# Local central limit theorem for gradient field models

Wei Wu  Mathematics Department, NYU Shanghai & NYU-ECNU Institute of
Mathematical Sciences, China<EMAIL_ADDRESS>
###### Abstract.

We consider the gradient field model in
$\left[-N,N\right]^{2}\cap\mathbb{Z}^{2}$ with a uniformly convex interaction
potential. Naddaf-Spencer [22] and Miller [20] proved that the macroscopic
averages of linear statistics of the field converge to a continuum Gaussian
free field. In this paper we prove the distribution of $\phi(0)/\sqrt{\log N}$
converges uniformly in $\mathbb{R}$ to a Gaussian density, with a Berry-Esseen
type bound. This implies the distribution of $\phi(0)$ is sufficiently
‘Gaussian like’ between $[-\sqrt{\log N},\sqrt{\log N}]$.

##  1\. Introduction

In this paper we study a two dimensional gradient interface field with a
nearest neighbor potential $V$. Explicitly, let
$Q_{N}:=\left[-N,N\right]^{2}\cap\mathbb{Z}^{2}$ and let the boundary
$\partial Q_{N}$ consist of the vertices in $Q_{N}$ that are connected to
$\mathbb{Z}^{2}\setminus Q_{N}$ by an edge. The gradient field on $Q_{N}$ with
zero boundary condition is a random field denoted by $\phi^{Q_{N},0}$, whose
distribution is given by the Gibbs measure

(1.1) $d\mu_{N}=Z_{N}^{-1}\exp\left[-\sum_{v\in
Q_{N}}\sum_{i=1}^{2}V\left(\nabla_{i}\phi\left(v\right)\right)\right]\prod_{v\in
Q_{N}\backslash\partial Q_{N}}d\phi\left(v\right)\prod_{v\in\partial
Q_{N}}\delta_{0}\left(\phi\left(v\right)\right),$

where
$\nabla_{i}\phi\left(v\right)=\phi\left(v+e_{i}\right)-\phi\left(v\right)$,
$e_{1}=(1,0)$ and $e_{2}=(0,1)$, and we set $\phi\left(v\right)=0$ for all
$v\in\mathbb{Z}^{2}\setminus Q_{N}$. Here $Z_{N}$ is the normalizing constant
ensuring that $\mu_{N}$ is a probability measure, i.e.
$\mu_{N}(\mathbb{R}^{|Q_{N}|})=1$. We denote expectation and variance with
respect to $\mu_{N}$ by $\langle\cdot\rangle_{\mu_{N}}$ and
$\operatorname{var}_{\mu_{N}}$, respectively.

We assume the interaction potential $V:\mathbb{R}\to\mathbb{R}$ in (1.1)
satisfies the following:

  1. (i)

_Symmetry_ : for every $t\in\mathbb{R}$, we have $V(t)=V(-t)$.

  2. (ii)

_Uniform convexity_ : for every $t\in\mathbb{R}$, we have $0<\lambda\leq
V^{\prime\prime}(t)\leq\Lambda<\infty$.

  3. (iii)

_Regularity_ : $V\in C^{2,1}(\mathbb{R})$. In other words,
$\mathsf{V}^{\prime\prime}$ is Lipschitz continuous with Lipschitz constant
$L$.

The Gibbs measure (1.1) was introduced in the 1970s by Brascamp, Lieb and
Lebowitz [11], in the name of anharmonic crystals. Since then, numerous
efforts have been made to study the large-scale (macroscopic) statistical
behavior of the field $\nabla\phi$. Notable progress was made by Naddaf and
Spencer [22], who studied the infinite volume limit of the Gibbs measure (1.1)
(the infinite volume Gibbs states were rigorously characterized by Funaki and
Spohn [15]), and proved a central limit for (rescaled) linear functions of
$\nabla\phi$. More precisely, they consider, for $R\geq 1$, the random
variable

