{m-2})\\\
&>(M-1)(a_{m-1}-b_{m-2})+M(b_{m-2}-b_{m-3})\end{split}$

and repeat the same steps for $M(b_{m-2}-b_{m-3})$ as we did in (3.38) for
$M(b_{m-1}-b_{m-2})$. After telescoping we get the lower bound in (3.37).
Moreover, by triangle inequality we have that

$\Big{|}\,\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}-\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}\,\Big{|}\leq\big{|}y^{(i_{m})}_{m-1}-x^{(i_{m})}_{m-1}\big{|}+\big{|}x^{(i_{m})}_{m-1}-y^{(i_{m})}_{m-2}\big{|}+\dots+\big{|}x^{(i_{m})}_{\mathsf{p}(m)+1}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}\,.$
(3.39)

Note that by the diffusivity constraints of
$\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}$ we have that

$\begin{split}\big{|}y^{(i_{m})}_{m-1}-x^{(i_{m})}_{m-1}\big{|}+\big{|}x^{(i_{m})}_{m-1}-y^{(i_{m})}_{m-2}\big{|}+\dots+&\big{|}x^{(i_{m})}_{\mathsf{p}(m)+1}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}\\\
\leq&R\cdot\bigg{(}\sum_{k=\mathsf{p}(m)}^{m-2}\sqrt{a_{k+1}-b_{k}}+\sum_{k=\mathsf{p}(m)+1}^{m-1}\sqrt{b_{k}-a_{k}}\bigg{)}\,.\end{split}$
(3.40)

By Cauchy-Schwarz on the right hand side of (3.40), (3.37) and the fact that
$m\leq r$, we furthermore have that

$\begin{split}\Big{(}\sum_{k=\mathsf{p}(m)}^{m-2}\sqrt{a_{k+1}-b_{k}}+\sum_{k=\mathsf{p}(m)+1}^{m-1}\sqrt{b_{k}-a_{k}}\Big{)}&\leq\sqrt{2r-1}\,\bigg{(}\sum_{k=\mathsf{p}(m)}^{m-2}(a_{k+1}-b_{k})+\sum_{k=\mathsf{p}(m)+1}^{m-1}(b_{k}-a_{k})\bigg{)}^{1/2}\\\
&=\sqrt{2r-1}\cdot\sqrt{b_{m-1}-b_{\mathsf{p}(m)}}\\\
&\leq\sqrt{2r-1}\cdot\sqrt{\frac{a_{m}-b_{m-1}}{M-1}}\,.\end{split}$

Thus, taking into account (3.39) we conclude that

$\Big{|}\,\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}-\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}\,\Big{|}\leq
R\cdot\sqrt{2r-1}\cdot\sqrt{\frac{a_{m}-b_{m-1}}{M-1}}\,.$ (3.41)

Now, we are ready to show approximation (3.35). By (3.36) we have

$\begin{split}&\frac{q_{a_{m}-b_{m-1}}\big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{)}}{q_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{)}}\\\
&=e^{-\frac{|x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}|^{2}}{a_{m}-b_{m-1}}+\frac{|x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}|^{2}}{a_{m}-b_{\mathsf{p}(m)}}}\cdot\bigg{(}\frac{a_{m}-b_{\mathsf{p}(m)}}{a_{m}-b_{m-1}}\bigg{)}\cdot
e^{O\Big{(}\frac{1}{a_{m}-b_{m-1}}+\frac{|x^{(i_{m})}_{m}-y_{m-1}^{(i_{m})}|^{4}+|x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}|^{4}}{(a_{m}-b_{m-1})^{3}}\Big{)}}\,.\end{split}$
(3.42)

Let us look at each term on the right hand side of (3.42), separately. First,
by (3.41) we have

$\begin{split}e^{-\frac{\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}^{2}}{a_{m}-b_{m-1}}+\frac{\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}^{2}}{a_{m}-b_{\mathsf{p}(m)}}}&\leq
e^{\frac{1}{a_{m}-b_{m-1}}\Big{(}\,\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}^{2}-\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}^{2}\,\Big{)}}\\\
&=e^{\frac{1}{a_{m}-b_{m-1}}\,\Big{(}\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}+\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}\Big{)}\cdot\Big{(}\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}-\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}\Big{)}}\,.\end{split}$

Using (3.41) we have

$\begin{split}\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}+\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}&\leq
2\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}+R\sqrt{2r-1}\sqrt{\frac{a_{m}-b_{m-1}}{M-1}}\\\
&\leq 2R\sqrt{a_{m}-b_{m-1}}+R\sqrt{2r-1}\sqrt{\frac{a_{m}-b_{m-1}}{M-1}}\\\
&=R\sqrt{a_{m}-b_{m-1}}\Big{(}2+\sqrt{\frac{2r-1}{M-1}}\Big{)}\,.\end{split}$

Therefore, by (3.41) we get

$e^{\frac{1}{a_{m}-b_{m-1}}\,\Big{(}\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}+\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}\Big{)}\cdot\Big{(}\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}-\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}\Big{)}}\leq
e^{R^{2}\Big{(}2+\sqrt{\frac{2r-1}{M-1}}\Big{)}\sqrt{\frac{2r-1}{M-1}}}\,.$

