[2]footnote

# Moments of the 2d directed polymer in the subcritical regime and a
generalisation of the Erdös-Taylor theorem

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

We compute the limit of the moments of the partition function
$Z_{N}^{\beta_{N}}$ of the directed polymer in dimension $d=2$ in the
subcritical regime, i.e. when the inverse temperature is scaled as
$\beta_{N}\sim\hat{\beta}\sqrt{\tfrac{\pi}{\log N}}$ for
$\hat{\beta}\in(0,1)$. In particular, we establish that for every
$h\in\mathbb{R}$,
$\lim_{N\to\infty}{\mathbb{E}}\big{[}\big{(}Z_{N}^{\beta_{N}}\big{)}^{h}\big{]}=\big{(}\frac{1}{1-\hat{\beta}^{2}}\big{)}^{\frac{h(h-1)}{2}}.$
We also identify the limit of the moments of the averaged field
$\tfrac{\sqrt{\log
N}}{N}\sum_{x\in\mathbb{Z}^{2}}\varphi(\tfrac{x}{\sqrt{N}})\big{(}Z_{N}^{\beta_{N}}(x)-1\big{)}$,
for $\varphi\in C_{c}(\mathbb{R}^{2})$, as those of a gaussian free field.

As a byproduct, we identify the limiting probability distribution of the total
pairwise collisions between $h$ independent, two dimensional random walks
starting at the origin. In particular, we derive that

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

where ${\mathsf{L}}^{(i,j)}_{N}$ denotes the collision local time by time $N$
between copies $i,j$ and $\Gamma$ denotes the Gamma distribution. This
generalises a classical result of Erdös-Taylor [ET60].

###### Key words and phrases:

two dimensional subcritical directed polymer, planar random walk collisions,
Erdös-Taylor theorem, Schrödinger operators with point interactions

###### 2010 Mathematics Subject Classification:

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

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Related literature.
    2. 1.2 Outline of the paper
  2. 2 Auxiliary tools
    1. 2.1 Partition functions and chaos expansion
    2. 2.2 Renewal representation
    3. 2.3 Some useful results
  3. 3 Expansion of moments and integral inequalities
    1. 3.1 Chaos expansion of moments
    2. 3.2 Integral inequalities for the operators $\widehat{\mathsf{Q}}^{I;J}_{N,0}$ and $\widehat{\mathsf{U}}^{I}_{N,0}$
  4. 4 Proofs of Theorems 1.1, 1.4, 1.5 and 1.3.
  5. A Some technical estimates

##  1\. Introduction and main results

Let $S=(S_{n})_{n\geq 0}$ be a two dimensional simple random walk and
$\big{(}\omega(n,z)\big{)}_{(n,z)\in\mathbb{N}\times\mathbb{Z}^{2}}$ a space-
time field of i.i.d. random variables with ${\mathbb{E}}[\omega]=0$,
${\mathbb{E}}[\omega^{2}]=1$ and
$\lambda(\beta):=\log{\mathbb{E}}[e^{\beta\omega}]<\infty$ for all $\beta>0$.
We use the notation $\mathrm{P}_{a,x}$ and $\mathrm{E}_{a,x}$ to denote the
probability and the expectation with respect to the distribution of the random
walk when the walk starts from $x\in\mathbb{Z}^{2}$ at time $a\in\mathbb{N}$.
If either $a$ or $x$ are zero, we will omit them from the subscripts. We
consider the (point-to-point) partition function

$\displaystyle
Z_{N}^{\beta}(x,y)=\mathrm{E}_{x}\bigg{[}e^{\sum_{n=1}^{N-1}\beta\omega(n,S_{n})-\lambda(\beta)}\,\mathbb{1}_{\\{S_{N}=y\\}}\bigg{]}\,$
(1.1)

of the directed polymer, i.e. random walk, in the random environment $\omega$,
at inverse temperature $\beta>0$. We also denote the point-to-line partition
function

$\displaystyle Z_{N}^{\beta}(x):=\sum_{y\in\mathbb{Z}^{2}}Z_{N}^{\beta}(x,y),$
(1.2)

and simply write $Z_{N}^{\beta}$ if $x=0$.

It was observed in [CSZ17b] that, while for any fixed $\beta>0$,
$Z_{N}^{\beta}\xrightarrow[N\to\infty]{}0,\,\,{\mathbb{P}}-\text{a.s.}$
[CSY03, C17], an intermediate disorder regime with a phase transition arises
when one scales the inverse temperature like

$\displaystyle\beta_{N}\sim\hat{\beta}\sqrt{\frac{\pi}{\log N}}\qquad\text{
with }\hat{\beta}>0\,.$ (1.3)

In particular, it was shown in [CSZ17b], that for
$\beta_{N}\sim\hat{\beta}\sqrt{\frac{\pi}{\log N}}\text{ with
}\hat{\beta}\in(0,1)$,

$\displaystyle
Z_{N}^{\beta_{N}}\xrightarrow[N\to\infty]{(d)}\exp(\varrho_{\hat{\beta}}\mathsf{X}-\tfrac{1}{2}\varrho^{2}_{\hat{\beta}})\,,$

where $\mathsf{X}\sim{\mathcal{N}}(0,1)$ and
$\varrho^{2}_{\hat{\beta}}=\log\big{(}\frac{1}{1-\hat{\beta}^{2}}\big{)}$,
while for $\hat{\beta}\geq 1$, $Z_{N}^{\beta_{N}}$ converges in distribution
to $0$.

One can guess the emergence of such intermediate scaling as follows. Using
gaussian environment for simplicity one has that

$\displaystyle{\mathbb{E}}\Big{[}\big{(}Z^{\beta}_{N}\big{)}^{2}\Big{]}=\mathrm{E}^{\otimes
2}\Big{[}e^{\beta^{2}\,\mathsf{L}^{(1,2)}_{N}}\Big{]}=\mathrm{E}\Big{[}e^{\beta^{2}\,\mathsf{L}_{N}}\Big{]}\,,$
(1.4)

where $\mathrm{E}^{\otimes 2}$ denotes the law of two independent, $2d$ simple
random walks starting both at the origin,
$\mathsf{L}^{(1,2)}_{N}:=\sum_{n=1}^{N-1}\mathbb{1}_{\\{S^{1}_{n}=S^{2}_{n}\\}}$
denotes their collision local time up to time $N-1$ and
$\mathsf{L}_{N}:=\sum_{n=1}^{N-1}\mathbb{1}_{\\{S_{2n}=0\\}}$ denotes the
number of returns to zero, up to time $N-1$, of a single $2d$ simple random
walk starting at $0$. The second equality in (1.4) follows since
$\mathsf{L}^{1,2}_{N}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathsf{L}_{N}$.
A classical result of Erdös-Taylor [ET60] states that

$\displaystyle\frac{\pi}{\log
N}\,\mathsf{L}_{N}\xrightarrow[N\to\infty]{(d)}\text{Exp}(1)\,,$ (1.5)

where $\text{Exp}(1)$ denotes an exponential random variable of parameter $1$.
Thus, it is not hard to see that under (1.3), one has $\sup_{N\geq
1}{\mathbb{E}}\big{[}(Z^{\beta_{N}}_{N})^{2}\big{]}<\infty$ if and only if
$\hat{\beta}<1$.

It is an interesting and non-trivial question whether all moments of the
point-to-plane partition function $Z_{N}^{\beta_{N}}$ remain uniformly bounded
as $N\to\infty$ in the same regime of $\hat{\beta}$ where the second moment
remains uniformly bounded. Information on moments higher than two in the
subcritical regime has already appeared necessary in a number of situations,
in particular in proving tightness and regularity properties of the
approximations to the solutions of the $2d$-KPZ [CD20] or Edwards-Wilkinson
universality for the $2d$-KPZ [CSZ20, G20] and [T22] for the nonlinear SHE.
The lack of control on higher moments was resulting into restrictions to
strict subsets of the subcritical regime in [CD20, G20], while this was
circumvented in [CSZ20] by employing hypercontractivity to show, for any
$\hat{\beta}<1$, the uniform boundedness of moments up to certain order
$h(\hat{\beta})>2$ with $\lim_{\hat{\beta}\uparrow 1}h(\hat{\beta})=2$. The
case of the nonlinear SHE in the entire subcritical regime is still open [T22]
and the techniques here could be useful there. Moreover, having bounds on the
moments can be useful in obtaining finer results relating to the rate of
convergences [DG22+].

The first result of this work is to show that all moments of the point-to-
plane partition function $Z_{N}^{\beta_{N}}$ are uniformly bounded in the
whole subcritical regime $\hat{\beta}<1$ while, obviously, no moment higher
than one exists in the limit at $\hat{\beta}\geq 1$. Note that this is in
contrast to what happens in the weak disorder regime of $d\geq 3$ polymers,
where the moments are gradually reduced to just existence of the first moment
as the critical point is approached [BS10, J22].

Combining this with the distributional convergence (1.3) we can actually
compute the limit of all moments. In particular, our first theorem is stated
as:

###### Theorem 1.1.

Consider the point-to-line partition function $Z_{N}^{\beta_{N}}$ defined in
(1.2) with an intermediate disorder scaling $\beta_{N}$ as in (2.1), which is
asymptotically equivalent to (1.3). Then, for every $\hat{\beta}\in(0,1)$ and
$h\geq 0$, it holds that

$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\Big{[}\big{(}Z_{N}^{\beta_{N}}\big{)}^{h}\Big{]}=\bigg{(}\frac{1}{1-\hat{\beta}^{2}}\bigg{)}^{\frac{h(h-1)}{2}}=\bigg{(}\lim_{N\to\infty}{\mathbb{E}}\Big{[}\big{(}Z_{N}^{\beta_{N}}\big{)}^{2}\Big{]}\bigg{)}^{\frac{h(h-1)}{2}}\,.$
(1.6)

Furthermore, (1.6) is valid also for all $h<0$ if we assume that the law of
$\omega$ satisfies the following concentration property: There exists
$\gamma>1$ and constants $c_{1},c_{2}\in(0,\infty)$ such that for all
$n\in\mathbb{N}$, $(\omega_{1},\dots,\omega_{n})$ i.i.d. and all convex,
$1$-Lipschitz functions $f:\mathbb{R}^{n}\to\mathbb{R}$,

$\displaystyle{\mathbb{P}}\Big{(}\big{|}f(\omega_{1},\dots,\omega_{n})-M_{f}\big{|}\geq
t\Big{)}\leq c_{1}\,\exp\Big{(}-\frac{t^{\gamma}}{c_{2}}\Big{)}\,,$ (1.7)

where $M_{f}$ is a median of $f$.

###### Remark 1.2.

We note that (1.7) is satisfied if $\omega$ is bounded or has a density of the
form $\exp(-V+U)$ for $V,U:\mathbb{R}\to\mathbb{R}$, where $V$ is strictly
convex and $U$ is bounded, see [Led01].

Furthermore, we compute the asymptotics of the moments of the logarithmically
scaled and averaged field

$\displaystyle\frac{\sqrt{\log
N}}{N}\sum_{x\in\mathbb{Z}^{2}}\varphi(\tfrac{x}{\sqrt{N}})\big{(}Z_{N}^{\beta_{N}}(x)-1\big{)},$

for $\varphi\in C_{c}(\mathbb{R}^{2})$. In particular, we establish that

###### Theorem 1.3.

Let $\varphi\in C_{c}(\mathbb{R}^{2})$ and consider the centred and averaged
field with respect to $\varphi$, that is

$\displaystyle\widebar{Z}^{\beta_{N}}_{N}(\varphi,1):=\frac{1}{N}\sum_{x\in\mathbb{Z}^{2}}\varphi(\tfrac{x}{\sqrt{N}})\,\big{(}Z^{\beta_{N}}_{N}(x)-1\big{)}\,\,.$

Then, for every $h\in\mathbb{N}$ with $h\geq 2$ and $\hat{\beta}\in(0,1)$,

$\displaystyle\lim_{N\to\infty}(\log
N)^{\frac{h}{2}}\,{\mathbb{E}}\Big{[}\widebar{Z}^{\beta_{N}}_{N}(\varphi)^{h}\Big{]}=\begin{cases}\varrho_{\varphi}(\hat{\beta})^{h}\cdot(h-1)!!&,\text{
if $h$ is even}\\\ 0&,\text{ if $h$ is odd}\,,\end{cases}$

where $\varrho_{\varphi}(\hat{\beta})$ is defined as

$\displaystyle\varrho^{2}_{\varphi}(\hat{\beta}):=\frac{\pi\,\hat{\beta}^{2}}{1-\hat{\beta}^{2}}\int_{0}^{1}\mathrm{d}t\int_{(\mathbb{R}^{2})^{2}}\mathrm{d}x\,\mathrm{d}y\,\varphi(x)g_{t}(x-y)\varphi(y),$

with $g_{t}(x):=\frac{1}{2\pi t}\,e^{-|x|^{2}/2t}$ the two-dimensional heat
kernel.

Theorem 1.1 in combination with an analogous to (1.4) computation for the $h$
moment will, almost immediately, lead us to a generalisation of the Erdös-
Taylor theorem (see [ET60] and [GS09] for a quenched path generalisation). The
generalisation amounts to the quantity of total pairwise collision times of
$h$ (instead of just two as in [ET60, GS09]) independent, two-dimensional
simple planar random walks. More specifically, let $\Gamma(a,1)$ denote the
Gamma distribution, which is the law with density function
$\tfrac{1}{\Gamma(a)}x^{a-1}e^{-x}\,\mathbb{1}_{\\{x>0\\}}$, where in the last
expression $\Gamma(a)$ is the gamma function. Then,

###### Theorem 1.4.

Consider $h\in\mathbb{N}$ such that $h\geq 2$ and for $i=1,\dots,h$ let
$S^{(i)}=\big{(}S^{(i)}_{n}\big{)}_{n\geq 0}$ be independent simple random
walks in $\mathbb{Z}^{2}$ starting all from the origin at time zero. Moreover,
for $1\leq i<j\leq h$ let

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

denote the collision local time of $S^{(i)}$ and $S^{(j)}$ until time $N$.
Then

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

More precisely, if $Y_{N}:=\frac{\pi}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}^{(i,j)}_{N}$, $Y$ is a random variable with law
$\Gamma\big{(}\tfrac{h(h-1)}{2},1\big{)}$ and $M_{Y_{N}}(t)$, $M_{Y}(t)$
denote the associated moment generating functions, respectively, we have that

$\displaystyle M_{Y_{N}}(t)\xrightarrow[N\to\infty]{}M_{Y}(t)\,,$

for all $t\in(0,1):=I$, which is the maximum interval $I\subset(0,\infty)$
where $M_{Y}(t)<\infty,\,t\in I$.

The main step towards the above theorems is to establish that, in the
subcritical regime, the moments of the two-dimensional point-to-line partition
function $Z_{N}^{\beta_{N}}$ are uniformly bounded. To state the corresponding
theorem, let us briefly introduce the averaged partition functions. For test
functions $\varphi,\psi:\mathbb{R}^{2}\to\mathbb{R}$ such that $\varphi$ has
compact support and $\psi$ is bounded, we define the averaged partition
function to be

$\displaystyle
Z^{\beta_{N}}_{N}(\varphi,\psi):=\frac{1}{N}\sum_{x,y}\varphi(\tfrac{x}{\sqrt{N}})\,Z^{\beta_{N}}_{N}(x,y)\,\psi(\tfrac{y}{\sqrt{N}})\,.$
(1.8)

Moreover, introduce its centred version as
$\bar{Z}^{\beta_{N}}_{N}(\varphi,\psi):=Z^{\beta_{N}}_{N}(\varphi,\psi)-{\mathbb{E}}\big{[}Z^{\beta_{N}}_{N}(\varphi,\psi)\big{]}$
and, similarly, the centred version of the point-to-line partition function
(1.2) as
$\bar{Z}^{\beta_{N}}_{N}:=Z^{\beta_{N}}_{N}-{\mathbb{E}}[Z^{\beta_{N}}_{N}]$.

The key estimate of this paper is the following:

###### Theorem 1.5.

Let $\varphi,\psi:\mathbb{R}^{2}\to\mathbb{R}$ be such that $\varphi$ has
compact support and $\psi$ is bounded and consider the centred, averaged field
$\widebar{Z}_{N}^{\beta_{N}}(\varphi,\psi)$ with respect to $\varphi,\psi$, as
in (1.8). Let also $w:\mathbb{R}^{2}\to\mathbb{R}$ be a weight function such
that $\log w$ is Lipschitz continuous. Then, for every $h\in\mathbb{N}$ with
$h\geq 3$, $\hat{\beta}\in(0,1)$, there exist
$\mathsf{a}_{*}=\mathsf{a}_{*}(h,\hat{\beta},w)\in(0,1)$ and
$C=C(h,\hat{\beta},w)\in(0,\infty)$ such that for any $p,q\in(1,\infty)$ that
satisfy $\frac{1}{p}+\frac{1}{q}=1$ and $p\,q\leq\mathsf{a}_{*}\log N$, the
following inequality holds:

$\displaystyle\bigg{|}{\mathbb{E}}\Big{[}\widebar{Z}_{N}^{\beta_{N}}(\varphi,\psi)^{h}\Big{]}\bigg{|}\leq\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{\frac{h}{2}}\cdot\frac{1}{N^{h}}\cdot\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert\psi_{N}\right\rVert^{h}_{\infty}\left\lVert
w_{N}\right\rVert_{\ell^{q}}^{h}\,,$ (1.9)

where for $x\in\mathbb{Z}^{2}$ we have $\varphi_{N}(x):=\varphi(x/\sqrt{N})$,
$\psi_{N}(x):=\psi(x/\sqrt{N})$ and $w_{N}(x):=w(x/\sqrt{N})$. Moreover, for
$\widebar{Z}_{N}^{\beta_{N}}$ being the centred, point-to-line partition
function, it holds that

$\displaystyle\sup_{N\in\mathbb{N}}\bigg{|}{\mathbb{E}}\Big{[}\big{(}\widebar{Z}_{N}^{\beta_{N}}\big{)}^{h}\Big{]}\bigg{|}<\infty.$
(1.10)

Let us comment on estimate (1.9) and the significance of the precise
dependence of the constant of the inequality (1.9) in terms of $p,q$ and $N$
as $(pq/\log N)^{h/2}N^{-h}$ (the generic constant $C$ that appears in the
right-hand side of (1.9) does not depend on either $p$, $q$ or $N$): In the
case that $\varphi,\psi,w$ are nice, smooth functions, i.e. the partition
function field is averaged out in an essentially uniform way, then, by Riemann
summation, $N^{-h/p}\|\frac{\varphi_{N}}{w_{N}}\|_{\ell^{p}}^{h}$ will
converge to $\|\varphi\|^{h}_{L^{p}}$ and $N^{-h/p}\|w_{N}\|_{\ell^{q}}^{h}$
will converge to $\|w\|^{h}_{L^{q}}$ and then the right hand side of (1.9)
will capture the decay of correlations of the partition function field as
$(\log N)^{-h/2}$. In contrast to this scenario, if we want to estimate the
moments of a single partition function, i.e. if we only restrict to a single
starting point, say $Z^{\beta}_{N}(0)$, we would need to insert in (1.9) a
delta-like function $\varphi_{N}(x):=N\mathbb{1}_{\\{x=0\\}}$. This choice,
however, leads to a blow up in $N$ in the right hand side of (1.9) of the form
$\big{(}Cpq/(\log N)\big{)}^{h/2}\cdot N^{h/q}$. The idea in order to
neutralise the blow up, is to optimise over the choice of $p,q$, of the
corresponding $\ell^{p}$ and $\ell^{q}$ spaces by choosing $q:=\mathsf{a}\log
N$ (for certain parameter $\mathsf{a}$). This is why the explicit dependence
of the constant in the right-hand side in terms of $p,q$ is crucial and it is
remarkable that this concrete dependence leads to a sharp estimate after a
suitable optimisation. The extraction of the precise dependence on $p,q$ in
inequality (1.9) has been one of our main efforts in this work.