(1.2)
$F_{R}(\nabla\phi):=R^{-\frac{d}{2}}\sum_{x\in\mathbb{Z}^{d}}\sum_{i=1}^{d}f_{i}\left(\frac{x}{R}\right)\left(\phi(x+e_{i})-\phi(x)\right),$

where for each $i\in\\{1,\ldots,d\\}$, take
$f_{i}:{\mathbb{R}^{d}}\to\mathbb{R}$ to be a compactly supported, smooth,
deterministic function, and proved that the random variable $F_{R}$ converges
in law to a normal random variable. Later, the result of [22] has been
generalized to dynamical settings [16] and also to finite volume measures such
that the boundary condition has at most poly-logarithmic fluctuations [20]. We
also mention the related work [14] which characterizes the Wulff shape and the
large deviation principle for macroscopic profiles of $\nabla\phi$, and
results that are related to the extension of the Gibbs measure (1.1) to non-
uniformly convex settings [25, 9, 1, 21, 18, 6, 7], the study of the maximum
of the random field (1.1) [8, 26], and to spin models which is related to a
Gibbs measure with convex interaction by a duality transform [13].

A natural question arises is whether the Gaussian limit established in [22]
holds on much smaller scales, given that the microscopic interaction is not
Gaussian. Under an additional assumption on the ellipticity contrast for $V$,
namely $\Lambda<2\lambda$, Conlon and Spencer proved a stronger result [12],
which states that for the infinite volume gradient Gibbs measure $\mu$,

$\left|\log\left\langle\exp(t(\phi(0)-\phi(x)))\right\rangle_{\mu}-\frac{t^{2}}{2}\operatorname{var}_{\mu}(\phi(0)-\phi(x))\right|\leq
Ct^{3}\sup_{x\in\mathbb{R}}|V^{\prime\prime\prime}(x)|.$

The result suggests that under these assumptions of $V$, the pointwise
distribution of $\phi(x)-\phi(0)$ is close to a Gaussian, and one has to move
to a large deviation regime to see non-Gaussian tails.

In this paper we are able to bring down the scale to $R=1$, and prove a local
central limit theorem of the gradient Gibbs measure (1.1), under the
assumptions on the potential $V$ given at the beginning of the paper.

######  Theorem 1.1.

Let $\phi$ be sampled from the Gibbs measure (1.1). Assume the potential
$\mathsf{V}\left(\cdot\right)$ satisfies the coniditions (i) -(iii). Then the
density function $g_{N}$ of $\phi(0)/\sqrt{\log N}$ converges uniformly in
$\mathbb{R}$ to $\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}$.
Moreover, there exists $C<\infty$ such that
$|g_{N}(x)-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}|\leq\frac{C}{\sqrt{\log
N}}$

We remark here the same proof (with a slightly modification of the multiscale
argument in Section 5 ) also gives the local CLT at other locations inside the
bulk. Namely, for any $x_{N}\in Q_{N}$ such that the graph distance between
$x_{N}$ and $\partial Q_{N}$ tends to infinity, the density of
$\frac{\phi(x_{N})}{\sqrt{\log\text{dist}(x_{N},\partial Q_{N})}}$ converges
uniformly in $\mathbb{R}$ to the same Gaussian limit. The same proof also
works in the inifnite volume setting, gives the local CLT for
$\frac{\phi(0)-\phi(x)}{(\log|x|)^{\frac{1}{2}}}$ as $|x|\to\infty$.

Notice that if $\phi(0)$ can be written as $X_{1}+\cdots X_{\log N}$, where
$X_{i}$ are i.i.d random variables, then the Berry-Esseen Theorem gives
$|g_{N}(x)-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}|\leq\frac{C\mathbb{E}[|X_{1}|^{3}]}{\sqrt{\log
N}}$. We will see in Section 5 that the analogues of $X_{i}$ are increments of
harmonic averages of $\phi$, which are not independent but have certain
decoupling properties (thanks to [20]), and Theorem 1.1 gives a Berry-Esseen
type estimate for the density function. An immediate consequence of Theorem
1.1 is that the distribution of $\phi(0)$ is sufficiently spreadout in
$[-\sqrt{\log N},\sqrt{\log N}]$. Indeed, given any $a\in[-\sqrt{\log
N},\sqrt{\log N}]$, we apply Theorem 1.1 to obtain

$\int_{\frac{a}{\sqrt{\log N}}}^{\frac{a+1}{\sqrt{\log
N}}}g_{N}(x)\,dx-\int_{\frac{a}{\sqrt{\log N}}}^{\frac{a+1}{\sqrt{\log
N}}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}\,dx=O(\frac{1}{\log
N})$