Similarly, we can get a lower bound of

$e^{-\frac{\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}\big{|}^{2}}{a_{m}-b_{m-1}}+\frac{\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}^{2}}{a_{m}-b_{\mathsf{p}(m)}}}\geq
e^{-R^{2}\big{(}1-\frac{1}{M}\big{)}\Big{(}2+\sqrt{\frac{2r-1}{M-1}}\Big{)}\sqrt{\frac{2r-1}{M-1}}}\,,$

since $a_{m}-b_{\mathsf{p}(m)}<\big{(}1+\frac{1}{M-1}\big{)}(a_{m}-b_{m-1})$
by (3.37). The second term in (3.42) can be handled by (3.37) as

$1\leq\Big{(}\frac{a_{m}-b_{\mathsf{p}(m)}}{a_{m}-b_{m-1}}\Big{)}=\Big{(}1+\frac{b_{m-1}-b_{\mathsf{p}(m)}}{a_{m}-b_{m-1}}\Big{)}<1+\frac{1}{M-1}\xrightarrow{M\to\infty}1\,.$

For the last term in (3.42) we have that

$\begin{split}&\frac{\big{|}x^{(i_{m})}_{m}-y_{m-1}^{(i_{m})}\big{|}^{4}+\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}^{4}}{(a_{m}-b_{m-1})^{3}}\\\
&\qquad\qquad\qquad\qquad\leq\frac{\big{|}x^{(i_{m})}_{m}-y_{m-1}^{(i_{m})}\big{|}^{4}}{(a_{m}-b_{m-1})^{3}}+\frac{\Big{(}|x^{(i_{m})}_{m}-y_{m-1}^{(i_{m})}|+R\cdot\sqrt{2r-1}\cdot\sqrt{\frac{(a_{m}-b_{m-1})}{M-1}}\,\Big{)}^{4}}{(a_{m}-b_{m-1})^{3}}\\\
&\qquad\qquad\qquad\qquad\leq\frac{9\,R^{4}}{(a_{m}-b_{m-1})}+\frac{8R^{4}(2r-1)^{2}}{(a_{m}-b_{m-1})\cdot(M-1)}\,,\end{split}$

where we used (3.41) along with the inequality $(x+y)^{4}\leq 8(x^{4}+y^{4})$
for $x,y\in\mathbb{R}$. Therefore,

$e^{\frac{\big{|}x^{(i_{m})}_{m}-y_{m-1}^{(i_{m})}\big{|}^{4}+\big{|}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{|}^{4}}{(a_{m}-b_{m-1})^{3}}}\leq
e^{\frac{9\,R^{4}}{(a_{m}-b_{m-1})}+\frac{8R^{4}(2r-1)^{2}}{(a_{m}-b_{m-1})\cdot(M-1)}}\leq
e^{\frac{9\,R^{4}}{M}+\frac{8R^{4}(2r-1)^{2}}{M\cdot(M-1)}}\xrightarrow{M\to\infty}1\,,$

where we used in the last inequality that
$a_{m}-b_{m-1}>M(b_{m-1}-b_{m-2})\geq M$ by (3.23). ∎

$0:=a_{1}$$b_{1}$$a_{2}$$b_{2}$$a_{3}$$b_{3}$$\\{2,3\\}$$\\{1,2\\}$$\\{2,3\\}$$a_{4}$$b_{4}$$\\{3,4\\}$$0:=a_{1}$$b_{1}$$a_{2}$$b_{2}$$a_{3}$$b_{3}$$\\{2,3\\}$$\\{1,2\\}$$\\{2,3\\}$$a_{4}$$b_{4}$$\\{3,4\\}$
Figure 4. Figure 3 after rewiring. We use blue lines to represent the new
kernels produced by rewiring. The dashed lines represent remaining free
kernels from the rewiring procedure as well as kernels coming from using the
Chapman-Kolmogorov formula for the simple random walk.

###  3.5. Final step

Now that we have Lemma 3.6 at our disposal, we can prove the main
approximation result of this step. Recall from (3.32) that

$\begin{split}&H^{\mathsf{(main)}}_{r,N,R,M}\\\
&\quad=\sum_{(I_{1},...,I_{r})\,\in\,\mathcal{I}^{(2)}}\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\mathsf{U}^{I_{1}}_{b_{1}}(0,\boldsymbol{y}_{1})\prod_{i=2}^{r}Q_{a_{i}-b_{i-1}}^{I_{i-1};I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{x}_{i})\,\mathsf{U}^{I_{i}}_{b_{i}-a_{i}}(\boldsymbol{x}_{i},\boldsymbol{y}_{i})\,\sigma_{N}(I_{i})\,.\end{split}$

Define $H^{\mathsf{(rew)}}_{r,N,R,M}$ to be the resulting sum after rewiring
has been applied to every term of $H^{\mathsf{(main)}}_{r,N,R,M}$, that is,
given a sequence of partitions $(I_{1},\dots,I_{r})\in\mathcal{I}^{(2)}$ and
$(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}$,
we apply the kernel replacement (3.33) to all partitions $I_{1},\dots,I_{r}$
starting from $I_{r}$ and moving backward. We remind the reader that that we
may denote a partition $I=\\{i,j\\}\sqcup\bigsqcup_{k\neq
i,j}\\{k\\}\in{\mathcal{I}}^{(2)}$ by its non-trivial part $\\{i,j\\}$, see
subsection 2.2.