###  1.1. Related literature.

Let us explain how our methods and results fit the various aspects of the
literature as well as outline some prospects. Moments, Functional inequalities
and Schrödinger operators.  Inequality (1.9) controls the moments of the
partition function when starting and ending points are averaged out. A version
of (1.9) but for $2d$ polymer at the critical temperature and without the
important for our purposes dependence on $p,q$ was established in [CSZ21].
Such inequality was used there as an input to prove uniqueness of the scaling
limit of the polymer field at the critical temperature scaling. In the
continuum setting of the $2d$ stochastic heat equation (again at critical
temperature scaling), existence of all moments of the fields averaged by test
functions $\phi,\psi$, which are restricted to only belong to
$L^{2}(\mathbb{R}^{2})$ was proved in [GQT21].

Important input in all the above came from earlier works of Dell’Antonio-
Figari-Teta [DFT94] and Dimock-Rajeev [DR04] on the spectral theory of
Schrödinger operators with point interactions. The objective there was to
define, via renormalisation, self-adjoint extensions of many-body hamiltonians
of the form

$\displaystyle\Delta+\sum_{1\leq i<j\leq h}\delta(x_{i}-x_{j}),\qquad
x_{1},...,x_{h}\in\mathbb{R}^{2},$

with $\delta(\cdot)$ being the delta function on $\mathbb{R}^{2}$ and $\Delta$
the Laplacian on $(\mathbb{R}^{2})^{h}$. This problem has a rich history, as
outlined in [DFT94]. Of particular significance for our purposes is a
functional inequality of Dell’Antonio-Figari-Teta [DFT94], Proposition 3.1
(see also Lemma 5.1 in [GQT21]), which essentially states that the Green’s
function of a $2h-$dimensional Brownian motion
$\mathfrak{G}(\boldsymbol{x},\boldsymbol{y}),\boldsymbol{x}=(x_{1},...,x_{h})\in(\mathbb{R}^{2})^{h},\boldsymbol{y}=(y_{1},...,y_{h})\in(\mathbb{R}^{2})^{h}$,
when restricted to hyperplanes $\\{x_{i}=x_{j}\\}$ and $\\{y_{k}=y_{\ell}\\}$
with $(i,j)\neq(k,\ell)$ is bounded as an operator from
$L^{2}(\mathbb{R}^{2(h-1)})\to L^{2}(\mathbb{R}^{2(h-1)})$. For our purposes,
the relevant operator, in the discrete setting is defined in (3.6) and (3.12)
and the corresponding estimate that establishes the boundedness of the
operator in $\ell^{q}(\mathbb{Z}^{2(h-1)})$, $q>1$, is proved in Proposition
3.3. To prove this, we follow the approach of [CSZ21], working on the real
space rather than the Fourier space as [DFT94] (the latter is only suitable
for $L^{2}$ estimates) paying particular attention in extracting the precise
dependence on the parameters $p,q$ of the $\ell^{p},\ell^{q}$ spaces. As
already mentioned after Theorem 1.5, this is crucial in order to apply the
optimisation trick, which allows us to average over a delta-like function and
thus control the moment of the, more singular, single-starting point polymer
partition.

Collision local times. Theorem 1.4 led us to conjecture and prove in [LZ22]
the following multivariate extension of the Erdös-Taylor theorem:

###### Theorem 1.6 ([LZ22]).

For $h\geq 3$ the joint law of the normalised pairwise collision local times
$\Big{\\{}\frac{\,\pi\mathsf{L}^{(i,j)}_{N}\,}{\log N}\Big{\\}}_{1\leq i<j\leq
h}$ of $h$ independent, two-dimensional simple, symmetric random walks
converges in distribution to a vector of $\tfrac{h(h-1)}{2}$ independent
Exp$(1)$ random variables.

The conjecture emerged from Theorem 1.4 and the fact that a gamma distributed
variable $\Gamma(a,1),a\in\mathbb{N},$ may arise as a sum of $a$ independent
Exp$(1)$ random variables, The methods developed around Theorem 1.5 in the
present paper played an important role towards the proof of Theorem 1.6 as
they paved the way for a number of necessary approximations. Still, in order
to establish the independence of the collision times, one had to look
carefully at the structure of the collisions: There is an intrinsic
logarithmic scaling, already manifested in a sense in (1.5), which introduces
a certain separation of scales. In turn, this separation of scales leads to
the phenomenon (called rewiring in [LZ22]) in which the random walks forget
how they tracked collisions and behave as if they only followed the pairwise
collisions independently. We refer to [LZ22] for further details.

Let us mention that a reader who has followed the literature might have
thought initially that Theorem 1.6 could simply follow from an extension of
the work [CSZ17b] to joint convergence of partition functions
$Z_{N}^{\beta_{N,1}},...,Z_{N}^{\beta_{N,h}}$ at multiple temperatures
$\beta_{N,i}=\beta_{i}\sqrt{\frac{\pi}{\log N}}$ for $i=1,...,h$, together
with the moment estimates obtained here. However, this is not sufficient as
the computation of $\mathbb{E}[Z_{N}^{\beta_{N,1}}\cdots Z_{N}^{\beta_{N,h}}]$
gives rise only to a functional of the form $\mathrm{E}^{\otimes
h}[e^{\frac{\pi}{\log N}\sum_{1\leq i<j\leq
h}\beta_{i}\beta_{j}\mathsf{L}_{N}^{(i,j)}}]$, while one would need to have
$\binom{h}{2}$ independent parameters $\beta_{i,j},1\leq i<j\leq h$ instead of
just $\beta_{i}\beta_{j}$ to identify the joint distribution.

Theorems 1.4 and 1.6 create some interesting connections with phenomena from
planar Brownian motion. In particular, related to windings of planar Brownian
motion [Y91] and more general, so-called log-scaling laws [PY86], [Kn93]. A
paradigm in these studies is the following: Let $B^{1},...,B^{h}$ denote $h$
independent planar Brownian motions starting from distinct points
$z_{1},\dots,z_{h}\in\mathbb{R}^{2}$ and for each pair $1\leq i<j\leq h$, we
consider $\vartheta^{(i,j)}(t)$ to be the total winding angle of
$Z^{i,j}_{s}:=B_{s}^{i}-B_{s}^{j}$ around $0$. Yor’s theorem [Y91] asserts
that $\big{\\{}\frac{2}{\log t}\,\vartheta^{(i,j)}(t)\big{\\}}_{1\leq i<j\leq
h}\xrightarrow[N\to\infty]{(d)}\big{\\{}C^{(i,j)}\big{\\}}_{1\leq i<j\leq h}$,
where $\big{\\{}C^{(i,j)}\big{\\}}_{1\leq i<j\leq h}$ are $\tfrac{h(h-1)}{2}$
independent Cauchy distributed random variables. This result generalises to
multiple Brownian motions the classical Spitzer’s law [S58] and bears
resemblance to Theorem 1.6. The above works rely crucial on the continuum
methods and the power of stochastic calculus. In this regard, it would be
interesting to investigate the scope of the methods we develop here (as well
as in [LZ22]) for wider applications and beyond continuum aspects.

Statistics of log-correlated fields. Let us denote by
$\mathfrak{h}_{N}(x):=\sqrt{\log N}\big{(}\log Z^{\beta_{N}}_{N}\big{(}\lfloor
x\sqrt{N}\rfloor-{\mathbb{E}}\log Z^{\beta_{N}}_{N}\big{(}\lfloor
x\sqrt{N}\rfloor\big{)}$, for $x\in\mathbb{R}^{2}$. It is known [CSZ20, G20]
that, in the subcritical regime, $\mathfrak{h}_{N}(x)$ converges to the
gaussian free field. This, now, raises interesting questions [CZ21] about
asymptotic statistics of extrema of the field $\mathfrak{h}_{N}(x)$ as well as
whether the exponential of $\mathfrak{h}_{N}(x)$, normalised by its mean,
converges to a Gaussian Multiplicative Chaos (GMC). The activity in the field
of log-correlated fields and on questions of this type for various models is
very large, so we will only refer to some reviews [B17, BK22, BP21, DRSV27,
RV14] for further guidance.

Since $\exp\big{(}\sqrt{\log N}\log
Z^{\beta_{N}}_{N}\Big{)}=\big{(}Z^{\beta_{N}}_{N}\big{)}^{\sqrt{\log N}}$,
questions like the above appear naturally related to information on the
asymptotic of moments of the partition function of order $\sqrt{\log N}$.
Progress in this direction, has recently been made in [CZ21] where the authors
showed that there exists a $\hat{\beta}_{0}<1$ such that for
$\hat{\beta}\leq\hat{\beta}_{0}$ and for $h=h_{N}$ such that

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

then $\mathbb{E}\big{[}(Z^{\beta_{N}}_{N})^{h}\big{]}\leq
C\big{(}\frac{1}{1-\hat{\beta}^{2}}\big{)}^{\tfrac{h(h-1)}{2}(1+\varepsilon_{N})}$,
for $C=C(\hat{\beta})$ and $\varepsilon_{N}=\varepsilon_{N}(\hat{\beta})\to 0$
for $N\to\infty$. Tighter estimates might be needed in order to tackle the
above questions on extrema and approximation to GMC, see discussion [CZ21],
Section 4.

Finally, let us remark that in the continuum setting of the stochastic heat
equation

$\displaystyle\partial_{t}u_{\varepsilon}=\frac{1}{2}\Delta
u_{\varepsilon}+\beta_{\varepsilon}\,\xi^{(\varepsilon)}(t,x)u_{\varepsilon},$

with
$\xi^{(\varepsilon)}(t,x):=\tfrac{1}{\varepsilon}\int_{\mathbb{R}^{2}}j(\frac{x-y}{\varepsilon})\xi(t,x)\mathrm{d}x$
the mollified noise and
$\beta_{\varepsilon}=\frac{\hat{\beta}}{\sqrt{\log\varepsilon^{-1}}}$ with
$\hat{\beta}<\sqrt{2\pi}$, corresponding to the subcritical regime for the
stochastic heat equation, the recent work [CSZ22] yields that
${\mathbb{E}}u_{\varepsilon}^{h}\geq\big{(}{\mathbb{E}}u_{\varepsilon}^{2}\big{)}^{\frac{h(h-1)}{2}}$,
for any positive $h$, irrespective the dependence of $h$ in $\varepsilon$ (one
should think of a correspondence between $\varepsilon$ and $N$ as
$N=\varepsilon^{-2}$ ).

###  1.2. Outline of the paper

  * –

In Section 2 we set up the general framework (including a chaos and a renewal
representation) and recall some results that will be useful for proving the
main theorems.

  * –

In Section 3 we present the moment expansion and functional analytic
framework. We also prove the key operator norm estimates in subsection 3.2.

  * –

In Section 4 we present the proofs of Theorems 1.1, 1.4, 1.5 and 1.3.

  * –

Last, Appendix A contains some technical estimates we make use of in Section
3.

##  2\. Auxiliary tools

In this section we develop all the necessary machinery for the proof of the
main results.

###  2.1. Partition functions and chaos expansion

Let us start by denoting the transition probability kernel of the underlying,
two-dimensional, simple random walk $S$ by $q_{n}(x)$ for $n\in\mathbb{N}$ and
$x\in\mathbb{Z}^{2}$, that is $q_{n}(x):=\mathrm{P}(S_{n}=x)$. Recall from
(1.1) the definition of the point-to-plane partition function

$\displaystyle
Z^{\beta_{N}}_{N}(x):=\mathrm{E}_{x}\bigg{[}e^{\,\sum_{n=1}^{N-1}\,\big{(}\beta_{N}\,\omega(n,S_{n})-\lambda(\beta_{N})\,\big{)}}\bigg{]}\,,$

where $\beta_{N}$ is chosen so that

$\displaystyle\sigma^{2}_{N,\beta}:=e^{\lambda(2\beta_{N})-2\lambda(\beta_{N})}-1=\frac{\hat{\beta}^{2}}{R_{N}},$
(2.1)

where

$\displaystyle R_{N}:=\mathrm{E}^{\otimes
2}\Big{[}\sum_{n=1}^{N}\mathbb{1}_{\\{S_{n}^{(1)}=S_{n}^{(2)}\\}}\Big{]}=\sum_{n=1}^{N}\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)^{2}=\sum_{n=1}^{N}q_{2n}(0)\,,$
(2.2)

denotes the expected collisions until time $N$ of two independent, two-
dimensional, simple random walks, starting from the origin. Note that [ET60]

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

where $\alpha:=\gamma+\log 16-\pi\simeq 0.208$ and $\gamma\simeq 0.577$ is the
Euler constant. By Taylor expansion in (2.1), this implies the asymptotic
scaling of $\beta_{N}$ as $\beta_{N}\sim\hat{\beta}\sqrt{\frac{\pi}{\log N}}$
for $N\to\infty$.

We shall also need the definition of the point-to-point partition functions.
In particular, for $a,b\in\mathbb{N}$ with $a<b$ and $x,y\in\mathbb{Z}^{2}$,
we define the point-to-point partition function from the space-time point
$(a,x)$ to $(b,y)$ by

$\displaystyle
Z^{\beta_{N}}_{a,b}(x,y):=\mathrm{E}_{a,x}\bigg{[}e^{\,\sum_{n=a+1}^{b-1}\,\big{(}\beta_{N}\,\omega(n,S_{n})-\lambda(\beta_{N})\,\big{)}}\,\mathbb{1}_{\\{S_{b}=y\\}}\bigg{]}\,,$
(2.3)

Note that with these definitions,

$\displaystyle
Z^{\beta_{N}}_{N}(x)=\sum_{y\in\mathbb{Z}^{2}}Z^{\beta_{N}}_{0,N}(x,y)\,.$

Given $\varphi,\psi:\mathbb{R}^{2}\to\mathbb{R}$ such that $\varphi$ has
compact support and $\psi$ is bounded, we can further define the averaged
partition functions by,

$\displaystyle
Z^{\beta_{N}}_{a,b}(\varphi,y):=\sum_{x\in\mathbb{Z}^{2}}\varphi(\tfrac{x}{\sqrt{N}})\,Z^{\beta_{N}}_{a,b}(x,y)\,,$
$\displaystyle
Z^{\beta_{N}}_{a,b}(x,\psi):=\sum_{y\in\mathbb{Z}^{2}}\,Z^{\beta_{N}}_{a,b}(x,y)\,\psi(\tfrac{y}{\sqrt{N}})$

and

$\displaystyle
Z^{\beta_{N}}_{a,b}(\varphi,\psi):=\frac{1}{N}\sum_{x,y}\varphi(\tfrac{x}{\sqrt{N}})\,Z^{\beta_{N}}_{a,b}(x,y)\,\psi(\tfrac{y}{\sqrt{N}})\,.$
(2.4)

For $(a,x),(b,y)\in\mathbb{N}\times\mathbb{Z}^{2}$ with $a<b$, the mean of
each of the quantities above is computed as

$\begin{split}&{\mathbb{E}}\big{[}Z^{\beta_{N}}_{a,b}(\varphi,y)\big{]}=\,q^{N}_{a,b}(\varphi,y):=\sum_{x\in\mathbb{Z}^{2}}\varphi(\tfrac{x}{\sqrt{N}})\,q_{a,b}(x,y)\,,\\\
&{\mathbb{E}}\big{[}Z^{\beta_{N}}_{a,b}(x,\psi)\big{]}=\,q^{N}_{a,b}(x,\psi):=\sum_{y\in\mathbb{Z}^{2}}\,q_{a,b}(x,y)\,\psi(\tfrac{y}{\sqrt{N}})\,\end{split}$
(2.5)

and

$\displaystyle{\mathbb{E}}\big{[}Z^{\beta_{N}}_{a,b}(\varphi,\psi)\big{]}=q^{N}_{a,b}(\varphi,\psi):=\frac{1}{N}\sum_{x,y\in\mathbb{Z}^{2}}\varphi(\tfrac{x}{\sqrt{N}})\,q_{a,b}(x,y)\psi(\tfrac{y}{\sqrt{N}})\,.$

Next, we derive an expansion for the point-to-point partition function
$Z^{\beta_{N}}_{a,b}(x,y)$ as a multilinear polynomial, which goes by the name
of chaos expansion. This is the starting point of our analysis. Recalling
(2.3) we have

$\displaystyle\qquad
Z^{\beta_{N}}_{a,b}(x,y)=\mathrm{E}_{a,x}\bigg{[}\prod_{a<n<b}\prod_{z\in\mathbb{Z}^{2}}e^{\big{(}\,\beta_{N}\,\omega(n,z)-\lambda(\beta_{N})\,\big{)}\,\mathbb{1}_{\\{S_{n}=z\\}}}\,\mathbb{1}_{\\{S_{b}=y\\}}\bigg{]}$

and by using the fact that for $\lambda\in\mathbb{R}$,
$e^{\lambda\,\mathbb{1}_{\\{S_{n}=z\\}}}=1+(e^{\lambda}-1)\mathbb{1}_{\\{S_{n}=z\\}}$
we obtain