Thus

$\mathbb{P}(\phi(0)\in[a,a+1])=\frac{1}{\sqrt{2\pi\log
N}}e^{-\frac{1}{2\mathbf{g}}\frac{a^{2}}{\log N}}+O(\frac{1}{\log N})$

Why do we expect a local CLT at scale $O(1)$? The argument to prove a
macroscopic CLT in [22] was based on a beautiful observation that the scaling
limit can be derived from an elliptic homogenization problem via the Helffer-
Sjöstrand representation [17]. With Armstrong [5], we extend and quantify the
homogenization argument by Naddaf and Spencer, based on the quantitative
theory for homogenization developed by Armstrong, Kuusi and Mourrat [3, 4]. In
particular, we obtained the convergence of the Hessian of the surface tension
with an algebraic rate, resolve an open question posed by Funaki and Spohn
[15] regarding the $C^{2}$ regularity of surface tension, and the fluctuation-
dissipation conjecture of [16]. Following the approach of [5] and [4], we are
able to obtain a quantitative homogenization of the Helffer-Sjöstrand PDE,
thus estimate the covariance structure of $\mu_{N}$, as $N$ gets large, with a
high precision (we also refer to [2] for some further applications of the
quantitative homgenization ideas to the $\nabla\phi$-model). To obtain a
Gaussian limit of $\phi(0)/\sqrt{\log N}$, we apply the harmonic approximation
result by Miller [20], which enables us to write $\phi(0)$ as the sum of $\log
N$ increments, with certain decoupling properties. These are the two main
ingredients behind the proof of Theorem 1.1.

We remark here that the key estimate for proving Theorem 1.1 is the
characteristic function asymptotics for
$\left\langle\exp\left(i\frac{t}{\sqrt{\log
N}}\phi(0)\right)\right\rangle_{\mu_{N}}$, given in Lemma 2.1. If one has the
convergence stated in Lemma 2.1 for $t\in\mathbb{R}$, without quantifying the
rate of convergence (namely, a CLT for $\phi(0)/\sqrt{\log N}$), then it
implies a local CLT without any rate of convergence. The rate of convergence
for the local CLT depdends on the convergence rate and the validity of the $t$
region such that Lemma 2.1 holds. In this paper we present a proof with a
convergence rate using quantitative homogenization, and explain briefly in the
next section how it can be simplified if one only aims for a qualitative local
CLT.

The proof of Theorem 1.1 presented in this paper implies the asymptotics for
the characteristic function for the distribution of $\phi(0)$, namely there
exists some $\mathbf{g}=\mathbf{g}(V)>0$, such that

(1.3) $\left\langle\exp\left(is\phi(0)\right)\right\rangle_{\mu_{N}}\approx
e^{-\frac{s^{2}}{2}\mathbf{g}\log N},$

as long as $s=O((\log N)^{-\frac{1}{4}})$. Using the same major ingredients,
but with a more elaborate multiscale argument, we may improve the (1.3) so
that it holds for $s$ within a small neighborhood of the origin, with radius
independent of $N$. It would be very interesting to extend the characteristic
function estimates (1.3) beyond $|s|=1$. For this, we include an open question
posed by Tom Spencer.

Open Question: Consider the lattice dipole gas model, which is a special case
of the gradient Gibbs measure (1.1) with $V(x)=\frac{x^{2}}{2}+z\cos x$, with
$|z|<1$. Prove that the leading term in the asymptotic expansion of the
characteristic function

(1.4)
$\left\langle\exp\left(i\pi(\phi(0)-\phi(x))\right)\right\rangle_{\mu_{dipole}}\approx
e^{-\frac{\pi^{2}}{2}\mathbf{g}\log|x|},$

for some $\mathbf{g}>0$. Estimates of type (1.4) plays an important role in
the study of the (lattice) Coulomb gas and of the Coulomb gas representation
of the low temperature Abelian spin models.