###### Proposition 3.7.

Fix $0\leq r\leq K$. We have that

$H^{\mathsf{(rew)}}_{r,N,R,M}=e^{K\cdot
o_{M}(1)}\,H^{\mathsf{(main)}}_{r,N,R,M}$ (3.43)

and

$\begin{split}&H^{\mathsf{(rew)}}_{r,N,R,M}\\\
&=\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,{\mathcal{I}}^{(2)}}\,\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y^{(i_{k})}_{k}-x^{(i_{k})}_{k}\big{)}\cdot\prod_{\begin{subarray}{c}1\leq\ell\leq
h,\\\ \ell\neq
i_{k},j_{k}\end{subarray}}q_{b_{k}-a_{k}}(y^{(\ell)}_{k}-x^{(\ell)}_{k})\\\
&\qquad\qquad\times\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{)}\cdot\prod_{\begin{subarray}{c}1\leq\ell\leq
h,\\\ \ell\neq
i_{m},j_{m}\end{subarray}}q_{a_{m}-b_{m-1}}\big{(}x^{(\ell)}_{m}-y^{(\ell)}_{m-1}\big{)}\,.\end{split}$
(3.44)

  * Proof.

Equation (3.43) is a consequence of Lemma 3.6 and the fact that $r\leq K$,
while expansion (3.44) is a direct consequence of the rewiring procedure we
described in the previous step, see also Figures 3 and 4. ∎

Next, we derive upper and lower bounds for $H^{\mathsf{(rew)}}_{r,N,R,M}$. We
begin with the upper bound.

###### Proposition 3.8.

We have that

$\begin{split}H^{\mathsf{(rew)}}_{r,N,R,M}\leq\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
0:=x_{1},y_{1},\dots,x_{r},y_{r}\in\mathbb{Z}^{2}\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y_{k}-x_{k}\big{)}\,\\\
\times\,\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x_{m}-y_{\mathsf{p}(m)}\big{)}\,.\end{split}$

  * Proof.

Fix $r\geq 1$ and from (3.44) recall that

$\begin{split}&H^{\mathsf{(rew)}}_{r,N,R,M}\\\
&\quad=\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y^{(i_{k})}_{k}-x^{(i_{k})}_{k}\big{)}\cdot\prod_{\begin{subarray}{c}1\leq\ell\leq
h,\\\ \ell\neq
i_{k},j_{k}\end{subarray}}q_{b_{k}-a_{k}}(y^{(\ell)}_{k}-x^{(\ell)}_{k})\\\
&\qquad\qquad\qquad\times\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{)}\cdot\prod_{\begin{subarray}{c}1\leq\ell\leq
h,\\\ \ell\neq
i_{m},j_{m}\end{subarray}}q_{a_{m}-b_{m-1}}\big{(}x^{(\ell)}_{m}-y^{(\ell)}_{m-1}\big{)}\,.\end{split}$
(3.45)

For the sake of obtaining an upper bound on $H^{\mathsf{(rew)}}_{r,N,R,M}$ we
can sum $(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})$ in
(3.45) over $\mathsf{C}_{r,N}$, see definition in (3.2), instead of
$\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}$. We start the summation of the right
hand side of (3.45) from the end. Using that for $n\in\mathbb{N}$,
$\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)=1$ we deduce that

$\sum_{\begin{subarray}{c}y_{r}^{(\ell)}\in\mathbb{Z}^{2}:\\\ 1\leq\ell\leq
h,\,\ell\neq i_{r},j_{r}\end{subarray}}\,\prod_{\ell\neq
i_{r},j_{r}}q_{b_{r}-a_{r}}(y^{(\ell)}_{r}-x^{(\ell)}_{r})=1\,.$

We leave the sum
$\sum_{b_{r}\in[a_{r},N],\,y^{(i_{r})}_{r}\in\mathbb{Z}^{2}}U_{N}\big{(}b_{r}-a_{r},y_{r}^{(i_{r})}-x_{r}^{(i_{r})}\big{)}$
intact and move on to the time interval $[b_{r-1},a_{r}]$. We use again that
for $n\in\mathbb{N}$, $\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)=1$, to deduce that

$\sum_{\begin{subarray}{c}x_{r}^{(\ell)}\in\mathbb{Z}^{2}:\\\ 1\leq\ell\leq
h,\,\ell\neq i_{r},j_{r}\end{subarray}}\,\prod_{\ell\neq
i_{r},j_{r}}\,q_{a_{r}-b_{r-1}}(x^{(\ell)}_{r}-y^{(\ell)}_{r-1})=1\,.$

Again, we leave the sum
$\sum_{a_{r}\in\,(b_{r-1},b_{r}],\,x_{r}^{(i_{r})}\in\mathbb{Z}^{2}}\,q_{a_{r}-b_{\mathsf{p}(r)}}^{2}\big{(}x^{(i_{r})}_{r}-y^{(i_{r})}_{\mathsf{p}(r)}\big{)}$
intact. We can iterate this procedure inductively since due to rewiring all
the spatial variables $y^{(\ell)}_{r-1}$, $\ell\neq i_{r-1},j_{r-1}$ are free,
that is, there are no outgoing laces starting off $y^{(\ell)}_{r-1}$,
$\ell\neq i_{r-1},j_{r-1}$ at time $b_{r-1}$. The summations we have performed
correspond to getting rid of the dashed lines in Figure 4. Iterating this
procedure inductively then implies the following upper bound for
$H^{\mathsf{(rew)}}_{r,N,R,M}$.