$\displaystyle
Z^{\beta_{N}}_{a,b}(x,y)=\mathrm{E}_{a,x}\bigg{[}\prod_{a<n<b}\prod_{z\in\mathbb{Z}^{2}}\big{(}1+\xi(n,z)\mathbb{1}_{\\{S_{n}=z\\}}\big{)}\,\mathbb{1}_{\\{S_{b}=y\\}}\bigg{]}\,$
(2.6)

where $\xi(n,z):=e^{\beta_{N}\,\omega(n,z)-\lambda(\beta_{N})}-1$ are i.i.d.
random variables with

$\displaystyle{\mathbb{E}}[\xi]=0\,,\qquad{\mathbb{E}}[\xi^{2}]=e^{\lambda(2\beta_{N})-2\lambda(\beta_{N})}-1=:\sigma^{2}_{N,\hat{\beta}}\stackrel{{\scriptstyle
N\to\infty}}{{\sim}}\beta_{N}^{2}\,,\quad{\mathbb{E}}\big{[}|\xi|^{k}\big{]}\leq
C_{k}\,\sigma^{k}_{N,\hat{\beta}}\,\quad\text{for }k\geq 3\,,$ (2.7)

for some constants $C_{k}\in(0,\infty)$, $k\geq 3$. The asymptotic and the
bound in (2.7) follow by Taylor expansion. Expanding the product in (2.6)
yields the following expansion of $Z^{\beta_{N}}_{a,b}(x,y)$ as a multilinear
polynomial of the variables $\xi(n,z)$,

$\displaystyle Z^{\beta_{N}}_{a,b}(x,y)$ $\displaystyle=q_{a,b}(x,y)$
$\displaystyle+\sum_{k\geq 1}\sum_{\begin{subarray}{c}a<n_{1}<\dots<n_{k}<b\\\
z_{1},\dots,z_{k}\in\mathbb{Z}^{2}\end{subarray}}q_{a,n_{1}}(x,z_{1})\,\xi(n_{1},z_{1})\,\bigg{\\{}\prod_{j=2}^{k}q_{n_{j-1},n_{j}}(z_{j-1},z_{j})\,\xi(n_{j},z_{j})\bigg{\\}}\,q_{n_{k},b}(z_{k},y)\,,$
(2.8)

which also leads to

$\displaystyle Z^{\beta_{N}}_{a,b}(\varphi,\psi)$
$\displaystyle:=q_{a,b}^{N}(\varphi,\psi)$
$\displaystyle+\frac{1}{N}\sum_{k\geq
1}\sum_{\begin{subarray}{c}a<n_{1}<\dots<n_{k}<b\\\
z_{1},\dots,z_{k}\in\mathbb{Z}^{2}\end{subarray}}q^{N}_{a,n_{1}}(\varphi,z_{1})\,\xi(n_{1},z_{1})\,\bigg{\\{}\prod_{j=2}^{k}q_{n_{j-1},n_{j}}(z_{j-1},z_{j})\,\xi(n_{j},z_{j})\bigg{\\}}\,q^{N}_{n_{k},b}(z_{k},\psi)$

for the averaged point-to-point partition function. Using the notation

$\displaystyle\widebar{Z}^{\beta_{N}}_{N}(\varphi,\psi):=Z^{\beta_{N}}_{N}(\varphi,\psi)-{\mathbb{E}}\big{[}Z^{\beta_{N}}_{N}(\varphi,\psi)\big{]}$

for the centred averaged partition function we have that

$\displaystyle\widebar{Z}^{\beta_{N}}_{N}(\varphi,\psi):=\frac{1}{N}\sum_{k\geq
1}\sum_{\begin{subarray}{c}z_{1},z_{2},\dots,z_{k}\\\
0<n_{1}<\dots<n_{k}<N\end{subarray}}q^{N}_{0,n_{1}}(\varphi,z_{1})\,\xi(n_{1},z_{1})\bigg{\\{}\prod_{j=2}^{k}q_{n_{j-1},n_{j}}(z_{j-1},z_{j})\xi(n_{j},z_{j})\bigg{\\}}q^{N}_{n_{k},N}(z_{k},\psi)\,.$
(2.9)

For simplicity, we will denote
$Z^{\beta_{N}}_{N}(\varphi):=Z^{\beta_{N}}_{N}(\varphi,1)$ and
$\widebar{Z}^{\beta_{N}}_{N}(\varphi):=\widebar{Z}^{\beta_{N}}_{N}(\varphi,1)$.

###  2.2. Renewal representation

We will also need certain renewal representations for the second moment of the
point-to-point partition functions. These were introduced in [CSZ19b] but only
mainly studied in the context of the critical directed polymer therein. Let
$(a,x),(b,y)\in\mathbb{N}\times\mathbb{Z}^{2}$ with $a<b$. We define

$\displaystyle
U^{\beta_{N}}_{N}\big{(}(a,x),(b,y)\big{)}:=\sigma_{N,\hat{\beta}}^{2}\,{\mathbb{E}}\Big{[}Z^{\beta_{N}}_{a,b}(x,y)^{2}\Big{]}\,.$
(2.10)

By translation invariance

$\displaystyle
U^{\beta_{N}}_{N}\big{(}(a,x),(b,y)\big{)}=U^{\beta_{N}}_{N}(b-a,y-x):=\sigma_{N,\hat{\beta}}^{2}\,{\mathbb{E}}\Big{[}Z^{\beta_{N}}_{b-a}(y-x)^{2}\Big{]},$

therefore it suffices to work with $U^{\beta_{N}}_{N}(n,x)$. We furthermore
define $U_{N}^{\beta_{N}}(n,x):=\mathbb{1}_{\\{x=0\\}}$ if $n=0$. Using (2.1)
and (2.7) we derive the expansion

$\displaystyle U^{\beta_{N}}_{N}(n,x)$
$\displaystyle=\sigma^{2}_{N,\hat{\beta}}\,q^{2}_{n}(x)$ (2.11)
$\displaystyle+\sum_{k\geq
1}\sigma^{2(k+1)}_{N,\hat{\beta}}\sum_{\begin{subarray}{c}0<n_{1}<\dots<n_{k}<n\\\
z_{1},z_{2},\dots,z_{k}\in\mathbb{Z}^{2}\end{subarray}}q^{2}_{0,n_{1}}(0,z_{1})\Big{\\{}\prod_{j=2}^{k}q^{2}_{n_{j-1},n_{j}}(z_{j-1},z_{j})\Big{\\}}\,q^{2}_{n_{k},n}(z_{k},x)\,.$

Moreover, for $0\leq n\leq N$ we define

$\displaystyle
U^{\beta_{N}}_{N}(n):=\sum_{x\in\mathbb{Z}^{2}}U^{\beta_{N}}_{N}(n,x)\,.$
(2.12)

We will, now, recast $U^{\beta_{N}}_{N}(n,x)$ and $U^{\beta_{N}}_{N}(n)$ in a
renewal theory framework. We define a family of i.i.d. random vectors
$(\mathsf{t}^{(N)}_{i},\mathsf{x}^{(N)}_{i})_{i\geq 1}$, such that

$\displaystyle\mathrm{P}\Big{(}\,\big{(}\mathsf{t}^{(N)}_{1},\mathsf{x}^{(N)}_{1}\big{)}=(n,x)\Big{)}=\frac{q_{n}^{2}(x)}{R_{N}}\,\mathbb{1}_{\\{n\leq
N\\}}\,$

and moreover we let
$\tau^{(N)}_{k}:=\mathsf{t}_{1}^{(N)}+\dots+\mathsf{t}_{k}^{(N)}$ and
$S^{(N)}_{k}:=\mathsf{x}^{(N)}_{1}+\dots+\mathsf{x}^{(N)}_{k}$ if $k\geq 1$.
For $k=0$ we set $(\tau_{0},S_{0}):=(0,0)$. Using this framework we see by
(2.11) and (2.12) that

$\displaystyle U^{\beta_{N}}_{N}(n,x)=\sum_{k\geq
0}\hat{\beta}^{2k}\,\mathrm{P}\big{(}\tau_{k}^{(N)}=n,S^{(N)}_{k}=x\big{)}$

and

$\displaystyle U^{\beta_{N}}_{N}(n)=\sum_{k\geq
0}\hat{\beta}^{2k}\,\mathrm{P}\big{(}\tau_{k}^{(N)}=n\big{)}\,.$

Finally, we remark that

$\displaystyle\sum_{n=0}^{N}U_{N}^{\beta_{N}}(n)={\mathbb{E}}\Big{[}(Z^{\beta_{N}}_{N+1})^{2}\Big{]}\,.$
(2.13)

###  2.3. Some useful results

We will make use of the following results on the limiting distribution of
$Z_{N}^{\beta_{N}}$ and the fluctuations of
$\widebar{Z}_{N}^{\beta_{N}}(\varphi)$, which were established in [CSZ17b].

###### Theorem 2.1.

[CSZ17b]. Fix $\hat{\beta}\in(0,1)$ and let
$\varrho^{2}_{\hat{\beta}}:=\log\big{(}\frac{1}{1-\hat{\beta}^{2}}\big{)}$.
Then,

$\displaystyle
Z_{N}^{\beta_{N}}\xrightarrow[N\to\infty]{(d)}\exp\big{(}\varrho_{\hat{\beta}}\,\mathsf{X}-\tfrac{1}{2}\,\varrho^{2}_{\hat{\beta}}\big{)}\,,$

where $\mathsf{X}$ has a standard normal distribution $\mathcal{N}(0,1)$.

###### Theorem 2.2.

[CSZ17b]. Fix $\hat{\beta}\in(0,1)$ and $\varphi\in C_{c}(\mathbb{R}^{2})$.
Then,

$\displaystyle\sqrt{\log
N}\,\widebar{Z}_{N}^{\beta_{N}}(\varphi)\xrightarrow[N\to\infty]{(d)}{\mathcal{N}}\big{(}0,\varrho^{2}_{\varphi}(\hat{\beta})\big{)}\,,$

where
$\widebar{Z}_{N}^{\beta_{N}}(\varphi):=\widebar{Z}_{N}^{\beta_{N}}(\varphi,1)$
is defined in (2.4),

$\displaystyle\varrho^{2}_{\varphi}(\hat{\beta}):=\frac{\pi\,\hat{\beta}^{2}}{1-\hat{\beta}^{2}}\int_{0}^{1}\mathrm{d}t\int_{(\mathbb{R}^{2})^{2}}\mathrm{d}x\,\mathrm{d}y\,\varphi(x)g_{t}(x-y)\varphi(y)$

and $g_{t}(x):=\frac{1}{2\pi t}\,e^{-|x|^{2}/2t}$ denotes the two-dimensional
heat kernel.

##  3\. Expansion of moments and integral inequalities

We shall hereafter use the notation

$\displaystyle
M^{\varphi,\psi}_{N,h}:={\mathbb{E}}\Big{[}\widebar{Z}^{\beta_{N}}_{N}(\varphi,\psi)^{h}\Big{]}\,,$

for the $h^{th}$ centred moments of the averaged field (2.4).

###  3.1. Chaos expansion of moments

By (2.9) we have

$\displaystyle M^{\varphi,\psi}_{N,h}$ $\displaystyle=\frac{1}{N^{h}}$
$\displaystyle\,\times{\mathbb{E}}\Bigg{[}\bigg{(}\sum_{k\geq
1}\sum_{\begin{subarray}{c}z_{1},z_{2},\dots,z_{k}\in\mathbb{Z}^{2}\\\
0<n_{1}<\dots<n_{k}<N\end{subarray}}q^{N}_{0,n_{1}}(\varphi,z_{1})\xi(n_{1},z_{1})\bigg{\\{}\prod_{j=2}^{k}q_{n_{j-1},n_{j}}(z_{j-1},z_{j})\xi(n_{j},z_{j})\bigg{\\}}q^{N}_{n_{k},N}(z_{k},\psi)\bigg{)}^{h}\Bigg{]}\,.$
(3.1)

When $h\in\mathbb{N}$, the power $h$ on the right hand side of (3.1) can be
expanded as

$\displaystyle\sum_{k_{1},\dots,k_{h}\geq
1}\sum_{\begin{subarray}{c}(n^{(r)}_{i},z^{(r)}_{i})\in\,\mathbb{N}\times\mathbb{Z}^{2}\,\\\
1\leq i\leq k_{r}\,,1\leq r\leq h\end{subarray}}$
$\displaystyle\prod_{r=1}^{h}q^{N}_{0,n^{(r)}_{1}}(\varphi,z^{(r)}_{1})\xi(n^{(r)}_{1},z^{(r)}_{1})\,$
$\displaystyle\times$
$\displaystyle\bigg{\\{}\prod_{j=2}^{k_{r}}q_{n^{(r)}_{j-1},n^{(r)}_{j}}(z^{(r)}_{j-1},z^{(r)}_{j})\xi(n^{(r)}_{j},z^{(r)}_{j})\bigg{\\}}q^{N}_{n^{(r)}_{k},N}(z^{(r)}_{k_{r}},\psi)\,.$
(3.2)

Note that every term in that expansion contains a product of disorder
variables of the form

$\displaystyle\prod_{r=1}^{h}\,\prod_{j=1}^{k_{r}}\,\xi(n^{(r)}_{j},x^{(r)}_{j})\,.$
(3.3)

Therefore, after taking the expectation with respect to the environment and
taking into account that the $\xi$ variables have mean zero and are
independent if they are indexed by different space time points, see (2.7), we
see that the non-zero terms of the expansion of (3.1) will be those such that
for every point $(n^{(r)}_{j},x^{(r)}_{j})$, $1\leq j\leq k_{r},1\leq r\leq h$
there exists (at least one) $1\leq r^{\prime}\leq h,1\leq j\leq
k_{r^{\prime}}$ such that $r\neq r^{\prime}$ and
$(n^{(r)}_{j},x^{(r)}_{j})=(n^{(r^{\prime})}_{j^{\prime}},x^{(r^{\prime})}_{j^{\prime}})$,
that is, every disorder variable $\xi(n^{(r)}_{j},x^{(r)}_{j})$ should appear
at least twice in a product of disorder variables. Hence, a natural way to
parametrise the sum (3.1) is to sum over the space-time locations of these
coincidence points along with all the possible coincidence configurations. We
will also use iteratively the Chapman-Kolmogorov equation
$q_{t_{1},t_{2}}(x,y)=\sum_{z\in\mathbb{Z}^{2}}q_{t_{1},s}(x,z)\,q_{s,t_{2}}(z,y)$,
$t_{1}<s<t_{2}$, for the simple random walk, to break down ’long range jumps’,
appearing in (3.1) via their transition probabilities, into smaller jumps, so
that we can track the location of each random walk at each time $t$, see
Figure 1. Let us introduce the framework which will allow to formalise the
above.

$\scriptstyle I_{1}$$\scriptstyle I_{2}$$\scriptstyle I_{3}$$\scriptstyle
I_{4}$$\scriptstyle I_{5}$$\dots$$\scriptstyle
y_{2}^{1}=y_{3}^{1}$$\scriptstyle y_{2}^{2}=y_{3}^{2}$$\scriptstyle
y_{1}^{2}$$\scriptstyle y_{3}^{3}=y_{4}^{3}$$\scriptstyle
y_{2}^{3}$$\scriptstyle y_{1}^{3}$$\scriptstyle y_{4}^{2}$$\scriptstyle
y_{4}^{1}$$\scriptstyle y_{1}^{1}$$\scriptstyle y_{1}^{4}$$\scriptstyle
y_{2}^{4}$$\scriptstyle y_{3}^{4}=y^{4}_{4}$$\scriptstyle
y_{1}^{5}=y_{2}^{5}=y_{3}^{5}$$\scriptstyle y_{4}^{5}$ Figure 1.  A
diagrammatic representation of the expansion (3.15) for
${\mathbb{E}}\big{[}(\widebar{Z}_{N}^{\beta_{N}})^{4}\big{]}$. The horizontal
direction is the time direction, while the vertical lines correspond to
different time slices, $\\{n\\}\times\mathbb{Z}^{2}$, $n\in\mathbb{N}$. We use
straight lines to represent free evolution (3.6) and wiggly lines to represent
replica evolution, see (3.11). We use filled dots to represent space-time
points where disorder $\xi$ is sampled.

For $h\geq 3$, let $I\vdash\\{1,\dots,h\\}$ denote a partition $I=I_{1}\sqcup
I_{2}\sqcup\cdots\sqcup I_{m}$ of $\\{1,\dots,h\\}$ into disjoint subsets
$I_{1},\dots,I_{m}$ with cardinality $|I|=m$. Given $I\vdash\\{1,\dots,h\\}$,
we define the equivalence relation $\stackrel{{\scriptstyle I}}{{\sim}}$ such
that for $k,\ell\in\\{1,\dots,h\\}$, we have $k\stackrel{{\scriptstyle
I}}{{\sim}}\ell$ if $k$ and $\ell$ belong to the same component of the
partition $I$. For $\boldsymbol{x}=(x_{1},\dots,x_{h})\in(\mathbb{Z}^{2})^{h}$
and a partition $I$ we will denote $\boldsymbol{x}\sim I$ if $x_{k}=x_{\ell}$
for all $k\stackrel{{\scriptstyle I}}{{\sim}}\ell$. We shall also use the
notation
$(\mathbb{Z}^{2})^{h}_{I}:=\\{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}\colon\boldsymbol{x}\sim
I\\}$.