We summarize the characteristic function estimates needed for proving Theorem
1.1 in the next section, and introduce some notations in Section 3. In Section
4 we derive a central limit theorem for the linear statistics of $\nabla\phi$
with an algebraic rate of convergence, which quantifies the result of [22],
following the argument of [5] and [4]. In Section 5 we recall the harmonic
approximation result in [20], and use a multiscale argument to obtain the
precise characteristic function asymptotic of $\phi(0)$. Finally, we gave an
upper bound for the large $s$ characteristic function in Section 6, based on a
Mermin-Wagner type argument, and finishes the proof of Theorem 1.1 .

##  2\. Estimates for characteristic functions

Theorem 1.1 follows from the quantitative estimates of the characteristic
function below. The first lemma gives the precise estimate for the
characteristic function $\langle e^{is\phi(0)}\rangle_{\mu_{N}}$ for
$s=o((\log N)^{-\frac{1}{4}})$.

######  Lemma 2.1.

There exists $\mathbf{g}=\mathbf{g}(V)>0$, such that for $N$ sufficiently
large and $t=o((\log N)^{\frac{1}{4}})$, we have

(2.1) $\left\langle\exp\left(i\frac{t}{\sqrt{\log
N}}\phi(0)\right)\right\rangle_{\mu_{N}}=e^{-\frac{t^{2}}{2}\mathbf{g}}\left(1+O\left(\frac{t^{2}}{(\log
N)^{\frac{1}{2}}}\right)\right)$

######  Remark 2.2.

Lemma 2.1 quantifies the CLT for the pointwise field $\phi(0)/\sqrt{\log N}$,
namely for $t\in\mathbb{R}$,

(2.2) $\left\langle\exp\left(i\frac{t}{\sqrt{\log
N}}\phi(0)\right)\right\rangle_{\mu_{N}}=e^{-\frac{t^{2}}{2}\mathbf{g}}+o_{N}(1).$

The pointwise CLT (2.2) can be proved by combining the CLT for the macroscopic
average of the field established in [22, 20], and the multscale argument
presented in [8] or in Section 5 of this paper.

We also need (non-optimal) decay estimate of the characteristic function for
large $s$, summarized in the two lemmas below.

######  Lemma 2.3.

Let $\mathbf{g}=\mathbf{g}(V)>0$ be the same constant as in Lemma 2.1. There
exists $\varepsilon=\varepsilon(V)>0$ , such that for $|s|<\varepsilon$, we
have for $N$ sufficiently large,

(2.3) $\left\langle\exp\left(is\phi(0)\right)\right\rangle_{\mu_{N}}\leq
2e^{-\frac{s^{2}}{3}\mathbf{g}\log N}$

######  Remark 2.4.

If one only aims for the qualitative local CLT, then it suffices to prove a
weaker estimate, that there exist $c_{1}>0$ and $\varepsilon=\varepsilon(V)>0$
, such that for $|s|<\varepsilon$, we have for $N$ sufficiently large,

(2.4) $\left\langle\exp\left(is\phi(0)\right)\right\rangle_{\mu_{N}}\leq
2e^{-c_{1}s^{2}\log N}.$

As will be explained in Section 5, the proof of (2.4) is simpler, and the
quantitative CLT presented in Section 4 would not be needed.

######  Lemma 2.5.

There exists $\varepsilon_{1}>0$ and $C<\infty$, such that for
$s\in\mathbb{R}$, we have

(2.5)
$|\left\langle\exp\left(is\phi(0)\right)\right\rangle_{\mu_{N}}|\leq\min\\{1-\varepsilon_{1},\frac{C}{s^{2}}\\}^{\log
N}$

Before proving these lemmas, we now explain how they imply Theorem 1.1.

###### Proof of Theorem 1.1 with no rate.

To prove Theorem 1.1, arguing as the classical local CLT and write

(2.6) $\Psi_{N}(t):=\left\langle\exp\left(it\phi(0)/\sqrt{\log
N}\right)\right\rangle_{\mu_{N}}$

Then by inversion theorm,

$|g_{N}(x)-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}|=\left|\int_{-\infty}^{\infty}\frac{1}{2\pi}\left(\Psi_{N}(t)e^{itx}-e^{-\frac{1}{2}\mathbf{g}t^{2}}e^{itx}\right)\,dt\right|\leq\frac{1}{2\pi}\int_{-\infty}^{\infty}\left|\Psi_{N}(t)-e^{-\frac{1}{2}\mathbf{g}t^{2}}\right|\,dt$

We claim the right side above goes to zero by split the integral into three
parts:

\- For $|t|\leq a$, we apply (2.2) to conclude that $\Psi_{N}(t)$ goes to
$e^{-\frac{1}{2}\mathbf{g}t^{2}}$ uniformly for $t\in[-a,a]$. Thus

(2.7)
$\int_{-a}^{a}\left|\Psi_{N}(t)-e^{-\frac{1}{2}\mathbf{g}t^{2}}\right|\,dt\to
0$

as $N\to\infty$.