$\begin{split}H^{\mathsf{(rew)}}_{r,N,R,M}\leq\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N\,,\\\
0:=x_{1},y_{1},\dots,x_{r},y_{r}\in\mathbb{Z}^{2}\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y_{k}-x_{k}\big{)}\,\\\
\times\,\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x_{m}-y_{\mathsf{p}(m)}\big{)}\,.\end{split}$

∎

In the next proposition we derive complementary lower bounds for
$H^{\mathsf{(rew)}}_{r,N,R,M}$. Given $0\leq r\leq K$ and a sequence of
partitions
$\vec{I}=(\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\})\in\mathcal{I}^{(2)}$ we
define the set $\mathsf{C}^{\mathsf{(rew)}}_{r,N,R,M}(\vec{I}\,)$ to be
$\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}$ where for every $2\leq m\leq r$ we
replace the diffusivity constraint
$\left\lVert\boldsymbol{x}_{m}-\boldsymbol{y}_{m-1}\right\rVert_{\infty}\leq
R\sqrt{a_{m}-b_{m-1}}$ by the constraints

$\begin{split}&|x^{(\ell)}_{m}-y^{(\ell)}_{m-1}|\leq
R\sqrt{a_{m}-b_{m-1}},\,\ell\in\\{1,\dots,h\\}\smallsetminus\\{i_{m},j_{m}\\}\quad\text{and}\\\
&\quad|x^{(\ell^{\prime})}_{m}-y^{(\ell^{\prime})}_{\mathsf{p}(m)}|\leq
R\sqrt{1-\frac{1}{M}}\bigg{(}1-\sqrt{\frac{2K-1}{M-1}}\bigg{)}\sqrt{a_{m}-b_{\mathsf{p}(m)}},\,\,\ell^{\prime}\in\\{i_{m},j_{m}\\}\,.\end{split}$

This replacement transforms the diffusivity constraints imposed on the jumps
of two walks $\\{i_{m},j_{m}\\}$ from their respective positions at time
$b_{m-1}$ to time $a_{m}$, which is the time they (re)start colliding, to a
diffusivity constraint connecting their common position at time
$b_{\mathsf{p}(m)}$, which is the last time they collided before time $a_{m}$,
to their common position at time $a_{m}$ when they start colliding again.

We have the following Lemma.

###### Lemma 3.9.

Let $0\leq r\leq K$ and $M>2K$. For all
$\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}$
we have that

$\mathsf{C}^{\mathsf{(rew)}}_{r,N,R,M}(\vec{I}\,)\subset\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}\,.$

  * Proof.

Fix $0\leq r\leq K$, a sequence
$\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in{\mathcal{I}}^{(2)}$
and
$(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\in\mathsf{C}^{\mathsf{(rew)}}_{r,N,M,R}(\vec{I}\,)$.
Moreover, let $2\leq m\leq r$. By symmetry it suffices to prove that

$\begin{split}\mathds{1}_{\big{\\{}|x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}|\leq
R\sqrt{1-\frac{1}{M}}\Big{(}1-\sqrt{\frac{2r-1}{M-1}}\Big{)}\sqrt{a_{m}-b_{\mathsf{p}(m)}}\big{\\}}}\leq\mathds{1}_{\big{\\{}|x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}|\leq
R\sqrt{a_{m}-b_{m-1}}\big{\\}}}\,.\end{split}$

Indeed, by the definition of
$\mathsf{C}^{\mathsf{(rew)}}_{r,N,M,R}(\vec{I}\,)$ and (3.41) we have that for
$(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\in\mathsf{C}^{\mathsf{(rew)}}_{r,N,M,R}(\vec{I}\,)$,

$\Big{|}\,|x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}|-|x^{(i_{m})}_{m}-y^{(i_{m})}_{m-1}|\,\Big{|}\leq
R\cdot\sqrt{2r-1}\cdot\sqrt{\frac{a_{m}-b_{m-1}}{M-1}}\,.$ (3.46)

Moreover by (3.37) we have that

$a_{m}-b_{m-1}>(M-1)(b_{m-1}-b_{\mathsf{p}(m)})\Rightarrow
a_{m}-b_{\mathsf{p}(m)}>M(b_{m-1}-b_{\mathsf{p}(m)})\,.$

Therefore,

$a_{m}-b_{m-1}=a_{m}-b_{\mathsf{p}(m)}-(b_{m-1}-b_{\mathsf{p}(m)})>a_{m}-b_{\mathsf{p}(m)}-\frac{1}{M}(a_{m}-b_{\mathsf{p}(m)})=\Big{(}1-\frac{1}{M}\Big{)}(a_{m}-b_{\mathsf{p}(m)})\,.$
(3.47)

Combining inequalities (3.46) and (3.47) we get the result.