For $p\in(1,\infty)$ we define the $I$-restricted $\ell^{p}$ spaces
$\ell^{p}\big{(}(\mathbb{Z}^{2})^{h}_{I}\big{)}$ via the norm
$\|f\|_{\ell^{p}((\mathbb{Z}^{2})^{h}_{I})}:=\big{(}\sum_{x\in(\mathbb{Z}^{2})^{h}_{I}}|f(x)|^{p}\big{)}^{1/p}$
for functions $f\colon(\mathbb{Z}^{2})^{h}_{I}\to\mathbb{R}$. In shorthand, we
will often write $\ell^{p}_{I}$ or just $\ell^{p}$ if there is no risk of
confusion. For an integral operator
$\mathsf{T}:\ell^{q}\big{(}(\mathbb{Z}^{2})^{h}_{J}\big{)}\rightarrow\ell^{q}\big{(}(\mathbb{Z}^{2})^{h}_{I}\big{)}$,
we define the pairing

$\displaystyle\langle
f,\mathsf{T}g\rangle:=\sum_{\boldsymbol{x}\,\in(\mathbb{Z}^{2})^{h}_{I},\boldsymbol{y}\,\in(\mathbb{Z}^{2})^{h}_{J}}f(\boldsymbol{x})\mathsf{T}(\boldsymbol{x},\boldsymbol{y})g(\boldsymbol{y})\,.$
(3.4)

The operator norm will be given by

$\displaystyle\left\lVert\mathsf{T}\right\rVert_{\ell^{q}\to\ell^{q}}:=\sup_{\left\lVert
g\right\rVert_{\ell^{q}_{J}}\leq
1}\left\lVert\mathsf{T}g\right\rVert_{\ell^{q}_{I}}=\sup_{\left\lVert
f\right\rVert_{\ell^{p}_{I}}\leq 1,\,\left\lVert
g\right\rVert_{\ell^{q}_{J}}\leq 1}\langle f,\mathsf{T}g\rangle$ (3.5)

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

For two partitions $I,J\vdash\\{1,\dots,h\\}$ and
$\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}$ with
$\boldsymbol{x}\sim I$ and $\boldsymbol{y}\sim J$ we define the free evolution
subject to constraints $I,J$ as

$\displaystyle
Q^{I,J}_{n}(\boldsymbol{x},\boldsymbol{y}):=\mathbb{1}_{\\{\boldsymbol{x}\sim
I\\}}\,\prod_{i=1}^{h}q_{n}(y_{i}-x_{i})\,\mathbb{1}_{\\{\boldsymbol{y}\sim
J\\}}\,,\qquad\text{for $n\in\mathbb{N}$}\,.$ (3.6)

$Q^{I,*}_{n}$ and $Q^{*,J}_{n}$ will denote the particular cases where $I$ and
$J$, respectively, are the partitions consisting only of singletons, i.e.
$I=\\{1\\}\sqcup\cdots\sqcup\\{h\\}$. Moreover, if $I,J\vdash\\{1,\dots,h\\}$,
$\varphi,\psi:\mathbb{R}^{2}\to\mathbb{R}$ and $n\in\mathbb{N}$ we define

$\begin{split}Q_{n}^{*,J}(\varphi^{\otimes
h},\boldsymbol{y})&:=\prod_{i=1}^{h}q^{N}_{n}(\varphi,y_{i})\cdot\mathbb{1}_{\\{\boldsymbol{y}\sim
J\\}}\\\ Q_{n}^{I,*}(\boldsymbol{x},\psi^{\otimes
h})&:=\mathbb{1}_{\\{\boldsymbol{x}\sim
I\\}}\cdot\prod_{i=1}^{h}q^{N}_{n}(x_{i},\psi)\,,\end{split}$ (3.7)

see also (2.5). The mixed moment subject to a partition $I$ will be denoted by

$\displaystyle{\mathbb{E}}[\xi^{I}]:=\prod_{1\leq j\leq|I|,\,|I_{j}|\geq
2}{\mathbb{E}}[\xi^{|I_{j}|}]\,.$ (3.8)

Using this formalism, we can then write

$\displaystyle M^{\varphi,\psi}_{N,h}=$
$\displaystyle\frac{1}{N^{h}}\sum_{k\geq
1}\sum_{\begin{subarray}{c}0:=n_{0}<n_{1}<\dots<n_{k}\leq N,\\\
(I_{1},\dots,I_{k})\in\mathcal{I},\\\
m_{i}:=|I_{i}|<h,\,\,\boldsymbol{y}_{i}\in(\mathbb{Z}^{2})^{m_{i}}\end{subarray}}Q_{n_{1}}^{*,I_{1}}(\varphi^{\otimes
h},\boldsymbol{y}_{1}){\mathbb{E}}\Big{[}\xi^{I_{1}}\Big{]}$
$\displaystyle\qquad\qquad\qquad\times\prod_{i=2}^{k}Q_{n_{i}-n_{i-1}}^{I_{i-1},I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{y}_{i}){\mathbb{E}}\Big{[}\xi^{I_{i}}\Big{]}\cdot
Q_{N-n_{k}}^{I_{k},*}(\boldsymbol{y}_{k},\psi^{\otimes h})\,,$ (3.9)

where $\mathcal{I}$ is the set of all finite sequences of partitions of
$\\{1,\dots,h\\}$, $(I_{1},\dots,I_{k})$, which satisfy the following
condition: For every $r\in\\{1,\dots,h\\}$ there exists $1\leq i\leq k$ such
that the block $I_{i}$ that contains $r$ is non-trivial, i.e. it has
cardinality equal or larger than $2$. This restriction comes from the fact
that $M^{\varphi,\psi}_{N,h}$ are centred moments and therefore every term in
the expansion (3.1) contains the expected value of a product of disorder
variables (3.3), which is non-zero only if the product of disorder variables
does not contain standalone $\xi$ variables, see the discussion below (3.3).

Let $B=B(0,r)\subset\mathbb{R}^{2}$ be a ball containing the support of $\psi$
(allowing the possibility of $r=\infty$, in case
$\operatorname{supp}\psi=\mathbb{R}^{2}$). We then have that

$Q^{I_{k},*}_{N-n_{k}}\big{(}\boldsymbol{y}_{k},\psi^{\otimes h}\big{)}\leq
Q^{I_{k},*}_{N-n_{k}}\big{(}\boldsymbol{y}_{k},\left\lVert\psi\right\rVert^{h}_{\infty}\mathbb{1}^{\otimes
h}_{B}\big{)}\leq\frac{c}{N}\,\sum_{n_{k+1}\in\\{N+1,\dots,2N\\}}Q^{I_{k},*}_{n_{k+1}-n_{k}}\big{(}\boldsymbol{y}_{k},\left\lVert\psi\right\rVert^{h}_{\infty}\mathbb{1}^{\otimes
h}_{B}\big{)}\,,$

with the latter inequality following because the probability that a random
walk starting inside the ball $B(0,\sqrt{N}r)\subset\mathbb{R}^{2}$ at time
$N-n_{k}$ is inside $B(0,\sqrt{N}r)$ at time $n_{k+1}-n_{k}$ with
$n_{k+1}\in\\{N+1,\dots 2N\\}$ is uniformly bounded away from zero.

Thus,

$\displaystyle\big{|}\,M^{\varphi,\psi}_{N,h}\,\big{|}\leq$
$\displaystyle\,\frac{c\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h+1}}\sum_{k\geq
1}\sum_{\begin{subarray}{c}0:=n_{0}<n_{1}<\dots<n_{k+1}\leq 2N,\\\
(I_{1},\dots,I_{k})\in\mathcal{I},\\\
m_{i}:=|I_{i}|<h,\,\,\boldsymbol{y}_{i}\in(\mathbb{Z}^{2})^{m_{i}}\end{subarray}}Q_{n_{1}}^{*,I_{1}}(\varphi^{\otimes
h},\boldsymbol{y}_{1}){\mathbb{E}}\Big{[}|\xi|^{I_{1}}\Big{]}$
$\displaystyle\qquad\qquad\qquad\times\prod_{i=2}^{k}Q_{n_{i}-n_{i-1}}^{I_{i-1},I_{i}}(\boldsymbol{y}_{i-1},\boldsymbol{y}_{i}){\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\cdot
Q_{n_{k+1}-n_{k}}^{I_{k},*}(\boldsymbol{y}_{k},\mathbb{1}^{\otimes h}_{B})\,.$
(3.10)

We also need to define the replica evolution. For $I\vdash\\{1,\dots,h\\}$ of
the form $I=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$

$\displaystyle\mathsf{U}^{I}_{n}(\boldsymbol{x},\boldsymbol{y}):=\mathbb{1}_{\\{\boldsymbol{x},\boldsymbol{y}\sim
I\\}}\cdot U^{\beta_{N}}_{N}(n,y_{k}-x_{k})\cdot\prod_{i\neq
k,\ell}q_{n}(y_{i}-x_{i})\,,$ (3.11)

where $U^{\beta_{N}}_{N}(n,y_{k}-x_{k})$ is defined in (2.10). The replica
evolution operator will be used to contract consecutive appearances of the
same partition $I$, with $|I|=h-1$ in the right-hand side of (3.1). In
particular, note that if $I\vdash\\{1,\dots,h\\}$, such that $|I|=h-1$, then

$\displaystyle\mathsf{U}^{I}_{n}(\boldsymbol{x},\boldsymbol{y})=\sum_{k\geq
0}{\mathbb{E}}[\xi^{2}]^{k}\,\sum_{0:=n_{0}<n_{1}<\dots<n_{k}:=n}\,\,\sum_{\begin{subarray}{c}\boldsymbol{y}_{i}\in(\mathbb{Z}^{2})^{h}_{I}\,,1\leq
i\leq k-1,\\\
\boldsymbol{y}_{0}:=\boldsymbol{x},\,\boldsymbol{y}_{k}:=\boldsymbol{y}\end{subarray}}\,\prod_{i=1}^{k}Q^{I;I}_{n_{i}-n_{i-1}}(\boldsymbol{y}_{i-1},\boldsymbol{y}_{i})\,.$

To be able to estimate the right-hand side of (3.1) we will upper bound it by
enlarging the domain of the temporal sum in the right-hand side of (3.1) from
$1\leq n_{1}<\dots<n_{k+1}\leq 2N$ to $n_{i}-n_{i-1}\in\\{1,\dots,2N\\}$ for
all $1\leq i\leq k+1$. This enlargement of the domain of summation deconvolves
the temporal sum in the right-hand side of (3.1).

On this account, we introduce the discrete Laplace transforms of the operators
$\mathsf{Q}$ and $\mathsf{U}$,

$\displaystyle\mathsf{Q}_{N,\lambda}^{I,J}(\boldsymbol{y},\boldsymbol{z}):=\sum_{n=1}^{2N}e^{-\lambda\frac{n}{N}}\,Q^{I,J}_{n}(\boldsymbol{y},\boldsymbol{z}),\,\qquad\boldsymbol{y},\boldsymbol{z}\in(\mathbb{Z}^{2})^{h},$
(3.12)
$\displaystyle\mathsf{U}_{N,\lambda}^{I}(\boldsymbol{y},\boldsymbol{z}):=\sum_{n=0}^{2N}e^{-\lambda\frac{n}{N}}\,\mathsf{U}^{I}_{n}(\boldsymbol{y},\boldsymbol{z}),\qquad\quad\boldsymbol{y},\boldsymbol{z}\in(\mathbb{Z}^{2})^{h}\,,$
(3.13)

for $\lambda\geq 0$.

###### Remark 3.1.

In our case, it will be sufficient to work with $\lambda=0$, however, keeping
a non zero $\lambda$ and tracking the dependence of the estimate on it
(together with a closer tracking on the order $h$ of the moment) would be
necessary in order to extend our method so that to accommodate $h$ that grows
with $N$.

Let us define

$\displaystyle\mathsf{P}^{I;J}_{N,\hat{\beta}}=\begin{dcases}\mathsf{Q}^{I;J}_{N,0}\,,&\text{
if }|J|<h-1\\\ \mathsf{Q}^{I;J}_{N,0}\,\mathsf{U}^{J}_{N,0}\,,&\text{ if
}|J|=h-1\,.\end{dcases}$

Note that the appearance of the operator $\mathsf{U}^{J}_{N,0}$ is necessarily
preceded by a free evolution operator $\mathsf{Q}^{I;J}_{N,0}$, with
$|J|=h-1$, see also Figure 1. In view of (3.1) and the discussion above we can
now write

$\displaystyle\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq$
$\displaystyle\frac{c\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h+1}}\sum_{k\geq
1}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\Big{\langle}\varphi_{N}^{\otimes
h},\mathsf{P}^{*,I_{1}}_{N,\hat{\beta}}\,\mathsf{P}^{I_{1},I_{2}}_{N,\hat{\beta}}\cdots\,\mathsf{P}_{N,\hat{\beta}}^{I_{k},*}\mathbb{1}^{\otimes
h}_{\sqrt{N}B}\Big{\rangle}\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\,,$

where we recall the definition of the pairing $\langle\cdot\,,\cdot\rangle$
from (3.4) and note that the sum runs over partitions $I_{1},\dots,I_{k}$ such
that $I_{j}\neq I_{j+1}$ if $|I_{j}|=|I_{j+1}|=h-1$ for $1\leq j\leq k-1$.

Because of the assumption of Theorem 1.5 on $\psi$ being merely a bounded
function we will need to introduce weighted versions of the operators
$\mathsf{U}^{I}_{N,\lambda}$, $\mathsf{Q}^{I;J}_{N,\lambda}$ and
$\mathsf{P}^{I;J}_{N,\hat{\beta}}$. In particular, if
$w:\mathbb{R}^{2}\to\mathbb{R}$ is such that $\log w$ is Lipschitz continuous
with Lipschitz constant denoted by $C_{w}>0$, and if we denote by
$w_{N}(x)=w\big{(}\frac{x}{\sqrt{N}}\big{)}$, then we define for $\lambda\geq
0$,

$\displaystyle\widehat{\mathsf{Q}}^{I;J}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y}):=\frac{w_{N}^{\otimes
h}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}\,\mathsf{Q}^{I;J}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y})\,,$
$\displaystyle\widehat{\mathsf{U}}^{I}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y}):=\frac{w_{N}^{\otimes
h}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}\,\mathsf{U}^{I}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y})\,,$

where we recall that $w_{N}^{\otimes h}(\boldsymbol{x})=w_{N}(x_{1})\cdots
w_{N}(x_{h})$, if $\boldsymbol{x}=(x_{1},\dots,x_{h})$. We modify accordingly
the operator $\mathsf{P}^{I;J}_{N,\hat{\beta}}$ into a new operator
$\widehat{\mathsf{P}}^{I;J}_{N,\hat{\beta}}$,

$\displaystyle\widehat{\mathsf{P}}^{I;J}_{N,\hat{\beta}}=\begin{dcases}\widehat{\mathsf{Q}}^{I;J}_{N,0}&,\text{
if }|J|<h-1\\\
\widehat{\mathsf{Q}}^{I;J}_{N,0}\,\widehat{\mathsf{U}}^{J}_{N,0}&,\text{ if
}|J|=h-1\,.\end{dcases}$ (3.14)

Therefore, we can now write

$\begin{split}\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq&\frac{c\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h+1}}\sum_{k\geq
1}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\Big{\langle}\frac{\varphi_{N}^{\otimes
h}}{w_{N}^{\otimes
h}},\widehat{\mathsf{P}}^{*,I_{1}}_{N,\hat{\beta}}\,\widehat{\mathsf{P}}^{I_{1},I_{2}}_{N,\hat{\beta}}\cdots\,\widehat{\mathsf{P}}_{N,\hat{\beta}}^{I_{k},*}\mathbb{1}^{\otimes
h}_{\sqrt{N}B}w_{N}^{\otimes
h}\Big{\rangle}\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\,\\\
\leq&\frac{c\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h+1}}\sum_{k\geq
1}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\Big{\langle}\frac{\varphi_{N}^{\otimes
h}}{w_{N}^{\otimes
h}},\widehat{\mathsf{P}}^{*,I_{1}}_{N,\hat{\beta}}\,\widehat{\mathsf{P}}^{I_{1},I_{2}}_{N,\hat{\beta}}\cdots\,\widehat{\mathsf{P}}_{N,\hat{\beta}}^{I_{k},*}w_{N}^{\otimes
h}\Big{\rangle}\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\end{split}$
(3.15)

where we bounded the indicator function $\mathbb{1}^{\otimes h}_{\sqrt{N}B}$
by $1$ to obtain the second inequality. Passing to the operator norms (see
(3.5)) we estimate

$\displaystyle\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq$
$\displaystyle\frac{c\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h+1}}\sum_{k\geq
1}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\left\lVert\widehat{\mathsf{P}}^{*,I_{1}}_{N,\hat{\beta}}\frac{\varphi_{N}^{\otimes
h}}{w^{\otimes
h}_{N}}\right\rVert_{\ell^{p}}\,\prod_{i=2}^{k}\,\left\lVert\widehat{\mathsf{P}}_{N,\hat{\beta}}^{I_{i-1},I_{i}}\right\rVert_{\ell^{q}\to\ell^{q}}\,\left\lVert\widehat{\mathsf{P}}_{N,\hat{\beta}}^{I_{k},*}w_{N}^{\otimes
h}\right\rVert_{\ell^{q}}\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\,.$
(3.16)

This is the key expansion we will use for the Proof of Theorem 1.5.

###  3.2. Integral inequalities for the operators
$\widehat{\mathsf{Q}}^{I;J}_{N,0}$ and $\widehat{\mathsf{U}}^{I}_{N,0}$

At this point, we will prove the key estimates about the operators
$\widehat{\mathsf{Q}}^{I;J}_{N,0},\widehat{\mathsf{U}}^{I}_{N,0}$ that we will
need along the way. In what follows we shall use the letter $C$ to denote
constants that may depend only on $h,\hat{\beta}$ and $w$ but not on $p$ and
$q$. We will also use the letter $c$ to denote absolute constants, i.e.
constants that do not depend on $h,\hat{\beta},w$ or $p,q$. Their value may
change from line to line.

We start this subsection by stating a lemma on the operator
$\mathsf{Q}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y}):=\sum_{n=1}^{2N}e^{-\tfrac{\lambda
n}{N}}Q_{n}(\boldsymbol{x},\boldsymbol{y})$ from [CSZ21] which we will need in
the sequel.

###### Lemma 3.2 ([CSZ21]).

Let $N\geq 1$, $h\geq 2$ and
$\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}$. Then, there exists a
constant $C\in(0,\infty)$ such that uniformly in $N$,
$\boldsymbol{x},\boldsymbol{y}$ and $\lambda\geq 0$,

$\displaystyle\mathsf{Q}_{N,\lambda}(\boldsymbol{x},\boldsymbol{y})\leq\begin{dcases}\,\,\frac{C}{\big{(}1+|\boldsymbol{x}-\boldsymbol{y}|^{2}\big{)}^{h-1}}&\text{
for all }\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}\,,\\\
\,\,\frac{C}{N^{h-1}}\,\exp\bigg{(}\frac{-|\boldsymbol{x}-\boldsymbol{y}|^{2}}{C\,N}\bigg{)}&\text{
if }|\boldsymbol{x}-\boldsymbol{y}|>\sqrt{N}\,.\end{dcases}$

The next proposition contains the central estimate. It is on the operator norm
of operator $\widehat{\mathsf{Q}}^{I;J}_{N,0}$, as an operator from an
$\ell^{q}\to\ell^{q}$, containing the explicit dependence on the parameters
$p,q$.