\- For $a\leq|t|\leq\varepsilon\sqrt{\log N}$, we apply (2.4), which yields

(2.8) $\Psi_{N}(t)\leq Ce^{-c_{1}t^{2}}$

Thus

(2.9) $\int_{a\leq|t|\leq\varepsilon\sqrt{\log
N}}\left|\Psi_{N}(t)-e^{-\frac{1}{2}\mathbf{g}t^{2}}\right|\,dt\leq
2C\int_{a}^{\infty}e^{-c_{1}t^{2}}\,dt\to 0$

if we take $a\to\infty$.

\- For $|t|\geq\varepsilon\sqrt{\log N}$, we use the bound (2.5) which implies
(let $s=t/\sqrt{\log N}$)

$\int_{|t|>\varepsilon\sqrt{\log N}}\left|\Psi_{N}(t)\right|\,dt=\sqrt{\log
N}\int_{|s|>\varepsilon}\left|\Psi_{N}(s)\right|\,ds\\\ \leq\sqrt{\log
N}\left(\int_{C>|s|>\varepsilon}(1-\varepsilon_{1})^{\log
N}\,ds+\int_{|s|>C}(\frac{C}{s^{2}})^{\log N}\,ds\right)$

which goes to $0$ as $N\to\infty$. And we conclude Theorem 1.1. ∎

###### Quantitative proof of Theorem 1.1.

To quantify the rate of convergence for the local CLT, take
$a_{N}=\sqrt{\frac{3}{2\mathbf{g}}\log\log N}$ in the proof above.

\- For $|t|\leq a_{N}$, we apply Lemma 2.1 to obtain a rate of convergence
that

(2.10)
$\int_{-a_{N}}^{a_{N}}\left|\Psi_{N}(t)-e^{-\frac{1}{2}\mathbf{g}t^{2}}\right|\,dt\leq\int_{-a_{N}}^{a_{N}}Ce^{-\frac{t^{2}}{2}\mathbf{g}}\frac{t^{2}}{(\log
N)^{\frac{1}{2}}}\,dt\leq\frac{C}{(\log N)^{\frac{1}{2}}}$

for $N$ sufficiently large.

\- For $a_{N}\leq|t|\leq\varepsilon\sqrt{\log N}$, we apply Lemma 2.3, which
yields

(2.11) $\Psi_{N}(t)\leq 2e^{-\frac{1}{3}\mathbf{g}t^{2}}$

Thus

(2.12) $\int_{a_{N}\leq|t|\leq\varepsilon\sqrt{\log
N}}\left|\Psi_{N}(t)-e^{-\frac{1}{2}\mathbf{g}t^{2}}\right|\,dt\leq
3\int_{a_{N}}^{\infty}e^{-\frac{1}{3}\mathbf{g}t^{2}}\,dt\leq\frac{C}{(\log
N)^{\frac{1}{2}}}$

for $N$ sufficiently large.

Combine with the estimates for $|t|\geq\varepsilon\sqrt{\log N}$ in the
qualitative proof above, we conclude
$|g_{N}(x)-\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2\mathbf{g}}x^{2}}|=O((\log
N)^{-\frac{1}{2}})$. ∎

Lemma 2.1 and 2.3 will be proved in Section 5.3, whereas Lemma 2.5 will be
proved in Section 6.