∎

###### Proposition 3.10.

Let $0\leq r\leq K$. For $M>2K$ we have that

$\displaystyle H^{\mathsf{(rew)}}_{r,N,R,M}$
$\displaystyle\quad\geq(1-e^{-cR^{2}})^{2Kh}\,\sum_{\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
a_{i+1}-b_{i}>M(b_{i}-b_{i-1}),\,1\leq i\leq r-1,\\\
0:=x_{1},y_{1},\dots,x_{r},y_{r}\in\mathbb{Z}^{2}\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y_{k}-x_{k}\big{)}\,$
$\displaystyle\quad\times\,\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x_{m}-y_{\mathsf{p}(m)}\big{)}\cdot\mathds{1}_{\big{\\{}|y_{k}-x_{k}|\leq
R\sqrt{b_{k}-a_{k}},\,|x_{m}-y_{\mathsf{p}(m)}|\leq
R\,C_{K,M}\sqrt{a_{m}-b_{\mathsf{p}(m)}}\big{\\}}},$

with $C_{K,M}:=\sqrt{1-\frac{1}{M}}\bigg{(}1-\sqrt{\frac{2K-1}{M-1}}\bigg{)}$.

  * Proof.

Recall from (3.44) that

$\begin{split}H^{\mathsf{(rew)}}_{r,N,R,M}&=\sum_{\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(main)}}_{r,N,R,M}}\\\
&\qquad\qquad\times\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y^{(i_{k})}_{k}-x^{(i_{k})}_{k}\big{)}\cdot\prod_{\ell\neq
i_{k},j_{k}}q_{b_{k}-a_{k}}(y^{(\ell)}_{k}-x^{(\ell)}_{k})\\\
&\qquad\qquad\quad\times\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{)}\cdot\prod_{\ell\neq
i_{m},j_{m}}q_{a_{m}-b_{m-1}}\big{(}x^{(\ell)}_{m}-y^{(\ell)}_{m-1}\big{)}\,.\end{split}$

By Lemma 3.9 we have that

$\begin{split}H^{\mathsf{(rew)}}_{r,N,R,M}&\geq\sum_{\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{(\vec{a},\vec{b},\vec{\boldsymbol{x}},\vec{\boldsymbol{y}})\,\in\,\mathsf{C}^{\mathsf{(rew)}}_{r,N,R,M}(\vec{I}\,)}\\\
&\qquad\qquad\times\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y^{(i_{k})}_{k}-x^{(i_{k})}_{k}\big{)}\cdot\prod_{\ell\neq
i_{k},j_{k}}q_{b_{k}-a_{k}}(y^{(\ell)}_{k}-x^{(\ell)}_{k})\\\
&\qquad\qquad\quad\times\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x^{(i_{m})}_{m}-y^{(i_{m})}_{\mathsf{p}(m)}\big{)}\cdot\prod_{\ell\neq
i_{m},j_{m}}q_{a_{m}-b_{m-1}}\big{(}x^{(\ell)}_{m}-y^{(\ell)}_{m-1}\big{)}\,.\end{split}$
(3.48)

The first step in getting a lower bound for $H^{\mathsf{(rew)}}_{r,N,R,M}$ is
to get rid of the dashed lines, see Figure 4. We follow the steps we took in
the proof of Proposition 3.8 for the upper bound. In particular, we start the
summation of (3.48) beginning from the end. Using that for $n\in\mathbb{N}$
and $R\in(0,\infty)$

$\sum_{z\in\mathbb{Z}^{2}:\,|z|\leq
R\sqrt{n}}q_{n}(z)=1-\sum_{z\in\mathbb{Z}^{2}:\,|z|>R\sqrt{n}}q_{n}(z)\geq
1-e^{-cR^{2}}\,,$ (3.49)

by (3.13), we get that

$\sum_{\begin{subarray}{c}y_{r}^{(\ell)}\in\mathbb{Z}^{2}:|y_{r}^{(\ell)}-x^{(\ell)}_{r}|\leq
R\sqrt{b_{r}-a_{r}},\,\\\ 1\leq\ell\leq h,\,\ell\neq
i_{r},j_{r}\end{subarray}}\prod_{\ell\neq
i_{r},j_{r}}q_{b_{r}-a_{r}}(y^{(\ell)}_{r}-x^{(\ell)}_{r})\geq(1-e^{-cR^{2}})^{h}\,.$

We leave the sum

$\sum_{\begin{subarray}{c}b_{r}\in[a_{r},N],\\\
y^{(i_{r})}_{r}\in\mathbb{Z}^{2}:\,|y^{(i_{r})}_{r}-x_{r}^{(i_{r})}|\leq
R\sqrt{b_{r}-a_{r}}\end{subarray}}U_{N}\big{(}b_{r}-a_{r},y_{r}^{(i_{r})}-x_{r}^{(i_{r})}\big{)}$

as is and move on to the time interval $[b_{r-1},a_{r}]$. We use (3.49) to
deduce that

$\sum_{\begin{subarray}{c}x_{r}^{(\ell)}\in\mathbb{Z}^{2}:|x_{r}^{(\ell)}-y^{(\ell)}_{r-1}|\leq
R\sqrt{a_{r}-b_{r-1}},\,\\\ 1\leq\ell\leq h,\,\ell\neq
i_{r},j_{r}\end{subarray}}\,\prod_{\ell\neq
i_{r},j_{r}}\,q_{a_{r}-b_{r-1}}(x^{(\ell)}_{r}-y^{(\ell)}_{r-1})\geq(1-e^{-cR^{2}})^{h}\,.$