###### Proposition 3.3.

Let $p,q\in(1,\infty)$ such that $\tfrac{1}{p}+\tfrac{1}{q}=1$. There exists a
constant $C=C(h,w)\in(0,\infty)$, independent of $p$ and $q$, such that for
all $I,J\vdash\\{1,\dots,h\\}$ with $1\leq|I|,|J|\leq h-1$ and $I\neq J$ when
$|I|=|J|=h-1$,

$\displaystyle\left\lVert\widehat{\mathsf{Q}}^{I;J}_{N,0}\right\rVert_{\ell^{q}\to\ell^{q}}\leq
C\,p\,q\,.$ (3.17)

  * Proof.

Let $I,J\vdash\\{1,\dots,h\\}$ with $1\leq|I|,|J|\leq h-1$ and $I\neq J$ when
$|I|=|J|=h-1$ and consider
$f\in\ell^{p}\big{(}(\mathbb{Z}^{2})^{h}_{I}\big{)}$,
$g\in\ell^{q}\big{(}(\mathbb{Z}^{2})^{h}_{J}\big{)}$. In view of (3.5), in
order to prove (3.17), we need to prove that there exists a constant
$C\in(0,\infty)$ such that

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}f(\boldsymbol{x})\mathsf{Q}^{I;J}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}g(\boldsymbol{y})\leq C\,p\,q\,\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}\,.$ (3.18)

Let

$\displaystyle
E_{N}:=\Big{\\{}(\boldsymbol{x},\boldsymbol{y})\in(\mathbb{Z}^{2})^{h}_{I}\times(\mathbb{Z}^{2})^{h}_{J}:|\boldsymbol{x}-\boldsymbol{y}|\leq
C_{0}\sqrt{N}\Big{\\}}\,.$ (3.19)

for some $C_{0}>0$ to be determined. By the second inequality in Lemma 3.2 and
the Lipschitz condition on $\log w$, we can choose $C_{0}$ large enough so
that for all $(\boldsymbol{x},\boldsymbol{y})\in E_{N}^{c}$ we have

$\displaystyle\mathsf{Q}^{I;J}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}\leq\frac{C}{N^{h-1}}\exp\big{(}-\tfrac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}\big{)}\,.$

Therefore, on $E_{N}^{c}$ we have that

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}^{c}}f(\boldsymbol{x})\mathsf{Q}^{I;J}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}g(\boldsymbol{y})\leq\frac{C}{N^{h-1}}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}^{c}}f(\boldsymbol{x})\exp\big{(}-\tfrac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}\big{)}g(\boldsymbol{y})\,$

and by Hölder’s inequality,

$\displaystyle\frac{1}{N^{h-1}}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}^{c}}f(\boldsymbol{x})\exp\big{(}-\tfrac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}\big{)}g(\boldsymbol{y})$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{N^{h-1}}\,\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}|f(\boldsymbol{x})|^{p}\exp\Big{(}-\tfrac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}\Big{)}\Bigg{)}^{\frac{1}{p}}\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}|g(\boldsymbol{y})|^{q}\exp\Big{(}-\tfrac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}\Big{)}\Bigg{)}^{\frac{1}{q}}$
$\displaystyle\leq$
$\displaystyle\,C\,N^{\textstyle\frac{|J|}{p}+\frac{|I|}{q}-(h-1)}\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}$
$\displaystyle\leq$ $\displaystyle\,C\,\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}\,,$ (3.20)

where the inequality in the last line of (Proof.) follows by the assumption
$|I|,|J|\leq h-1$. Thus,

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}^{c}}f(\boldsymbol{x})\mathsf{Q}^{I;J}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}g(\boldsymbol{y})\leq C\,\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}\,,$

for a constant $C\in(0,\infty)$.

Let us now estimate the sum over $(\boldsymbol{x},\boldsymbol{y})\in E_{N}$.
Recalling that $\log w$ is Lipschitz with Lipschitz constant $C_{w}$ and
(3.19), we get that

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}f(\boldsymbol{x})\mathsf{Q}^{I;J}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w^{\otimes
h}_{N}(\boldsymbol{y})}g(\boldsymbol{y})\leq
e^{C_{w}\,C_{0}}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}f(\boldsymbol{x})\mathsf{Q}^{I;J}_{N,0}(\boldsymbol{x},\boldsymbol{y})g(\boldsymbol{y})\,.$

Therefore, using the first inequality of Lemma 3.2, the key step is to show
that there exists a constant $C\in(0,\infty)$ that may depend on $h$ and $w$
but not on $p$ and $q$, such that

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{f(\boldsymbol{x})g(\boldsymbol{y})}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\leq
C\,p\,q\left\lVert f\right\rVert_{\ell^{p}}\left\lVert
g\right\rVert_{\ell^{q}}\,.$ (3.21)

By assumption there exist $1\leq k,\ell\leq h$ such that
$k\stackrel{{\scriptstyle I}}{{\sim}}\ell$ and $1\leq m,n\leq h$ such that
$m\stackrel{{\scriptstyle J}}{{\sim}}n$. Since we have assumed that $I\neq J$
when $|I|=|J|=h-1$, we may assume without loss of generality that $m\neq
k,\ell$. Let $a\in\big{(}0,\min\\{p^{-1},q^{-1}\\}\big{)}$ to be determined
later. By multiplying and dividing by
$\frac{1+|x_{m}-x_{n}|^{2a}}{1+|y_{k}-y_{\ell}|^{2a}}$ and using Hölder’s
inequality, the left-hand side of (3.21) is upper bounded by

$\displaystyle\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}$
$\displaystyle\frac{|f(\boldsymbol{x})|^{p}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\cdot\frac{\big{(}1+|x_{m}-x_{n}|^{2a}\big{)}^{p}}{\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{p}}\Bigg{)}^{\frac{1}{p}}$
$\displaystyle\times\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{|g(\boldsymbol{y})|^{q}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\cdot\frac{\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{q}}{\big{(}1+|x_{m}-x_{n}|^{2a}\big{)}^{q}}\Bigg{)}^{\frac{1}{q}}\,.$
(3.22)

By symmetry, it is enough to bound one of the two factors in (Proof.). By
triangle inequality and the fact that $m\stackrel{{\scriptstyle J}}{{\sim}}n$,
which means that $y_{m}=y_{n}$, we have

$\displaystyle|x_{m}-y_{m}|^{2}+|x_{n}-y_{n}|^{2}\geq\frac{|x_{m}-x_{n}|^{2}+|x_{n}-y_{n}|^{2}}{4}\,.$

Therefore,

$\displaystyle\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{|f(\boldsymbol{x})|^{p}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{j}|^{2}\Big{)}^{h-1}}\cdot\frac{\big{(}1+|x_{m}-x_{n}|^{2a}\big{)}^{p}}{\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{p}}\Bigg{)}^{\frac{1}{p}}$
(3.23) $\displaystyle\leq$
$\displaystyle\,4^{\frac{h-1}{p}}\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\,(1+|x_{m}-x_{n}|^{2a})^{p}\,\,$
$\displaystyle\qquad\qquad\qquad\times\sum_{\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+\sum_{i\neq
m}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{p}}\Bigg{)}^{\frac{1}{p}}\,.$

By using (A.1) of Lemma A.1 and summing successively the $y_{i}$ variables for
$i\neq k,\ell$ we obtain that

$\displaystyle\sum_{\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+\sum_{i\neq
m}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}(1+|y_{k}-y_{\ell}|^{2a})^{p}}$
$\displaystyle\leq$
$\displaystyle\,c^{|J|-2}\sum_{y_{k},\,y_{\ell}\in\mathbb{Z}^{2}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+|y_{k}-x_{k}|^{2}+|y_{\ell}-x_{\ell}|^{2}\Big{)}^{h+1-|J|}\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{p}}\,.$

We make a change of variables $w_{1}=y_{k}-y_{\ell}$ and
$w_{2}=y_{k}+y_{\ell}-2x_{k}$ and observe that
$\frac{w^{2}_{1}+w^{2}_{2}}{2}=|y_{k}-x_{k}|^{2}+|y_{\ell}-x_{\ell}|^{2}$,
where we used that $k\stackrel{{\scriptstyle I}}{{\sim}}\ell$ thus
$x_{k}=x_{\ell}$. Therefore, we have

$\displaystyle
c^{|J|-2}\sum_{y_{k},\,y_{\ell}\in\mathbb{Z}^{2}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+|y_{k}-x_{k}|^{2}+|y_{\ell}-x_{\ell}|^{2}\Big{)}^{h+1-|J|}\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{p}}$
$\displaystyle\leq$
$\displaystyle\,2^{h+1-|J|}\,c^{|J|-2}\sum_{w_{1},\,w_{2}\in\mathbb{Z}^{2}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+|w_{1}|^{2}+|w_{2}|^{2}\Big{)}^{h+1-|J|}\big{(}1+|w_{1}|^{2a}\big{)}^{p}}\,.$

By summing $w_{2}$ and using (A.1) of Lemma A.1 we have,

$\displaystyle
2^{h+1-|J|}\,c^{|J|-2}\sum_{w_{1},\,w_{2}\in\mathbb{Z}^{2}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+|w_{1}|^{2}+|w_{2}|^{2}\Big{)}^{h+1-|J|}\big{(}1+|w_{1}|^{2a}\big{)}^{p}}$
$\displaystyle\leq$
$\displaystyle\,2^{h+1-|J|}\,c^{|J|-1}\,\,\sum_{w_{1}\in\mathbb{Z}^{2}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+|w_{1}|^{2}\Big{)}^{h-|J|}\big{(}1+|w_{1}|^{2a}\big{)}^{p}}$

By (A.2) of Lemma A.1 we have that

$\displaystyle
2^{h+1-|J|}\,c^{|J|-1}\,\sum_{w_{1}\in\mathbb{Z}^{2}}\frac{1}{\Big{(}1+|x_{m}-x_{n}|^{2}+|w_{1}|^{2}\Big{)}^{h-|J|}\big{(}1+|w_{1}|^{2a}\big{)}^{p}}$
$\displaystyle\leq$
$\displaystyle\,2^{h+1-|J|}\,c^{|J|}\,\frac{1}{ap(1-ap)}\frac{1}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap+h-1-|J|}}$
$\displaystyle\leq$
$\displaystyle\,2^{h+1-|J|}\,c^{|J|}\frac{1}{ap(1-ap)}\frac{1}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap}}\,,$

where in the last inequality we used that $|J|\leq h-1$ by assumption.
Therefore, the right-hand side of (3.23) is bounded by

$\displaystyle\bigg{(}4^{h-1}\,2^{h+1}\,\Big{(}\frac{c}{2}\Big{)}^{|J|}\frac{1}{ap(1-ap)}\,\bigg{)}^{\frac{1}{p}}\cdot\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\,\frac{(1+|x_{m}-x_{n}|^{2a})^{p}}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap}}\Bigg{)}^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\bigg{(}2^{3h-1}\,\Big{(}\frac{c}{2}\Big{)}^{|J|}\frac{1}{ap(1-ap)}\,\bigg{)}^{\frac{1}{p}}\cdot\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\,\frac{(1+|x_{m}-x_{n}|^{2a})^{p}}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap}}\Bigg{)}^{\frac{1}{p}}\,.$
(3.24)

Note furthermore, that

$\displaystyle\frac{\big{(}1+|x_{m}-x_{n}|^{2a}\big{)}^{p}}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap}}\leq\frac{2^{p}\max\big{\\{}1,|x_{m}-x_{n}|\big{\\}}^{2ap}}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap}}\leq
2^{p}\,,$

therefore,

$\displaystyle\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\,\frac{(1+|x_{m}-x_{n}|^{2a})^{p}}{\big{(}1+|x_{m}-x_{n}|^{2}\big{)}^{ap}}\Bigg{)}^{\frac{1}{p}}\leq
2\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\,\Bigg{)}^{\frac{1}{p}}=\,2\left\lVert
f\right\rVert_{\ell^{p}}\,.$

Hence, setting

$\displaystyle
C^{J}_{p,h}:=2\cdot\Bigg{(}2^{3h-1}\,\Big{(}\frac{c}{2}\Big{)}^{|J|}\frac{1}{ap(1-ap)}\Bigg{)}^{\frac{1}{p}}\,$

and recalling (3.23), (Proof.) we get that

$\displaystyle\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{|f(\boldsymbol{x})|^{p}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\cdot\frac{\big{(}1+|x_{m}-x_{n}|^{2a}\big{)}^{p}}{\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{p}}\Bigg{)}^{\frac{1}{p}}\leq\,C^{J}_{p,h}\,\left\lVert
f\right\rVert_{\ell^{p}}\,.$ (3.25)

By symmetry we also obtain that

$\displaystyle\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{|g(\boldsymbol{y})|^{q}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\cdot\frac{\big{(}1+|y_{k}-y_{\ell}|^{2a}\big{)}^{q}}{\big{(}1+|x_{m}-x_{n}|^{2a}\big{)}^{q}}\Bigg{)}^{\frac{1}{q}}\leq
C^{I}_{q,h}\,\left\lVert g\right\rVert_{\ell^{q}}\,,$ (3.26)

with

$\displaystyle
C^{I}_{q,h}:=2\cdot\Bigg{(}2^{3h-1}\,\Big{(}\frac{c}{2}\Big{)}^{|I|}\frac{1}{aq(1-aq)}\Bigg{)}^{\frac{1}{q}}\,.$

Consequently, recalling (3.21) and using (3.25), (3.26) we deduce that

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{f(\boldsymbol{x})g(\boldsymbol{y})}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\leq\,C^{J}_{p,h}\,C^{I}_{q,h}\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}\,.$

We optimise by choosing $a=(p\,q)^{-1}$ so as to obtain

$\displaystyle
C^{J}_{p,h}=2\cdot\Bigg{(}2^{3h-1}\,\Big{(}\frac{c}{2}\Big{)}^{|J|}\,p\,q\,\Bigg{)}^{\frac{1}{p}}\,\text{
and
}\,C^{I}_{q,h}=2\cdot\Bigg{(}2^{3h-1}\,\Big{(}\frac{c}{2}\Big{)}^{|I|}\,p\,q\,\Bigg{)}^{\frac{1}{q}}\,,$

which implies that

$\displaystyle
C^{J}_{p,h}\,C^{I}_{q,h}=2^{3h+1}\,\Big{(}\frac{c}{2}\Big{)}^{\frac{|J|}{p}+\frac{|I|}{q}}\,p\,q\,.$

Noting that
$\big{(}\frac{c}{2}\big{)}^{\frac{|J|}{p}+\frac{|I|}{q}}\leq\max\Big{\\{}1,\big{(}\frac{c}{2}\big{)}^{h-1}\Big{\\}}$,
we deduce that there exists $C=C(h,w)\in(0,\infty)$ such that

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{J}}\frac{f(\boldsymbol{x})g(\boldsymbol{y})}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\leq
C\,p\,q\,\left\lVert f\right\rVert_{\ell^{p}}\left\lVert
g\right\rVert_{\ell^{q}}\,,$

which together with (Proof.) imply (3.18). ∎

The next proposition is the analogue of Proposition 3.3 for the boundary
operators.

###### Proposition 3.4.

Let $p,q\in(1,\infty)$ such that $\frac{1}{p}+\frac{1}{q}=1$. There exists a
constant $C=C(h,w)\in(0,\infty)$, independent of $p$ and $q$, such that for
all $I\vdash\\{1,\dots,h\\}$ with $|I|\leq h-1$ and
$g\in\ell^{q}(\mathbb{Z}^{2})$,

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

  * Proof.

Let $I\vdash\\{1,\dots,h\\}$ with $|I|\leq h-1$. In order to prove Proposition
3.4, we need to show that

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I},\,\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}f(\boldsymbol{x})\mathsf{Q}^{I;*}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w_{N}^{\otimes
h}(\boldsymbol{x})}{w_{N}^{\otimes h}(\boldsymbol{y})}g^{\otimes
h}(\boldsymbol{y})\leq C\,p\,N^{\frac{1}{p}}\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert^{h}_{\ell^{q}}\,.$

for any $f\in\ell^{p}\big{(}(\mathbb{Z}^{2})^{|I|}\big{)}$. The proof of this
Proposition is a modification of the proof of Proposition 3.3. Let

$\displaystyle
E_{N}:=\Big{\\{}(\boldsymbol{x},\boldsymbol{y})\in(\mathbb{Z}^{2})^{h}_{I}\times(\mathbb{Z}^{2})^{h}:|\boldsymbol{x}-\boldsymbol{y}|\leq
C_{0}\sqrt{N}\Big{\\}}\,.$

For $(\boldsymbol{x},\boldsymbol{y})\in E_{N}^{c}$, following (Proof.) we have

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}^{c}}f(\boldsymbol{x})\frac{w_{N}^{\otimes
h}(\boldsymbol{x})}{w_{N}^{\otimes
h}(\boldsymbol{y})}\mathsf{Q}^{I;*}_{N,0}(\boldsymbol{x},\boldsymbol{y})g^{\otimes
h}(\boldsymbol{y})\leq$
$\displaystyle\,C\,N^{\textstyle\frac{h}{p}+\frac{|I|}{q}-(h-1)}\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}^{h}$
$\displaystyle\leq$ $\displaystyle\,C\,N^{\frac{1}{p}}\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert^{h}_{\ell^{q}}\,,$

since $|I|\leq h-1$. Therefore, in light of the first inequality of Lemma 3.2,
it remains to show that

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{f(\boldsymbol{x})g^{\otimes
h}(\boldsymbol{y})}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{j}|^{2}\Big{)}^{h-1}}\leq
C\,p\,N^{\frac{1}{p}}\,\left\lVert f\right\rVert_{\ell^{p}}\left\lVert
g\right\rVert^{h}_{\ell^{q}}\,.$ (3.27)

We can assume without loss of generality that $1\stackrel{{\scriptstyle
I}}{{\sim}}2$, that is $x_{1}=x_{2}$. We multiply and divide by the factor
$\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}\Big{)}^{\frac{1}{q}}$
in (3.27) and apply Hölder’s inequality, namely

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{f(\boldsymbol{x})g^{\otimes
h}(\boldsymbol{y})}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}$
$\displaystyle\leq$
$\displaystyle\Bigg{(}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{|f(\boldsymbol{x})|^{p}\,\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\Bigg{)}^{\frac{1}{p}}\,\,\,$
$\displaystyle\qquad\qquad\qquad\times\Bigg{(}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{|g^{\otimes
h}(\boldsymbol{y})|^{q}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}\Bigg{)}^{\frac{1}{q}}\,.$
(3.28)