##  3\. Preliminaries and Notation

Given a set $U\subseteq Q_{R}$, we let $\mathcal{E}(U)$ denote the set of
directed edges on $U$ and $U^{\circ}$ the _interior_ of $U$. Define
$\Omega_{0}(U)$ to be the set of functions $\phi:U\to\mathbb{R}$ such that
$\phi=0$ on $\partial U$. Given $e=(x,y)\in\mathcal{E}(U)$ and
$\phi\in\mathbb{R}^{U}$, we define $\nabla\phi(e):=\phi(y)-\phi(x)$. The
formal adjoint $\nabla^{*}$ of $\nabla$, which is the discrete version of the
negative of the divergence operator, is defined for functions
$\mathbf{g}:\mathcal{E}(U)\to\mathbb{R}$ by

(3.1) $\left(\nabla^{*}\mathbf{g}\right)(x):=\sum_{e\ni x}\mathbf{g}(e),\quad
x\in U^{\circ}.$

The average of a function $f:U\to\mathbb{R}$ on $U$ is denoted as
$(f)_{U}:=\frac{1}{|U|}\sum_{x\in U}f(x)$.

We define, for each $x\in U$, the basis element $\omega_{x}\in\Omega_{0}(U)$
by

$\omega_{x}(y):=\left\\{\begin{aligned} &1&\mbox{if}\ x=y,\\\ &0&\mbox{if}\
x\neq y,\end{aligned}\right.$

and the differential operator $\partial_{x}$ by

(3.2) $\partial_{x}u(\phi):=\lim_{h\to
0}\frac{1}{h}\left(u(\phi+h\omega_{x})-u(\phi)\right).$

Define $L^{p}(\mu)$ to be the set of measurable functions
$u:\Omega_{0}(U)\to\mathbb{R}$ such that

$\left\|u\right\|_{L^{p}(\mu)}:=\left(\int_{\Omega}\left|u(\phi)\right|^{p}\,d\mu(\phi)\right)^{\frac{1}{p}}<+\infty.$

We define $H^{1}(\mu)$ to be

$\left\|u\right\|_{H^{1}(\mu)}:=\left(\left\|u\right\|_{L^{2}(\mu)}^{2}+\sum_{x\in
U^{\circ}}\left\|\partial_{x}u\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}}.$

We let $H^{-1}(\mu)$ denote the dual space of $H^{1}(\mu)$, that is, the
closure of $C^{\infty}(\Omega_{0}(U))$ functions under the norm

$\left\|w\right\|_{H^{-1}(\mu)}:=\sup\left\\{\int_{\Omega}u(\phi)w(\phi)\,d\mu(\phi)\,:\,u\in
H^{1}(\mu),\ \left\|u\right\|_{H^{1}(\mu)}\leq 1\right\\}.$

We define the space $L^{2}(U,\mu)=L^{2}(U;L^{2}(\mu))$ to be the set of
measurable functions $u:U\times\Omega_{0}(U)\to\mathbb{R}$ with respect to the
norm

$\left\|u\right\|_{L^{2}(U,\mu)}:=\left(\sum_{x\in
U}\left\|u(x,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}}.$

We also define $H^{1}(U,\mu)$ by the norm

$\left\|u\right\|_{H^{1}(U,\mu)}:=\left(\sum_{x\in
U}\left\|u(x,\cdot)\right\|_{H^{1}(\mu)}^{2}+\sum_{e\in\mathcal{E}(U)}\left\|\nabla
u(e,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}}$

The subset $H^{1}_{0}(U,\mu)\subseteq H^{1}(U,\mu)$ consists of those
functions $u\in H^{1}(U,\mu)$ which satisfy $u(x,\phi)=0$ for every $\partial
U\times\Omega_{0}(U)$.

We define $H^{-1}(U,\mu)$ to be the dual space of $H^{1}_{0}(U,\mu)$. That is,
$H^{-1}(U,\mu)$ is the closure of smooth functions with respect to the norm

$\left\|w\right\|_{H^{-1}(U,\mu)}:=\sup\left\\{\sum_{x\in
U}\int_{\Omega_{0}(U)}u(x,\phi)w(x,\phi)\,d\mu(\phi)\,:\,u\in
H^{1}_{0}(U,\mu),\ \left\|u\right\|_{H^{1}(U,\mu)}\leq 1\right\\}.$

It is sometimes convenient to work with the volume-normalized versions of the
$L^{2}$ and Sobolev norms, defined by