Again, we leave the sum
$\sum_{a_{r}\in\,(b_{r-1},b_{r}],\,x_{r}^{(i_{r})}\in\mathbb{Z}^{2}}\,q_{a_{r}-b_{\mathsf{p}(r)}}^{2}\big{(}x^{(i_{r})}_{r}-y^{(i_{r})}_{\mathsf{p}(r)}\big{)}$
intact. We can continue this procedure since due to rewiring all the spatial
variables $y^{(\ell)}_{r-1}$, $\ell\neq i_{r-1},j_{r-1}$ are free, i.e. there
are no outgoing laces starting off $y^{(\ell)}_{r-1}$, $\ell\neq
i_{r-1},j_{r-1}$ at time $b_{r-1}$, and there are no diffusivity constraints
linking $x^{(i_{r})}_{r}=x_{r}^{(j_{r})}$ with
$y^{(i_{r})}_{r-1},y^{(j_{r})}_{r-1}$ by definition of
$\mathsf{C}^{\mathsf{(rew)}}_{r,N,R,M}(\vec{I})$. Iterating this procedure we
obtain that

$\begin{split}&H^{\mathsf{(rew)}}_{r,N,R,M}\\\
&\geq(1-e^{-cR^{2}})^{2Kh}\,\sum_{\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
a_{i+1}-b_{i}>M(b_{i}-b_{i-1}),\,1\leq i\leq r-1,\\\
0:=x_{1},y_{1},\dots,x_{r},y_{r}\in\mathbb{Z}^{2}\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y_{k}-x_{k}\big{)}\,\\\
&\qquad\times\,\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x_{m}-y_{\mathsf{p}(m)}\big{)}\cdot\mathds{1}_{\big{\\{}|y_{k}-x_{k}|\leq
R\sqrt{b_{k}-a_{k}},\,|x_{m}-y_{\mathsf{p}(m)}|\leq
R\,C_{K,M}\sqrt{a_{m}-b_{\mathsf{p}(m)}}\big{\\}}},\end{split}$

with $C_{K,M}=\sqrt{1-\frac{1}{M}}\bigg{(}1-\sqrt{\frac{2K-1}{M-1}}\bigg{)}$.
∎

###### Proposition 3.11.

We have that

$\lim_{K\to\infty}\lim_{M\to\infty}\lim_{R\to\infty}\lim_{N\to\infty}\sum_{r=0}^{K}H^{\mathsf{(rew)}}_{r,N,R,M}=\prod_{1\leq
i<j\leq h}\frac{1}{1-\beta_{i,j}}\,.$

  * Proof.

We are going to prove this Proposition via means of the lower and upper bounds
established in Propositions 3.8 and 3.10. By Proposition 3.8 we have that

$\begin{split}H^{\mathsf{(rew)}}_{r,N,R,M}\leq\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N\,,\\\
0:=x_{1},y_{1},\dots,x_{r},y_{r}\in\mathbb{Z}^{2}\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y_{k}-x_{k}\big{)}\,\\\
\times\,\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x_{m}-y_{\mathsf{p}(m)}\big{)}\,.\end{split}$
(3.50)

Summing the spatial points on the right hand side of (3.50) we obtain that

$\displaystyle H^{\mathsf{(rew)}}_{r,N,R,M}$ (3.51)
$\displaystyle\leq\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq
N}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k}\big{)}\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}q_{2(a_{m}-b_{\mathsf{p}(m)})}(0)\,.$

Note that the right hand sides of (3.50) and (3.51) describe a system of
$\binom{h}{2}$ pairs of random walks which collide only between themselves.
The times $a_{i}\leq b_{i}$ mark when a pair of random walks starts and
terminates colliding (temporarily) before the next pair starts colliding. The
order of these collision events is encoded in
$\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}$.
Using (2.1) and (2.7) one can deduce that

$\ \sum_{r\geq 0}H^{\mathsf{(rew)}}_{r,N,R,M}\leq\prod_{1\leq i<j\leq
h}\mathrm{E}\Big{[}e^{\frac{\pi\beta_{i,j}}{\log
N}\,\mathsf{L}_{N}^{(i,j)}}\Big{]}=\big{(}1+o_{N}(1)\big{)}\prod_{1\leq
i<j\leq h}\frac{1}{1-\beta_{i,j}}\,.$ (3.52)

Next, by Proposition 3.10 we have that

$\begin{split}&H^{\mathsf{(rew)}}_{r,N,R,M}\\\
&\geq(1-e^{-cR^{2}})^{2Kh}\,\sum_{\vec{I}=\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
a_{i+1}-b_{i}>M(b_{i}-b_{i-1}),\,1\leq i\leq r-1,\\\
0:=x_{1},y_{1},\dots,x_{r},y_{r}\in\mathbb{Z}^{2}\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k},y_{k}-x_{k}\big{)}\,\\\
&\quad\times\,\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q^{2}_{a_{m}-b_{\mathsf{p}(m)}}\big{(}x_{m}-y_{\mathsf{p}(m)}\big{)}\cdot\mathds{1}_{\big{\\{}|y_{k}-x_{k}|\leq
R\sqrt{b_{k}-a_{k}},\,|x_{m}-y_{\mathsf{p}(m)}|\leq
R\,C_{K,M}\sqrt{a_{m}-b_{\mathsf{p}(m)}}\big{\\}}},\end{split}$ (3.53)