By triangle inequality and using that $x_{1}=x_{2}$ we have that

$\displaystyle\displaystyle|x_{1}-y_{1}|^{2}+|x_{2}-y_{2}|^{2}\geq\frac{|y_{1}-y_{2}|^{2}+|x_{2}-y_{2}|^{2}}{4}\,,$

therefore

$\displaystyle\Bigg{(}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{|g^{\otimes
h}(\boldsymbol{y})|^{q}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}\Bigg{)}^{\frac{1}{q}}$
$\displaystyle\leq$
$\displaystyle\,4^{\frac{h-1}{q}}\Bigg{(}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{|g^{\otimes
h}(\boldsymbol{y})|^{q}}{\Big{(}1+|y_{1}-y_{2}|^{2}+\sum_{i=2}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}\Bigg{)}^{\frac{1}{q}}\,.$
(3.29)

We sum the $x_{i}$ variables for $i>2$ successively, so that by inequality
(A.1) of Lemma A.1,

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}:\,(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{1}{\Big{(}1+|y_{1}-y_{2}|^{2}+\sum_{i=2}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}$
$\displaystyle\leq$
$\displaystyle\,c^{|I|-1}\frac{1}{\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}\sum_{\begin{subarray}{c}x_{2}\in\mathbb{Z}^{2}\\\
|x_{2}-y_{2}|\leq
C_{0}\sqrt{N}\end{subarray}}\frac{1}{\big{(}1+|y_{1}-y_{2}|^{2}+|x_{2}-y_{2}|^{2}\big{)}^{h-|I|}}\,.$
(3.30)

We also note that since $|I|\leq h-1$,

$\displaystyle\sum_{\begin{subarray}{c}x_{2}\in\mathbb{Z}^{2}\\\
|x_{2}-y_{2}|\leq
C_{0}\sqrt{N}\end{subarray}}\frac{1}{\big{(}1+|y_{1}-y_{2}|^{2}+|x_{2}-y_{2}|^{2}\big{)}^{h-|I|}}$
$\displaystyle\leq\sum_{\begin{subarray}{c}x_{2}\in\mathbb{Z}^{2}\\\
|x_{2}-y_{2}|\leq
C_{0}\sqrt{N}\end{subarray}}\frac{1}{1+|y_{1}-y_{2}|^{2}+|x_{2}-y_{2}|^{2}}$
$\displaystyle\leq\,c\log\bigg{(}1+\frac{C_{0}^{2}N}{1+|y_{1}-y_{2}|^{2}}\bigg{)}\,,$
(3.31)

where the last inequality in (Proof.) follows from inequality (A.10) of Lemma
A.2. Thus, taking into account (Proof.) and (Proof.) we deduce that

$\displaystyle\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}:\,(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{1}{\Big{(}1+|y_{1}-y_{2}|^{2}+\sum_{i=2}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}\leq
c^{|I|}\leq c^{h-1}\,,$

since $|I|\leq h-1$. By (Proof.) we obtain that

$\displaystyle\Bigg{(}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{|g^{\otimes
h}(\boldsymbol{y})|^{q}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}}\Bigg{)}^{\frac{1}{q}}$
$\displaystyle\leq\,(4c)^{\frac{h-1}{q}}\Bigg{(}\sum_{\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}|g^{\otimes
h}(\boldsymbol{y})|^{q}\Bigg{)}^{\frac{1}{q}}$
$\displaystyle=(4c)^{\frac{h-1}{q}}\left\lVert
g\right\rVert^{h}_{\ell^{q}}\,.$ (3.32)

On the other hand, for the first term in (Proof.), using that $x_{1}=x_{2}$,
by (A.1) of Lemma A.1, we have that

$\displaystyle\sum_{\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}}\frac{\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\leq
c^{h-2}\sum_{\begin{subarray}{c}y_{1},y_{2}\in\mathbb{Z}^{2}\\\
|y_{1}-x_{1}|,|y_{2}-x_{1}|\leq
C_{0}\sqrt{N}\end{subarray}}\frac{\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{\Big{(}1+|x_{1}-y_{1}|^{2}+|x_{1}-y_{2}|^{2}\Big{)}}\,.$
(3.33)

We make the change of variables $w_{1}:=y_{1}-y_{2}$ and
$w_{2}:=y_{1}+y_{2}-2x_{1}$, so that $|w_{1}|,|w_{2}|\leq 2C_{0}\sqrt{N}$ and
$|w_{1}|^{2}+|w_{2}|^{2}=2|y_{1}-x_{1}|^{2}+2|y_{2}-x_{1}|^{2}$. Note that
then,

$\displaystyle c^{h-2}\sum_{\begin{subarray}{c}y_{1},y_{2}\in\mathbb{Z}^{2}\\\
|y_{1}-x_{1}|,|y_{2}-x_{1}|\leq
C_{0}\sqrt{N}\end{subarray}}\frac{\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{\Big{(}1+|x_{1}-y_{1}|^{2}+|x_{1}-y_{2}|^{2}\Big{)}}\leq\,2c^{h-2}\sum_{\begin{subarray}{c}w_{1},w_{2}\in\mathbb{Z}^{2}\\\
|w_{1}|,|w_{2}|\leq
2\,C_{0}\sqrt{N}\end{subarray}}\frac{\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|w_{1}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{1+|w_{1}|^{2}+|w_{2}|^{2}}\,.$

Next, we sum over $w_{2}$ and use inequality (A.10) of Lemma A.2 to obtain

$\displaystyle\,2c^{h-2}\sum_{\begin{subarray}{c}w_{1},w_{2}\in\mathbb{Z}^{2}\\\
|w_{1}|,|w_{2}|\leq
2\,C_{0}\sqrt{N}\end{subarray}}\frac{\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|w_{1}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{1+|w_{1}|^{2}+|w_{2}|^{2}}\leq\,2c^{h-1}\sum_{\begin{subarray}{c}w_{1}\in\mathbb{Z}^{2}\\\
|w_{1}|\leq
2\,C_{0}\sqrt{N}\end{subarray}}\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|w_{1}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}+1}\,.$
(3.34)

By (A.12) of Lemma A.2 and noting that $\tfrac{p}{q}+1=p$ we have

$\displaystyle\sum_{\begin{subarray}{c}w_{1}\in\mathbb{Z}^{2}\\\ |w_{1}|\leq
2C_{0}\sqrt{N}\end{subarray}}\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|w_{1}|^{2}}\Big{)}\Big{)}^{p}\leq
c\,C_{0}^{2}\,N\,p^{p}\,.$ (3.35)

Therefore, by (3.33), (3.34) and (3.35) we have that

$\displaystyle\Bigg{(}\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{|f(\boldsymbol{x})|^{p}\,\Big{(}\log\Big{(}1+\frac{C^{2}_{0}N}{1+|y_{1}-y_{2}|^{2}}\Big{)}\Big{)}^{\frac{p}{q}}}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{i}|^{2}\Big{)}^{h-1}}\Bigg{)}^{\frac{1}{p}}\leq$
$\displaystyle\Big{(}2\,c^{h}\,C_{0}^{2}\Big{)}^{\frac{1}{p}}\,N^{\frac{1}{p}}\,p\,\Bigg{(}\sum_{\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\Bigg{)}^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\Big{(}2\,c^{h}\,C_{0}^{2}\Big{)}^{\frac{1}{p}}\,N^{\frac{1}{p}}\,p\,\left\lVert
f\right\rVert_{\ell^{p}}\,.$ (3.36)

Taking into account (Proof.), (Proof.) and (Proof.) we obtain that there
exists $C=C(h,w)\in(0,\infty)$ such that

$\displaystyle\sum_{(\boldsymbol{x},\boldsymbol{y})\in
E_{N}}\frac{f(\boldsymbol{x})g^{\otimes
h}(\boldsymbol{y})}{\Big{(}1+\sum_{i=1}^{h}|x_{i}-y_{j}|^{2}\Big{)}^{h-1}}\leq
C\,p\,N^{\frac{1}{p}}\,\left\lVert f\right\rVert_{\ell^{p}}\left\lVert
g\right\rVert^{h}_{\ell^{q}}\,,$

which concludes the proof of (3.27) and thus, the proof of Proposition 3.4. ∎

###### Proposition 3.5.

Let $p,q\in(1,\infty)$ such that $\frac{1}{p}+\frac{1}{q}=1$. There exists a
constant $C=C(h,\hat{\beta},w)\in(0,\infty)$, independent of $p$ and $q$, such
that for all $I\vdash\\{1,\dots,h\\}$ with $|I|=h-1$,

$\displaystyle\left\lVert\widehat{\mathsf{U}}_{N,0}^{I}\right\rVert_{\ell^{q}\to\ell^{q}}\leq
C\,.$

  * Proof.

Using (3.5) it suffices to prove that if
$f\in\ell^{p}\big{(}(\mathbb{Z}^{2})^{h}_{I}\big{)}$,
$g\in\ell^{q}\big{(}(\mathbb{Z}^{2})^{h}_{I}\big{)}$, then we have

$\displaystyle\sum_{\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}}f(\boldsymbol{x})\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w_{N}^{\otimes
h}(\boldsymbol{y})}g(\boldsymbol{y})\leq C\left\lVert
f\right\rVert_{\ell^{p}}\left\lVert g\right\rVert_{\ell^{q}}\,.$

By the Lipschitz condition on $\log w$ we first have

$\displaystyle\sum_{\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}}f(\boldsymbol{x})\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})\frac{w^{\otimes
h}_{N}(\boldsymbol{x})}{w_{N}^{\otimes
h}(\boldsymbol{y})}g(\boldsymbol{y})\leq\sum_{\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}}f(\boldsymbol{x})\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})e^{C_{w}\frac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}}g(\boldsymbol{y})\,,$

which by Hölder’s inequality is bounded by

$\displaystyle\Bigg{(}\sum_{\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}}|f(\boldsymbol{x})|^{p}\,\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})e^{C_{w}\frac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}}\Bigg{)}^{\frac{1}{p}}\cdot\Bigg{(}\sum_{\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}}|g(\boldsymbol{y})|^{q}\,\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})e^{C_{w}\frac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}}\Bigg{)}^{\frac{1}{q}}\,.$

Therefore, in order to conclude the proof of (3.5) it suffices to prove that
there exists a constant $C$ such that uniformly in
$\boldsymbol{x}\in(\mathbb{Z}^{2})^{h}_{I}$,

$\displaystyle\sum_{\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}}\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})e^{C_{w}\frac{|\boldsymbol{x}-\boldsymbol{y}|}{\sqrt{N}}}\leq
C\,.$ (3.37)

Recall from (3.11) that if $I$ is of the form
$I=\\{k,\ell\\}\sqcup\bigsqcup_{j\neq k,\ell}\\{j\\}$ then for
$\boldsymbol{x},\boldsymbol{y}\in(\mathbb{Z}^{2})^{h}_{I}$ the operator
$\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})$ is defined as

$\displaystyle\mathsf{U}^{I}_{N,0}(\boldsymbol{x},\boldsymbol{y})=\sum_{n=0}^{2N}\mathsf{U}^{I}_{n}(\boldsymbol{x},\boldsymbol{y})=\mathbb{1}_{\\{\boldsymbol{x},\boldsymbol{y}\sim
I\\}}\cdot\sum_{n=0}^{2N}U^{\beta_{N}}_{N}(n,y_{k}-x_{k})\cdot\prod_{i\neq
k,\ell}q_{n}(y_{i}-x_{i})\,.$

Therefore, in view of (3.37), we shall prove that uniformly in $0\leq n\leq
2N$,

$\displaystyle\sum_{z\in\mathbb{Z}^{2}}U^{\beta_{N}}_{N}(n,z)e^{C_{w}\frac{|z|}{\sqrt{N}}}\leq
C\,U^{\beta_{N}}_{N}(n)$ (3.38)

and

$\displaystyle\sum_{z\in\mathbb{Z}^{2}}q_{n}(z)e^{C_{w}\frac{|z|}{\sqrt{N}}}\leq
C\,q_{n}(z)\,.$ (3.39)

Inequality (3.39) follows easily by the local CLT, see [LL10] and Gaussian
concentration. For the sake of the presentation, we will prove (3.38) for
$0\leq n\leq N$, that is,

$\displaystyle\sum_{z\in\mathbb{Z}^{2}}U^{\beta_{N}}_{N}(n,z)e^{C_{w}\frac{|z|}{\sqrt{N}}}\leq
C\,U^{\beta_{N}}_{N}(n)\,,\qquad\forall\,\,0\leq n\leq N\,.$ (3.40)

Note that, by (2.13) we have,

$\displaystyle\sum_{n=0}^{N}U^{\beta_{N}}_{N}(n)\leq{\mathbb{E}}\Big{[}(Z_{N+1}^{\beta_{N}})^{2}\Big{]}\leq\frac{C}{1-\hat{\beta}^{2}}\,.$
(3.41)

Moreover, following the renewal framework we introduced in Section 2, we have

$\displaystyle\sum_{z\in\mathbb{Z}^{2}}U^{\beta_{N}}_{N}(n,z)e^{C_{w}\frac{|z|}{\sqrt{N}}}=$
$\displaystyle\sum_{k\geq
0}{\hat{\beta}}^{2k}\,\mathrm{E}\bigg{[}e^{C_{w}\frac{|S^{(N)}_{k}|}{\sqrt{N}}};\,\tau_{k}^{(N)}=n\bigg{]}$
(3.42) $\displaystyle=$ $\displaystyle\sum_{k\geq
0}{\hat{\beta}}^{2k}\sum_{n_{1}+\dots+n_{k}=n}\,\mathrm{E}\bigg{[}e^{C_{w}\frac{|S^{(N)}_{k}|}{\sqrt{N}}}\,\Big{|}\,\mathsf{t}^{(N)}_{i}=n_{i}\,,1\leq
i\leq
k\bigg{]}\prod_{i=1}^{k}\mathrm{P}\big{(}\mathsf{t}_{i}^{(N)}=n_{i}\big{)}\,.$

Therefore, in order to establish (3.40) it suffices to prove that there exists
$C\in(0,\infty)$, such that for all $k\geq 1$,

$\displaystyle\mathrm{E}\bigg{[}e^{C_{w}\frac{|S^{(N)}_{k}|}{\sqrt{N}}}\,\Big{|}\,\mathsf{t}^{(N)}_{i}=n_{i}\,,1\leq
i\leq k\bigg{]}\leq C\,.$ (3.43)

We note that when we condition on the times
$\big{(}\mathsf{t}^{(N)}_{i}\big{)}_{1\leq i\leq k}$, the space increments
$\big{(}\mathsf{x}^{(N)}_{i}\big{)}_{1\leq i\leq k}$ are independent with
distribution

$\displaystyle\mathrm{P}\big{(}\mathsf{x}^{(N)}_{1}=x\,\big{|}\,\mathsf{t}_{1}^{(N)}=n_{1}\big{)}=\frac{q_{n_{1}}^{2}(x)}{q_{2n_{1}}(0)}\,\mathbb{1}_{\\{n_{1}\leq
N\\}}\,.$

Let $\lambda\geq 0$ and $(\xi_{i})_{1\leq i\leq k}$ independent random
variables such that
$\xi_{i}\stackrel{{\scriptstyle\text{law}}}{{=}}\mathsf{x}_{i}^{(N)}\big{|}\,\mathsf{t}^{(N)}_{i}=n_{i}$.
We will show that

$\displaystyle\mathrm{E}\Big{[}e^{\lambda|\sum_{i=1}^{k}{\xi_{i}}|}\,\Big{]}\leq
2e^{4c\lambda^{2}n}\,,$

for some $c>0$. Therefore, taking $\lambda=\frac{C_{w}}{\sqrt{N}}$ will lead
to (3.43). To this end, for each $1\leq i\leq k$, let
$\xi_{i,1},\xi_{i,2}\in\mathbb{Z}$ be the two components of
$\xi_{i}\in\mathbb{Z}^{2}$ .Then we can find $c>0$ such that

$\displaystyle\mathrm{E}\big{[}e^{\pm\,\lambda\,\xi_{i,j}}\big{]}\leq
e^{c\lambda^{2}n_{i}}$

for $j=1,2$, since by the local CLT we have

$\displaystyle\mathrm{P}(\xi_{i}=x)=\frac{q_{n_{i}}^{2}(x)}{q_{2n_{i}}(0)}\leq\Big{(}\frac{\sup_{x\in\mathbb{Z}^{2}}q_{n_{i}}(x)}{q_{2n_{i}}(0)}\Big{)}\,q_{n_{i}}(x)\leq
C^{\prime}q_{n_{i}}(x)\,$

and $q_{n_{i}}(x)=2(g_{{n_{i}}/2}(x)+o(1))$, thus $q_{n_{i}}$ has Gaussian
tail decay. By Cauchy-Schwarz we

$\displaystyle\mathrm{E}\Big{[}e^{\lambda|\sum_{i=1}^{k}{\xi_{i}}|}\,\Big{]}\leq\mathrm{E}\Big{[}e^{2\lambda|\sum_{i=1}^{k}\xi_{i,1}|}\Big{]}^{\frac{1}{2}}\Big{[}e^{2\lambda|\sum_{i=1}^{k}\xi_{i,2}|}\Big{]}^{\frac{1}{2}}\,.$

Also, by the inequality $e^{|x|}\leq e^{x}+e^{-x}$ and independence, we obtain
for $j=1,2$

$\displaystyle\mathrm{E}\Big{[}e^{2\lambda|\sum_{i=1}^{k}\xi_{i,j}|}\Big{]}^{\frac{1}{2}}\leq\Bigg{(}\prod_{i=1}^{k}\mathrm{E}[e^{2\lambda\xi_{i,j}}]+\prod_{i=1}^{k}\mathrm{E}[e^{-2\lambda\xi_{i,j}}]\Bigg{)}^{\frac{1}{2}}\leq\Big{(}2e^{4c\lambda^{2}n}\Big{)}^{\frac{1}{2}}\,,$

therefore,

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

Given the inequality above and choosing $\lambda=\frac{C_{w}}{\sqrt{N}}$ we
get that

$\displaystyle\mathrm{E}\bigg{[}e^{C_{w}\frac{|S^{(N)}_{k}|}{\sqrt{N}}}\,\Big{|}\,\mathsf{t}^{(N)}_{i}=n_{i}\,,1\leq
i\leq k\bigg{]}\leq 2e^{4c\,C_{w}^{2}}\,,$

since $1\leq n\leq N$. Therefore, recalling (3.41) and (3.42),we have

$\displaystyle\sum_{\begin{subarray}{c}z\in\mathbb{Z}^{2},\\\ 0\leq n\leq
N\end{subarray}}U^{\beta_{N}}_{N}(n,z)e^{C_{w}\frac{|z|}{\sqrt{N}}}\leq\,2e^{4c\,C_{w}^{2}}\,\sum_{n=0}^{N}U^{\beta_{N}}_{N}(n)\leq
2e^{4c\,C_{w}^{2}}\,{\mathbb{E}}\Big{[}({Z_{N+1}^{\beta_{N}}})^{2}\Big{]}\leq
C\,,$

for a constant $C=C(h,\hat{\beta},w)\in(0,\infty)$. ∎

##  4\. Proofs of Theorems 1.1, 1.4, 1.5 and 1.3.

We are now in a position to prove the main results. We begin with Theorem 1.5.