$\left\|u\right\|_{\underline{L}^{2}(U,\mu)}:=\left(\frac{1}{|U|}\sum_{x\in
U}\left\|u(x,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}},$
$\left\|u\right\|_{\underline{H}^{1}(U,\mu)}:=\left(\frac{1}{|U|}\sum_{x\in
U}\left\|u(x,\cdot)\right\|_{H^{1}(\mu)}^{2}+\frac{1}{|U|}\sum_{e\in\mathcal{E}(U)}\left\|\nabla
u(e,\cdot)\right\|_{L^{2}(\mu)}^{2}\right)^{\frac{1}{2}},$
$\left\|w\right\|_{\underline{H}^{-1}(U,\mu)}\\\
:=\sup\left\\{\frac{1}{|U|}\sum_{x\in
U}\int_{\Omega_{0}(U)}u(x,\phi)w(x,\phi)\,d\mu(\phi)\,:\,u\in
H^{1}_{0}(U,\mu),\ \left\|u\right\|_{\underline{H}^{1}(U,\mu)}\leq
1\right\\}.$

We notice that the formal adjoint of $\partial_{x}$ with respect to $\mu_{N}$,
which we denote as $\partial_{x}^{*}$, is given by

$\partial_{x}^{*}w:=-\partial_{x}w+\sum_{y\sim
x}V^{\prime}(\phi(y)-\phi(x)-\xi\cdot(y-x))w(\phi).$

This can be easily checked by the identity for all $u,v\in H^{1}(\mu_{N})$
that

$\left\langle(\partial_{x}u)v\right\rangle_{\mu_{N}}=\left\langle
u(\partial^{*}_{x}v)\right\rangle_{\mu_{N}}.$

We also have the commutator identity

(3.3) $\left[\partial_{x},\partial_{y}^{*}\right]=-\mathds{1}_{\\{x\sim
y\\}}V^{\prime\prime}\big{(}\phi(y)-\phi(x)-\xi\cdot(y-x)\big{)}+\mathds{1}_{\\{x=y\\}}\sum_{e\ni
x}V^{\prime\prime}\left(\nabla\phi(e)-\nabla\ell_{\xi}(e)\right).$

Define the Witten Laplacian $\mathcal{L}_{\mu_{N}}$ as

$\mathcal{L}_{\mu_{N}}F=-\sum_{x\in
Q_{N}^{\circ}}\partial_{x}^{*}\partial_{x}F,$

For every cube $Q\subseteq\mathbb{Z}^{d}$ and $u,v\in H^{1}(Q,\mu)$, we define

(3.4) $\mathsf{B}_{\mu,Q}\left[u,v\right]:=\frac{1}{|Q|}\sum_{y\in
Q_{N}^{\circ}}\sum_{x\in
Q^{\circ}}\left(\partial_{y}u(x,\cdot),\partial_{y}v(x,\cdot)\right)_{\mu}+\frac{1}{|Q|}\sum_{e\in\mathcal{E}(Q)}\left\langle\nabla
u(e,\cdot)V^{\prime\prime}(e,\cdot)\nabla v(e,\cdot)\right\rangle_{\mu}$

and

$\displaystyle\mathsf{E}_{\mu,Q,\mathbf{f}}\left[u\right]$
$\displaystyle:=\frac{1}{2}\mathsf{B}_{\mu,Q}\left[u,u\right]-\frac{1}{|Q|}\sum_{e\in\mathcal{E}(Q)}\left\langle\mathbf{f}(e,\phi)\nabla
u(e,\cdot)\right\rangle_{\mu}.$

For $D\subseteq Q_{R}$, and $f:\partial D\to\mathbb{R}$, define the
$\nabla\phi$ measure on $D$ with Dirichlet boundary condition $f$ by

(3.5) $d\mu_{D}^{f}=Z_{D}^{-1}\exp\left[-\sum_{x\in
D}\sum_{i=1}^{2}V\left(\nabla_{i}\phi\left(x\right)\right)\right]\prod_{x\in
D\backslash\partial D}d\phi\left(x\right)\prod_{x\in\partial
D}\delta_{0}\left(\phi\left(x\right)-f\left(x\right)\right).$

Here $Z_{D}$ is the normalizing constant ensuring that $\mu_{D}^{f}$ is a
probability measure. We denote expectation and variance with respect to
$\mu_{D}^{f}$ by $\mathbb{E}^{D,f}$ and $\operatorname{var}_{D,f}$,
respectively.