Lifting the diffusivity conditions imposed on the right-hand side of (3.53)
can be done using arguments already present in Lemma 3.4. More specifically,
we use that for $0\leq m\leq N$, $w\in\mathbb{Z}^{2}$ and $1\leq i<j\leq h$,

$\begin{split}\sum_{\begin{subarray}{c}n\in[m,N],\,\\\
z\in\mathbb{Z}^{2}:\,|z-w|\leq
R\sqrt{n-m}\end{subarray}}U^{\beta_{i,j}}_{N}\big{(}n-m,z-w\big{)}&=\sum_{n\in[m,N]}U^{\beta_{i,j}}_{N}\big{(}n-m\big{)}-\sum_{\begin{subarray}{c}n\in[m,N],\,\\\
z\in\mathbb{Z}^{2}:\,|z-w|>R\sqrt{n-m}\end{subarray}}U^{\beta_{i,j}}_{N}\big{(}n-m,z-w\big{)}\\\
&\geq\sum_{n\in[m,N]}U^{\beta_{i,j}}_{N}\big{(}n-m\big{)}-e^{-\kappa
R}\sum_{n\in[m,N]}U^{\beta_{i,j}}_{N}\big{(}n-m\big{)}\\\
&\geq\big{(}1-e^{-\kappa
R}\big{)}\,\sum_{n\in[m,N]}U^{\beta_{i,j}}_{N}\big{(}n-m\big{)}\,,\end{split}$

where in the first inequality we used (3.22) from Lemma 3.4 with a suitable
constant $\kappa(\bar{\beta})\in(0,\infty)$. Similarly we have that

$\begin{split}\sum_{\begin{subarray}{c}n\in[m,N],\,\\\
z\in\mathbb{Z}^{2}:|z-w|\leq
R\,C_{K,M}\,\sqrt{n-m}\end{subarray}}q^{2}_{n-m}(z-w)=&\sum_{n\in[m,N]}q_{2(n-m)}(0)-\sum_{\begin{subarray}{c}n\in[m,N],\,\\\
z\in\mathbb{Z}^{2}:|z-w|>R\,C_{K,M}\,\sqrt{n-m}\end{subarray}}q^{2}_{n-m}(z-w)\\\
\geq&\big{(}1-e^{-\kappa\,R^{2}\,C^{2}_{K,M}}\big{)}\sum_{n\in[m,N]}q_{2(n-m)}(0)\end{split}$

by tuning the constant $\kappa$ if needed. Therefore, we finally obtain that

$\begin{split}&H^{\mathsf{(rew)}}_{r,N,R,M}\geq(1-e^{-cR^{2}})^{2Kh}\,(1-e^{-\kappa\,R})^{K}\,(1-e^{-\kappa\,R^{2}\,C^{2}_{K,M}})^{K}\times\\\
&\,\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\
a_{i+1}-b_{i}>M(b_{i}-b_{i-1}),\,1\leq i\leq
r-1\,.\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k}\big{)}\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q_{2(a_{m}-b_{\mathsf{p}(m)})}(0)\,.\end{split}$ (3.54)

The last restriction we need to lift is the restriction
$a_{i+1}-b_{i}>M(b_{i}-b_{i-1}),\,1\leq i\leq r-1$. This can be done via the
arguments used in Proposition 3.5, so we do not repeat it here, but only note
that there exists a constant
$\widetilde{C}_{K}=\widetilde{C}_{K}(\bar{\beta},h)\in(0,\infty)$ such that
for all $0\leq r\leq K$, the corresponding sum to the right hand side of
(3.54), but with its temporal range of summation be such that there exists
$1\leq i\leq r-1:\,a_{i+1}-b_{i}\leq M(b_{i}-b_{i-1})$, satisfies the bound

$\begin{split}&\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{\begin{subarray}{c}0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq N,\\\ \exists\,1\leq i\leq
r-1:\,a_{i+1}-b_{i}\leq
M(b_{i}-b_{i-1})\end{subarray}}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k}\big{)}\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q_{2(a_{m}-b_{\mathsf{p}(m)})}(0)\\\
&\leq\widetilde{C}_{K}\cdot\varepsilon_{N,M}\,,\end{split}$

where $\varepsilon_{N,M}$ is such that $\lim_{N\to\infty}\varepsilon_{N,M}=0$
for any fixed $M\in(0,\infty)$. Therefore, the resulting resulting lower bound
on $H^{\mathsf{(rew)}}_{r,N,R,M}$ will be then

$\begin{split}H^{\mathsf{(rew)}}_{r,N,R,M}&\geq(1-e^{-cR^{2}})^{2Kh}\,(1-e^{-\kappa\,R})^{K}\,(1-e^{-\kappa\,R^{2}\,C^{2}_{K,M}})^{K}\\\
&\times\Bigg{(}\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\,\sum_{0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq
N}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k}\big{)}\,\\\
&\,\qquad\qquad\times\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q_{2(a_{m}-b_{\mathsf{p}(m)})}(0)-\widetilde{C}_{K}\cdot\varepsilon_{N,M}\Bigg{)}\,.\end{split}$
(3.55)