  * Proof of Theorem 1.5.

We first prove (1.9). Recall from (3.16) that

$\displaystyle\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq$
$\displaystyle\frac{c\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h+1}}\sum_{k\geq
1}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\left\lVert\widehat{\mathsf{Q}}^{*;I_{1}}_{N,0}\frac{\varphi_{N}^{\otimes
h}}{w^{\otimes
h}_{N}}\right\rVert_{\ell^{p}}\prod_{i=2}^{k}\,\left\lVert\widehat{\mathsf{P}}_{N,\hat{\beta}}^{I_{i-1};I_{i}}\right\rVert_{\ell^{q}\to\ell^{q}}\,\left\lVert\widehat{\mathsf{Q}}_{N,0}^{I_{k};*}w_{N}^{\otimes
h}\right\rVert_{\ell^{q}}\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\,.$
(4.1)

By Proposition 3.4, we have the following bounds on the boundary operator
norms

$\left\lVert\widehat{\mathsf{Q}}^{*;I_{1}}_{N,0}\frac{\varphi_{N}^{\otimes
h}}{w^{\otimes h}_{N}}\right\rVert_{\ell^{p}}\leq
C\,q\,N^{\frac{1}{q}}\,\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\quad\text{
and }\quad\left\lVert\widehat{\mathsf{Q}}_{N,0}^{I_{k};*}w_{N}^{\otimes
h}\right\rVert_{\ell^{q}}\leq C\,p\,N^{\frac{1}{p}}\,\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\,,$ (4.2)

for a constant $C=C(h,w)\in(0,\infty)$. By Propositions 3.3 and 3.5 we also
have that for all $2\leq i\leq k$, there exists a constant
$C=C(h,\hat{\beta},w)\in(0,\infty)$, such that

$\left\lVert\widehat{\mathsf{P}}_{N,\hat{\beta}}^{I_{i-1};I_{i}}\right\rVert_{\ell^{q}\to\ell^{q}}\leq
C\,p\,q\,.$ (4.3)

By inserting the bounds (4.2) and (4.3) in (4.1) we obtain that

$\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq\,\frac{\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h}}\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\sum_{k\geq
1}(C\,p\,q)^{k}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\,\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\,.$
(4.4)

We now distinguish two cases depending on the range of $k$.

(Case 1). If $k>\lfloor\tfrac{h}{2}\rfloor$ we use the bound

$\prod_{i=1}^{k}{\mathbb{E}}\big{[}|\xi|^{I_{i}}\big{]}\leq\big{(}\tfrac{C}{\log
N}\big{)}^{k}\,,$

which is a consequence of the fact that
${\mathbb{E}}\big{[}|\xi|^{I_{i}}\big{]}\leq
C\,\sigma^{2}_{N,\hat{\beta}}=O(1/\log N)$, see (3.8) and (2.7). Therefore, in
this case

$\sum_{k>\lfloor\frac{h}{2}\rfloor}(C\,p\,q)^{k}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\,\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\leq\sum_{k>\lfloor\frac{h}{2}\rfloor}\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{k}\,,$ (4.5)

for a constant $\widetilde{C}=\widetilde{C}(h,\hat{\beta},w)\in(0,\infty)$,
which also incorporates the fact that the number of possible choices for a
sequence of partitions $(I_{1},\dots,I_{k})$ is bounded by $C^{k}$ where
$C=C(h)$ is some positive constant.

(Case 2). The second case is when $1\leq k\leq\lfloor\tfrac{h}{2}\rfloor$, for
which we claim that there exists a constant $C=C(h,\hat{\beta})\in(0,\infty)$
such that

$\prod_{i=1}^{k}{\mathbb{E}}\big{[}|\xi|^{I_{i}}\big{]}\leq C^{k}\,(\log
N)^{-\frac{h}{2}}\,.$

To see this fix $1\leq k\leq\lfloor\tfrac{h}{2}\rfloor$ and
$(I_{1},\dots,I_{k})\in\mathcal{I}$, and let $I_{i}=\bigsqcup_{1\leq
j\leq|I_{i}|}I_{i,j}$. By (3.8) and (2.7), we have that

$\prod_{i=1}^{k}{\mathbb{E}}\big{[}|\xi|^{I_{i}}\big{]}\leq
C^{k}\,(\sigma_{N,\hat{\beta}})^{\sum_{1\leq i\leq k}\sum_{1\leq
j\leq|I_{i}|;|I_{i,j}|\geq 2}|I_{i,j}|}\,.$

From the definition of $\mathcal{I}$ (see below (3.1)), we have that

$\sum_{1\leq i\leq k}\sum_{1\leq j\leq|I_{i}|;|I_{i,j}|\geq 2}|I_{i,j}|\geq
h\,,$

since every $r\in\\{1,\dots,h\\}$ necessarily belongs to a non-trivial block
of some partition $I_{i},1\leq i\leq k$, see the discussion below (3.1).
Therefore, as in the derivation of (4.5), we have that there exists a constant
$\widetilde{C}=\widetilde{C}(h,\hat{\beta},w)\in(0,\infty)$ such that

$\sum_{1\leq
k\leq\lfloor\frac{h}{2}\rfloor}(C\,p\,q)^{k}\sum_{(I_{1},\dots,I_{k})\in\mathcal{I}}\,\prod_{i=1}^{k}{\mathbb{E}}\Big{[}|\xi|^{I_{i}}\Big{]}\leq(\log
N)^{-\frac{h}{2}}\sum_{1\leq
k\leq\lfloor\frac{h}{2}\rfloor}(\widetilde{C}\,p\,q)^{k}\,.$ (4.6)

Combining estimates (4.5) and (4.6) we deduce from (4.4) that

$\begin{split}\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq\,C\,\frac{\left\lVert\psi\right\rVert^{h}_{\infty}}{N^{h}}\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\Bigg{(}\sum_{k>\lfloor\frac{h}{2}\rfloor}\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{k}+(\log N)^{-\frac{h}{2}}\sum_{1\leq
k\leq\lfloor\frac{h}{2}\rfloor}(\widetilde{C}\,p\,q)^{k}\Bigg{)}\,.\end{split}$
(4.7)

Let $p,q>1$, conjugate exponents, that satisfy the growth condition

$\frac{\widetilde{C}\,p\,q}{\log N}<\frac{1}{2}\,.$ (4.8)

In particular, $p\,q\leq\mathsf{a}_{*}\log N$ with
$\mathsf{a}_{*}=\mathsf{a}_{*}(h,\hat{\beta},w)\in(0,1)$ defined as
$\mathsf{a}_{*}:=(2\widetilde{C})^{-1}$. We then have that

$\sum_{k>\lfloor\frac{h}{2}\rfloor}\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{k}\leq 2\,\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{\lfloor\frac{h}{2}\rfloor+1}$ (4.9)

by summing the tail of the geometric series, which is possible due to the
growth condition (4.8) imposed on $p,q$. On the other hand, we have that

$\displaystyle(\log N)^{-\frac{h}{2}}\sum_{1\leq
k\leq\lfloor\frac{h}{2}\rfloor}(\widetilde{C}\,p\,q)^{k}$
$\displaystyle\leq(\log
N)^{-\frac{h}{2}}\cdot\frac{(\widetilde{C}\,p\,q)^{\lfloor\frac{h}{2}\rfloor+1}-\widetilde{C}\,p\,q}{\widetilde{C}\,p\,q-1}$
$\displaystyle\leq(\log
N)^{-\frac{h}{2}}\cdot\frac{(\widetilde{C}\,p\,q)^{\lfloor\frac{h}{2}\rfloor+1}}{\widetilde{C}\,p\,q-1}$
$\displaystyle\leq 2(\log
N)^{-\frac{h}{2}}\,(\widetilde{C}\,p\,q)^{\lfloor\frac{h}{2}\rfloor}\,,$
(4.10)

since $\widetilde{C}\,p\,q-1>(\widetilde{C}\,p\,q)/2$, ($pq\geq 4$ because
$\tfrac{1}{p}+\tfrac{1}{q}=1$ and we can choose $\widetilde{C}>1$). Combining
estimates (4.9) and (4) we obtain that

$\begin{split}\Bigg{(}\sum_{k>\lfloor\frac{h}{2}\rfloor}\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{k}+(\log N)^{-\frac{h}{2}}\sum_{1\leq
k\leq\lfloor\frac{h}{2}\rfloor}(\widetilde{C}\,p\,q)^{k}\Bigg{)}&\leq
2\,\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{\lfloor\frac{h}{2}\rfloor+1}+2(\log
N)^{-\frac{h}{2}}\,(\widetilde{C}\,p\,q)^{\lfloor\frac{h}{2}\rfloor}\\\ &\leq
4\,\Big{(}\frac{\widetilde{C}\,p\,q}{\log
N}\Big{)}^{\frac{h}{2}}\,,\end{split}$

by using that $\tfrac{\widetilde{C}\,p\,q}{\log N}\leq\tfrac{1}{2}$ and
$\lfloor\tfrac{h}{2}\rfloor\leq\tfrac{h}{2}<\lfloor\tfrac{h}{2}\rfloor+1$.
Inserting this bound to (4.7) we finally obtain that

$\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq\,\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{\frac{h}{2}}\,\frac{1}{N^{h}}\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert\psi\right\rVert^{h}_{\infty}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\,,$ (4.11)

for a constant $C=C(h,\hat{\beta},w)>\widetilde{C}$, which establishes (1.9).

Let us now prove (1.10). By choosing
$\varphi:=\delta_{0}^{\scriptscriptstyle(N)}:=N\,\mathbb{1}_{\\{x=0\\}}$,
$\psi\equiv 1$ and $w(x)=e^{-|x|}$, we deduce from (1.9) that

$\displaystyle\Big{|}{\mathbb{E}}\big{[}(\widebar{Z}_{N}^{\beta_{N}})^{h}\big{]}\Big{|}\leq\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{\frac{h}{2}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}=\Big{(}\frac{C\,p\,q}{\log
N}\Big{)}^{\frac{h}{2}}\cdot
N^{\frac{h}{q}}\cdot\frac{1}{N^{\frac{h}{q}}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\,.$ (4.12)

Since $w(x)=e^{-|x|}$ is decreasing in the radial direction we have

$\displaystyle\frac{1}{N^{\frac{h}{q}}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\leq\bigg{(}\frac{1}{N}+\frac{1}{N}\int_{\mathbb{R}^{2}}\,e^{-q\frac{|x|}{\sqrt{N}}}\,\mathrm{d}x\bigg{)}^{\frac{h}{q}}=\bigg{(}\frac{1}{N}+\int_{\mathbb{R}^{2}}\,e^{-q|x|}\,\mathrm{d}x\bigg{)}^{\frac{h}{q}}=\bigg{(}\frac{1}{N}+\frac{2\pi}{q^{2}}\bigg{)}^{\frac{h}{q}}\leq
e^{(2\pi h)/q^{3}}\,.$ (4.13)

We choose $q=q_{N}:=\mathsf{a}\log N$ with
$\mathsf{a}=\mathsf{a}(h,\hat{\beta},w)\in(0,1)$ small enough such that
$\tfrac{C\,p\,q}{\log N}<\tfrac{1}{2}$ (and therefore (4.8) is satisfied). For
this choice of $q$ we have by (4.13) that

$\displaystyle\frac{1}{N^{\frac{h}{q}}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\leq\,e^{O\big{(}(\log N)^{-3}\big{)}}\leq
C\,.$ (4.14)

Furthermore, again with $q=q_{N}=\mathsf{a}\log N$ and thus $p=p_{N}=1+o(1)$,
since $\frac{1}{p}+\frac{1}{q}=1$, we get

$\displaystyle\Big{(}\frac{C\,p\,q}{\log N}\Big{)}^{\frac{h}{2}}\cdot
N^{\frac{h}{q}}\leq
2^{-\frac{h}{2}}\,\exp\big{(}\tfrac{h}{\mathsf{a}}\big{)}<\infty\,,$ (4.15)

since $\tfrac{C\,p\,q}{\log N}<\tfrac{1}{2}$. We note that the parameter
$\mathsf{a}=\mathsf{a}(h,\hat{\beta},w)$ on the right-hand side of (4.15)
depends non-trivially on $h$, and therefore the order of the bound in (4.15)
is not just exponential in $h$. However, we also note that the dependence on
$h$ deduced from (4.15) does not capture the true growth of the moments, which
is as in (1.6). Finally, by (4.12), (4.14) and (4.15), we obtain that

$\displaystyle\sup_{N\in\mathbb{N}}{\mathbb{E}}\big{[}(\widebar{Z}^{\beta_{N}}_{N})^{h}\big{]}<\infty\,.$

∎

  * Proof of Theorem 1.1.

By binomial expansion, for $h\in\mathbb{N}$ we have that

$\displaystyle{\mathbb{E}}\big{[}(Z^{\beta_{N}}_{N})^{h}\big{]}=\sum_{k=0}^{h}\binom{h}{k}{\mathbb{E}}\big{[}(\widebar{Z}^{\beta_{N}}_{N})^{k}\big{]}\leq\sum_{k=0}^{h}\binom{h}{k}\Big{|}{\mathbb{E}}\big{[}(\widebar{Z}^{\beta_{N}}_{N})^{k}\big{]}\Big{|}\,.$

Therefore, by estimate (1.10) of Theorem 1.5, for every $h\geq 3$ we obtain
that
$\sup_{N\in\mathbb{N}}{\mathbb{E}}\big{[}(Z^{\beta_{N}}_{N})^{h}\big{]}<\infty$.
Hence, for every $h\geq 0$ the sequence
$\Big{\\{}(Z^{\beta_{N}}_{N})^{h}\Big{\\}}_{N\geq 1}$ is uniformly integrable
and therefore, by Theorem 2.1 for every $h\geq 0$,

$\displaystyle\lim_{N\to\infty}{\mathbb{E}}\big{[}(Z^{\beta_{N}}_{N})^{h}\big{]}={\mathbb{E}}\big{[}\exp\big{(}\varrho_{\hat{\beta}}\,h\,\mathsf{X}-\tfrac{1}{2}\varrho^{2}_{\hat{\beta}}\,h\big{)}\big{]}=\exp\Big{(}\tfrac{h(h-1)}{2}\,\varrho^{2}_{\hat{\beta}}\Big{)}=\bigg{(}\frac{1}{1-\hat{\beta}^{2}}\bigg{)}^{\frac{h(h-1)}{2}}\,.$

As can be seen in [CSZ20], section 3, (1.7) implies that for all $h>0$,

$\displaystyle\sup_{N\in\mathbb{N}}{\mathbb{E}}\big{[}(Z^{\beta_{N}}_{N})^{-h}\big{]}<\infty,$

which in combination with Theorem 2.1 implies the convergence of negative
moments. ∎

  * Proof of Theorem 1.4.

We note that if we choose the law of the environment $\omega$ to be Gaussian,
i.e. $\omega\sim{\mathcal{N}}(0,1)$, then for $h\in\mathbb{N}$

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

Therefore, by Theorem 1.1 we have that

$\displaystyle\mathrm{E}^{\otimes
h}\bigg{[}\exp\Big{(}\frac{\hat{\beta}^{2}\,\pi}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}^{(i,j)}_{N}\Big{)}\bigg{]}\xrightarrow{N\to\infty}\bigg{(}\frac{1}{1-\hat{\beta}^{2}}\bigg{)}^{\frac{h(h-1)}{2}}\,,$
(4.16)

for all $\hat{\beta}\in[0,1)$. The right-hand side of (4.16) is equal to
$M_{Y}(\hat{\beta}^{2})$, where $M_{Y}(t):=\mathrm{E}[e^{tY}]$ denotes the
moment generating function of a random variable $Y$ with law
$\Gamma\big{(}\frac{h(h-1)}{2},1\big{)}$. By exercise 9, chapter 4 in [K97],
(4.16) implies the convergence of $\frac{\pi}{\log N}\sum_{1\leq i<j\leq
h}\mathsf{L}^{(i,j)}_{N}$ in law, to a
$\Gamma\big{(}\frac{h(h-1)}{2},1\big{)}$ distribution. ∎

  * Proof of Theorem 1.3.

We are going to show that for all $h\in\mathbb{N}$ with $h\geq 3$ we have that

$\displaystyle\sup_{N\in\mathbb{N}}\,(\log
N)^{\frac{h}{2}}\,|M^{\varphi,\psi}_{N,h}|<\infty\,.$ (4.17)

In that case we obtain uniform integrability of $(\log
N)^{\frac{h}{2}}\cdot\big{(}\widebar{Z}^{\beta_{N}}_{N}(\varphi,\psi)\big{)}^{h}$
for all $h\in\mathbb{N}$ and the convergence of moments in Theorem 1.3 follows
by Theorem 2.2. But, (4.17) is an immediate consequence of (1.9) of Theorem
1.5. Indeed, let us fix $p,q\in(1,\infty)$ such that
$\frac{1}{p}+\frac{1}{q}=1$. By (1.9) of Theorem 1.5 we have that

$\displaystyle(\log
N)^{\frac{h}{2}}\big{|}M^{\varphi,\psi}_{N,h}\big{|}\leq\,(C\,p\,q)^{\frac{h}{2}}\frac{1}{N^{h}}\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\left\lVert\psi_{N}\right\rVert^{h}_{\infty}.$
(4.18)

Furthermore, by Riemann approximation we have that

$\displaystyle\frac{1}{N^{h}}\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\left\lVert\psi_{N}\right\rVert^{h}_{\infty}=\frac{1}{N^{\frac{h}{p}}}\left\lVert\frac{\varphi_{N}}{w_{N}}\right\rVert^{h}_{\ell^{p}}\frac{1}{N^{\frac{h}{q}}}\left\lVert
w_{N}\right\rVert^{h}_{\ell^{q}}\left\lVert\psi_{N}\right\rVert^{h}_{\infty}\leq\,C\,\left\lVert\frac{\varphi}{w}\right\rVert^{h}_{L^{p}}\left\lVert
w\right\rVert^{h}_{L^{q}}\left\lVert\psi\right\rVert^{h}_{\infty}\,.$ (4.19)

Therefore, by (4.18) and (4.19) we obtain that

$\displaystyle\sup_{N\in\mathbb{N}}\,(\log
N)^{\frac{h}{2}}\,|M^{\varphi,\psi}_{N,h}|<\infty\,,$

which concludes the proof. ∎

##  Appendix A Some technical estimates

We state here the integral estimates we used for proving Propositions 3.3 and
3.4.