Note that

$\begin{split}\prod_{1\leq i<j\leq
h}\mathrm{E}\Big{[}e^{\frac{\pi\beta_{i,j}}{\log
N}\,\mathsf{L}^{(i,j)}_{N}}\Big{]}&=\sum_{r=0}^{K}\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\sum_{0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq
N}\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k}\big{)}\\\
&\qquad\times\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}\cdot
q_{2(a_{m}-b_{\mathsf{p}(m)})}(0)+A^{(1)}_{N}+A_{N,K}^{(2)}\,,\end{split}$
(3.56)

where $A_{N}^{(1)}$ denotes the part of the chaos expansion of $\prod_{1\leq
i<j\leq h}\mathrm{E}\Big{[}e^{\frac{\pi\beta_{i,j}}{\log
N}\,\mathsf{L}^{(i,j)}_{N}}\Big{]}$ which contains multiple collisions for at
least some time $1\leq n\leq N$ and $A^{(2)}_{N,K}$ denotes the corresponding
sum on the right hand side of (3.56) but from $r=K+1$ to $\infty$, that is

$\begin{split}A^{(2)}_{N,K}=\sum_{r>K}\sum_{\big{(}\\{i_{1},j_{1}\\},\dots,\\{i_{r},j_{r}\\}\big{)}\in\,\mathcal{I}^{(2)}}\,\,\sum_{0:=a_{1}\leq
b_{1}<a_{2}\leq\dots<a_{r}\leq b_{r}\leq
N}&\prod_{k=1}^{r}U^{\beta_{i_{k},j_{k}}}_{N}\big{(}b_{k}-a_{k}\big{)}\\\
\times&\prod_{m=2}^{r}\sigma^{i_{m},j_{m}}_{N}q_{2(a_{m}-b_{\mathsf{p}(m)})}(0)\,.\end{split}$

Next, we will give bounds for $A^{(1)}_{N}$ and $A_{N,K}^{(2)}$. Beginning
with $A_{N,K}^{(2)}$, let
$\varrho_{K}:=\Big{\lfloor}\frac{K}{2\binom{h}{2}}\Big{\rfloor}\,$. Since we
are summing over $r>K$, there has to be a pair $1\leq i<j\leq h$ which has
recorded more than $\varrho_{K}$ collisions. We recall from (2.8) that
$U^{\beta}_{N}(\cdot)$ admits the renewal representation

$U_{N}^{\beta}(n)=\sum_{k\geq
0}(\sigma_{N}(\beta)\,R_{N})^{k}\,\mathrm{P}\big{(}\tau^{\scriptscriptstyle(N)}_{k}=n\big{)}\,.$

There are $\binom{h}{2}$ choices for the pair with more than $\varrho_{K}$
collisions. We can also use the bound (2.1) to bound the contribution of the
rest $\binom{h}{2}-1$ pairs in $A^{(2)}_{N,K}$. Therefore, we can then write

$A^{(2)}_{N,K}\leq\frac{\binom{h}{2}}{(1-\bar{\beta}^{\prime})^{\binom{h}{2}}}\sum_{k>\varrho_{K}}(\sigma_{N}(\bar{\beta})R_{N})^{k}\,\mathrm{P}(\tau^{\scriptscriptstyle(N)}_{k}\leq
N)\leq\frac{\binom{h}{2}}{(1-\bar{\beta}^{\prime})^{\binom{h}{2}}}\sum_{k>\varrho_{K}}(\bar{\beta}^{\prime})^{k}\xrightarrow{K\to\infty}0\,,$

uniformly in $N$, where $\bar{\beta}^{\prime}\in(\bar{\beta},1)$.

Similarly, for $A_{N}^{(1)}$, we can choose two pairs which collide at the
same time in
$\binom{h}{2}\cdot\Big{(}\binom{h}{2}-1\Big{)}\leq\binom{h}{2}^{2}$ ways and
we can use bound (2.1) to bound the contribution in $A^{(1)}_{N}$ of the rest
$\binom{h}{2}-2$ pairs. Therefore, we obtain that

$\begin{split}A_{N}^{(1)}\leq\frac{\binom{h}{2}^{2}}{(1-\bar{\beta}^{\prime})^{\binom{h}{2}}}\sum_{n\geq
0,\,x,y\,\in\mathbb{Z}^{2}}U^{\bar{\beta}}_{N}(n,x)\,U^{\bar{\beta}}_{N}(n,y)\leq\frac{\binom{h}{2}^{2}}{(1-\bar{\beta}^{\prime})^{\binom{h}{2}}}\sum_{n\geq
0}\big{(}U^{\bar{\beta}}_{N}\big{)}^{2}(n)\,.\end{split}$

By Proposition 1.5 of [CSZ19a] we get the estimate

$\mathrm{P}(\tau_{k}^{\scriptscriptstyle(N)}=n)\leq\frac{C\,k\,q_{2n}(0)}{R_{N}}\leq\frac{C^{\prime}\,k\,}{n(\log
N)}\,,$

where the second inequality follows by the local limit theorem. Therefore, by
(2.8) and the aforementioned estimate we get that

$\big{(}U^{\bar{\beta}}_{N}\big{)}^{2}(n)=\sum_{k,\ell\,\geq