###### Lemma A.1.

Let $\lambda\geq 1$, $p>1$, $a<\tfrac{1}{p}$. Then,

$\displaystyle\,\,\,\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}}\leq\frac{c}{\lambda^{r-1}}$
$\displaystyle\quad\text{ if }r\geq 2\,,$ (A.1)
$\displaystyle\,\,\,\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}\big{(}1+|y|^{2a}\big{)}^{p}}\leq\,\frac{c}{ap\,(1-ap)\lambda^{r-1+ap}}$
$\displaystyle\quad\text{ if }r\geq 1\,,$ (A.2)

for a constant $c\in(0,\infty)$, that does not depend on $\lambda,p,a$ or $r$.

  * Proof.

We note that since $\displaystyle
y\mapsto\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}}$ and $\displaystyle
y\mapsto\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}\big{(}1+|y|^{2a}\big{)}^{p}}$
are decreasing in the radial direction we have that

$\displaystyle\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}}\leq\frac{1}{\lambda^{r}}+\int_{\mathbb{R}^{2}}\,\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}}\,\mathrm{d}y\,$
(A.3)

and

$\displaystyle\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}\big{(}1+|y|^{2a}\big{)}^{p}}\leq\frac{1}{\lambda^{r}}+\int_{\mathbb{R}^{2}}\,\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}|y|^{2ap}}\,\mathrm{d}y\,.$
(A.4)

In order to prove (A.1), we switch to polar coordinates in (A.3), so that

$\displaystyle\int_{\mathbb{R}^{2}}\,\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}}\,\mathrm{d}y=2\pi\int_{0}^{\infty}\frac{\varrho}{(\lambda+\varrho^{2})^{r}}\,\mathrm{d}\varrho\,=\pi\cdot\frac{(\lambda+\varrho^{2})^{1-r}}{1-r}\bigg{|}_{\varrho=0}^{\varrho=\infty}=\frac{\pi}{r-1}\,\frac{1}{\lambda^{r-1}}\,.$
(A.5)

Therefore, by (A.3) and (A.5) we get that

$\displaystyle\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}}\leq\frac{1}{\lambda^{r}}+\frac{\pi}{r-1}\frac{1}{\lambda^{r-1}}=\frac{1}{\lambda^{r-1}}\bigg{(}\frac{1}{\lambda}+\frac{\pi}{r-1}\bigg{)}\,.$

Thus, since $r\geq 2$ and $\lambda\geq 1$ we conclude (A.1) with $c=\pi+1$.

For (A.2) we split the integral in (A.4) into two regions,

$\displaystyle\int_{\mathbb{R}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}|y|^{2ap}}\,\mathrm{d}y\,=\underbrace{\int_{|y|\leq\sqrt{\lambda}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}|y|^{2ap}}\,\mathrm{d}y\,}_{:=I_{1}}+\underbrace{\int_{|y|>\sqrt{\lambda}}\,\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}|y|^{2ap}}\,\mathrm{d}y\,}_{:=I_{2}}\,.$

First,

$\displaystyle
I_{1}\leq\frac{1}{\lambda^{r}}\int_{|y|\leq\sqrt{\lambda}}\,\frac{1}{|y|^{2ap}}\,\mathrm{d}y=\frac{2\pi}{\lambda^{r}}\int_{0}^{\sqrt{\lambda}}\,\frac{1}{\varrho^{2ap-1}}\,\mathrm{d}\varrho=\frac{\pi}{\lambda^{r}}\frac{\lambda^{1-ap}}{1-ap}=\frac{\pi}{1-ap}\frac{1}{\lambda^{r-1+ap}}\,.$

Similarly,

$\displaystyle
I_{2}\leq\int_{|y|>\sqrt{\lambda}}\,\frac{1}{|y|^{2r+2ap}}\mathrm{d}y=2\pi\int_{\sqrt{\lambda}}^{\infty}\frac{1}{\varrho^{2r+2ap-1}}\,\mathrm{d}\varrho\,=$
$\displaystyle\frac{\pi}{r-1+ap}\frac{-1}{\varrho^{2r+2ap-2}}\bigg{|}^{\varrho=\infty}_{\varrho=\sqrt{\lambda}}$
$\displaystyle=$
$\displaystyle\frac{\pi}{r-1+ap}\frac{1}{\lambda^{r-1+ap}}\,.$

Therefore,

$\displaystyle\int_{\mathbb{R}^{2}}\,\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}|y|^{2ap}}\,\mathrm{d}y\,=I_{1}+I_{2}\leq$
$\displaystyle\,\frac{\pi}{1-ap}\frac{1}{\lambda^{r-1+ap}}+\frac{\pi}{r-1+ap}\frac{1}{\lambda^{r-1+ap}}$
$\displaystyle=$
$\displaystyle\,\frac{\pi\,r}{(1-ap)(r-1+ap)}\frac{1}{\lambda^{r-1+ap}}\,.$
(A.6)

By (A.4) and (Proof.) we thus obtain

$\displaystyle\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}\big{(}1+|y|^{2a}\big{)}^{p}}\leq$
$\displaystyle\frac{1}{\lambda^{r}}+\int_{\mathbb{R}^{2}}\,\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}|y|^{2ap}}\,\mathrm{d}y$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{r}}+\frac{\pi\,r}{(1-ap)(r-1+ap)}\frac{1}{\lambda^{r-1+ap}}\,.$
(A.7)

Note that

$\displaystyle\frac{\pi\,r}{(1-ap)(r-1+ap)}\leq\frac{\pi}{(1-ap)ap}\,,$

since that inequality is equivalent to $(r-1)(ap-1)\leq 0$, which is valid
since we have assumed that $a\,p<1$ and $r\geq 2$. Therefore,

$\displaystyle\frac{1}{\lambda^{r}}+\frac{\pi\,r}{(1-ap)(r-1+ap)}\frac{1}{\lambda^{r-1+ap}}\leq\,\frac{1}{\lambda^{r}}+\frac{\pi}{(1-ap)ap}\frac{1}{\lambda^{r-1+ap}}=$
$\displaystyle\frac{1}{\lambda^{r-1+ap}}\bigg{(}\frac{1}{\lambda^{1-ap}}+\frac{\pi}{(1-ap)ap}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\lambda^{r-1+ap}}\bigg{(}1+\frac{\pi}{(1-ap)ap}\bigg{)}\,,$
(A.8)

since $\lambda\geq 1$ and $1-ap>0$, by assumption. Last, we have that

$\displaystyle
1+\frac{\pi}{(1-ap)ap}=\frac{\pi+ap(1-ap)}{ap(1-ap)}\leq\frac{1+\pi}{ap(1-ap)}\,.$
(A.9)

Hence, by (Proof.), (Proof.) and (A.9),

$\displaystyle\sum_{y\in\mathbb{Z}^{2}}\frac{1}{\big{(}\lambda+|y|^{2}\big{)}^{r}\big{(}1+|y|^{2a}\big{)}^{p}}\leq\frac{c}{(1-ap)\,ap\,\lambda^{r-1+ap}}\,,$

with $c=1+\pi$, thus concluding the proof of (A.2). ∎

###### Lemma A.2.

There exists a constant $c\in(0,\infty)$ such that uniformly in
$A,\lambda,p\geq 1$,

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\
|y|\leq\sqrt{A}\end{subarray}}\frac{1}{\lambda+|y|^{2}}\leq
c\log\Big{(}1+\frac{A}{\lambda}\Big{)}\,\,,$ (A.10)
$\displaystyle\bigg{(}\int_{1}^{A}\big{(}\log\big{(}\tfrac{A}{x}\big{)}\big{)}^{p}\,\mathrm{d}x\,\bigg{)}^{\frac{1}{p}}\leq
p\,A^{\frac{1}{p}}\,,\,$ (A.11)

and

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\ |y|\leq
2\sqrt{A}\end{subarray}}\Big{(}\log\Big{(}1+\frac{A}{1+|y|^{2}}\Big{)}\Big{)}^{p}\leq
c\,A\,p^{p}\,.$ (A.12)

  * Proof.

For (A.10), using the same reasoning as in the proof of Lemma A.1 we have

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\
|y|\leq\sqrt{A}\end{subarray}}\frac{1}{\lambda+|y|^{2}}\leq\frac{1}{\lambda}+\int_{|y|\leq\sqrt{A}}\,\frac{1}{\lambda+|y|^{2}}\,\mathrm{d}y\,.$
(A.13)

Switching to polar coordinates in (A.13) we have

$\displaystyle\int_{|y|\leq\sqrt{A}}\frac{1}{\lambda+|y|^{2}}\,\mathrm{d}y\,=2\pi\int_{0}^{\sqrt{A}}\frac{\varrho}{\lambda+\varrho^{2}}\,\mathrm{d}\varrho=\pi\,\log(\lambda+\varrho^{2})\Big{|}_{\varrho=0}^{\varrho=\sqrt{A}}=\pi\log\Big{(}1+\frac{A}{\lambda}\Big{)}\,.$
(A.14)

A simple computation shows that when $\lambda\geq 1$, one has that
$\frac{1}{\lambda}\leq 2\log\Big{(}1+\frac{1}{\lambda}\Big{)}\leq
2\log\Big{(}1+\frac{A}{\lambda}\Big{)}$, the latter following since $A\geq 1$,
by assumption. Therefore, by (A.13) and (A.14) we have that

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\
|y|\leq\sqrt{A}\end{subarray}}\frac{1}{\lambda+|y|^{2}}\leq\frac{1}{\lambda}+\int_{|y|\leq\sqrt{A}}\,\frac{1}{\lambda+|y|^{2}}\,\mathrm{d}y\leq(2+\pi)\log\Big{(}1+\frac{A}{\lambda}\Big{)}\,,$

which implies (A.10) with $c=2+\pi$.

Let us now prove (A.11) and (A.12). First, we prove (A.11). We have

$\displaystyle\frac{1}{A}\int_{1}^{A}\big{(}\log\big{(}\tfrac{A}{x}\big{)}\big{)}^{p}\,\mathrm{d}x\,\stackrel{{\scriptstyle
y=A\,e^{-u}}}{{=\joinrel=}}\int_{0}^{\log
A}e^{-u}\,u^{p}\,\mathrm{d}u\leq\,\Gamma(p+1)\leq p^{p}\,,$ (A.15)

since $\Gamma(p+1)\leq p^{p}$ for $p\geq 1$. After raising both sides of
(A.15) to the $\frac{1}{p}$ we get (A.11).

To prove (A.12) we first note that

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\ |y|\leq
2\sqrt{A}\end{subarray}}\Big{(}\log\Big{(}1+\frac{A}{1+|y|^{2}}\Big{)}\Big{)}^{p}\leq(\log(1+A))^{p}+\int_{|y|\leq
2\sqrt{A}}\,\Big{(}\log\Big{(}1+\frac{A}{1+|y|^{2}}\Big{)}\Big{)}^{p}\,\mathrm{d}y\,.$
(A.16)

Using polar coordinates in (A.16) we compute

$\displaystyle\int_{|y|\leq
2\sqrt{A}}\,\Big{(}\log\Big{(}1+\frac{A}{1+|y|^{2}}\Big{)}\Big{)}^{p}\,$
$\displaystyle\mathrm{d}y=2\pi\int_{0}^{2\sqrt{A}}\,\varrho\Big{(}\log\Big{(}1+\frac{A}{1+\varrho^{2}}\Big{)}\Big{)}^{p}\,\mathrm{d}\varrho\,$
$\displaystyle\stackrel{{\scriptstyle
u=1+\varrho^{2}}}{{=\joinrel=}}\,\pi\int_{1}^{1+4A}\Big{(}\log\Big{(}1+\frac{A}{u}\Big{)}\Big{)}^{p}\,\mathrm{d}u\,.$

Furthermore,

$\displaystyle\pi\int_{1}^{1+4A}\Big{(}\log\Big{(}1+\frac{A}{u}\Big{)}\Big{)}^{p}\,\mathrm{d}u\leq\pi\int_{1}^{1+4A}\Big{(}\log\Big{(}\frac{1+5A}{u}\Big{)}\Big{)}^{p}\,\mathrm{d}u\leq\,\pi\int_{1}^{1+5A}\Big{(}\log\Big{(}\frac{1+5A}{u}\Big{)}\Big{)}^{p}\mathrm{d}u\,.$

Note that by (A.11), we further have that

$\displaystyle\pi\int_{1}^{1+5A}\Big{(}\log\Big{(}\frac{1+5A}{u}\Big{)}\Big{)}^{p}\mathrm{d}u\leq(1+5A)\,\pi\,p^{p}\,\leq
6\,A\,\pi\,p^{p}\,,$

since $A\geq 1$. Combining this inequality with (A.16) we get that

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\ |y|\leq
2\sqrt{A}\end{subarray}}\Big{(}\log\Big{(}1+\frac{A}{1+|y|^{2}}\Big{)}\Big{)}^{p}\leq\log(1+A)^{p}+6\,A\,\pi\,p^{p}\,.$
(A.17)

We are going to prove that for all $A\geq 1$,

$\displaystyle(\log(1+A))^{p}\leq\frac{1}{\sqrt{e}-1}\,A\,p^{p}\,,$

thus deducing inequality (A.12), via (A.17), with
$c=\frac{1}{\sqrt{e}-1}+6\pi$. To this end, consider
$k_{p}(x):=\frac{(\log(1+x))^{p}}{x}$ for $x\geq 0$ and $p\geq 1$. We have
that

$\displaystyle
k^{\prime}_{p}(x):=\frac{(\log(1+x))^{p-1}}{x}\Big{(}\frac{p}{1+x}-\frac{\log(1+x)}{x}\Big{)}\,,$

therefore, $k_{p}$ is increasing in $[0,x_{p}]$ and decreasing in
$[x_{p},\infty)$, where $x_{p}\geq 0$ is the solution to the equation
$k^{\prime}_{p}(x_{p})=0$, or equivalently

$\displaystyle p=\frac{(1+x_{p})\,\log(1+x_{p})}{x_{p}}\,.$ (A.18)

By working with $g(x):=\frac{(1+x)\log(1+x)}{x}$, one can see that equation
(A.18) has a unique solution $x_{p}\geq 0$ for every $p\geq 1$, since
$g^{\prime}(x)>0$ for all $x>0$, $\lim_{x\downarrow 0}g(x)=1$ and
$\lim_{x\to\infty}g(x)=\infty$. We distinguish two cases:

Suppose first that $x_{p}\geq 1$. Then

$\displaystyle\log(1+x_{p})\leq p\leq 2\,\log(1+x_{p})\,,$ (A.19)

by (A.18) and since $x_{p}\geq 1$. Therefore, in this case, for all $x\geq 1$,

$\displaystyle\frac{(\log(1+x))^{p}}{x}=k_{p}(x)\leq
k_{p}(x_{p})=\frac{(\log(1+x_{p}))^{p}}{x_{p}}\leq\frac{p^{p}}{x_{p}}\leq\frac{p^{p}}{e^{\frac{p}{2}}-1}\,,$

where the last two inequalities follow by the first and second inequality in
(A.19), respectively. Since, $p\geq 1$ we have that
$e^{\frac{p}{2}}-1\geq\sqrt{e}-1$, thus we conclude that in the case where
$x_{p}\geq 1$ we have for all $x\geq 1$,

$\displaystyle
k_{p}(x)=\frac{(\log(1+x))^{p}}{x}\leq\frac{1}{\sqrt{e}-1}\,p^{p}\,.$ (A.20)

Moving to the second case, i.e. $0\leq x_{p}<1$, we have that since $k_{p}$ is
decreasing in $[1,\infty)\subset[x_{p},\infty)$, we have that for all $x\geq
1$,

$\displaystyle k_{p}(x)=\frac{(\log(1+x))^{p}}{x}\leq
k_{p}(1)=(\log(2))^{p}<1\,,$ (A.21)

since $p\geq 1$ and $\log 2<1$. Therefore, by (A.20), (A.21) and since $p\geq
1$, for all $x\geq 1$ and $p\geq 1$ we have that

$\displaystyle
k_{p}(x)\leq\max\Big{\\{}1,\frac{1}{\sqrt{e}-1}\Big{\\}}\,p^{p}=\frac{1}{\sqrt{e}-1}\,p^{p}\,.$
(A.22)

Recalling that $k_{p}(x)=\frac{(\log(1+x))^{p}}{x}$ and applying (A.22) to
(A.17) for $x=A\geq 1$ we get that

$\displaystyle\sum_{\begin{subarray}{c}y\in\mathbb{Z}^{2}\\\ |y|\leq
2\sqrt{A}\end{subarray}}\Big{(}\log\Big{(}1+\frac{A}{1+|y|^{2}}\Big{)}\Big{)}^{p}\leq(\log(1+A))^{p}+6\,A\,\pi\,p^{p}\leq\frac{1}{\sqrt{e}-1}\,A\,p^{p}+6\,A\,\pi\,p^{p}=c\,A\,p^{p}\,,$

with $c=\frac{1}{\sqrt{e}-1}+6\pi$, thus concluding the proof of (A.12). ∎

Acknowledgements. We would like to thank Francesco Caravenna and Rongfeng Sun
for many useful discussions. We would also like to thank Clément Cosco for
pointing to the problem of the convergence of the exponential of the rescaled
log-partition function to GMC, which motivated this work, and for making us
aware of [CZ21] prior to publication. We would also like to thank Mo Dick Wong
and Perla Sousi for useful discussions. D. L. acknowledges financial support
from EPRSC through grant EP/HO23364/1 as part of the MASDOC DTC at the
University of Warwick. N.Z. work was supported by EPSRC through grant
EP/R024456/1.

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