We finally present the Brascamp-Lieb inequality [10, 22], which states that
the variance of observables with respect to a log-concave measure is dominated
by that of a Gaussian measure. We denote the Green function for the discrete
Laplacian with zero Dirichlet boundary conditions in $Q_{L}$ by
$G_{Q_{L}}(x,y)$.

######  Proposition 3.1 (Brascamp-Lieb inequality for $\mu_{L}$).

For every $F\in H^{1}(\mu_{L})$,

(3.6)
$\operatorname{var}_{\mu_{L}}\left[F\right]\leq\frac{1}{\lambda}\sum_{x,y\in
Q_{L}^{\circ}}G_{Q_{L}}(x,y)\left\langle\left(\partial_{x}F\right)\left(\partial_{y}F\right)\right\rangle_{\mu_{L}}.$

For every $f\in\mathbb{R}^{Q_{L}}$, we have

(3.7) $\log\left\langle\exp(t\sum_{y\in
Q_{L}}\phi(y)f(y))\right\rangle_{\mu_{L}}\leq\frac{t^{2}}{2\lambda}\sum_{x,y\in
Q_{L}^{\circ}}G_{Q_{L}}(x,y)f(x)f(y)$

We sometimes denote by $\mu_{G,D}^{f}$ the finite volume Gaussian measure in
$D$ (i.e., the special case of (3.5) with $V(x)=\frac{1}{2}x^{2}$). We denote
the corresponding expectation and variance by $\mathbb{E}^{G,D,f}$ and
$\operatorname{var}_{G,D,f}$ respectively. When $f=0$ we will omit its
appearence on the supercripts.

##  4\. Quantitative convergence of the variance

A main ingredient for the refined estimate of $\Psi_{N}$, defined in (2.6) is
the following convergence of the variance of the linear statisitcs of
$\nabla\phi$, with an algebraic rate.

######  Theorem 4.1 (Quantitative convergence of variance).

Fix $R\in[1,\infty)$. Let $\phi$ be sampled from the finite volume Gibbs
measure $\mu_{R}$ with zero boundary condition (1.1). Let
$f_{R}:Q_{R}\to\mathbb{R}$ and $f\in L^{2}([0,1]^{2})$ be such that there
exists $\alpha>0$, so that $\|\nabla^{*}\cdot
f_{R}(\frac{\cdot}{R})-f(\cdot)\|_{L^{\infty}([0,1]^{2})}\leq R^{-\alpha}$.
Define the random variable

$\Phi_{R}\left(f\right):=R^{-d/2}\sum_{e\in\mathcal{E}(Q_{R})}\nabla\phi(e)\cdot
f_{R}\left(e\right).$

Then there exists $\mathbf{g}=\mathbf{g}_{V,f}>0$,
$\beta=\beta(d,\alpha,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C_{0}(d,\lambda,\Lambda)<\infty$ such that, for every $R\in[1,\infty)$,

$\left|\operatorname{var}_{\mu_{R}}[\Phi_{R}\left(f\right)]-\mathbf{g}\right|\leq
C_{0}R^{-\beta}\|\nabla^{*}\cdot f_{R}\|_{L^{\infty}}.$

######  Remark 4.2.

The central limit theorem for the $\nabla\phi$ model, i.e., the convergence of
$\Phi_{R}$ in distribution to a normal random variable, was established in
[22, 16, 20], without quantifying the rate of convergence.

######  Remark 4.3.

It will be clear from Theorem 4.4 below that $\mathsf{g}$ can be explicity
written as

$\mathsf{g}=\int_{[0,1]^{2}\times[0,1]^{2}}f(x)(\nabla^{*}\cdot{\overline{\mathbf{a}}}\nabla)^{-1}(x,y)f(y)\,dxdy,$

for some positive definite matrix
${\overline{\mathbf{a}}}={\overline{\mathbf{a}}}(V)$.

Theorem 4.1 follows from homogenization of an elliptic PDE based on the
convergence results of [5], as we explain below. The starting observation is
the variational characterization of the variance, known as the Helffer-
Sjöstrand representation (see [22, 5]), which gives

(4.1)
$\operatorname{var}_{\mu_{R}}[R^{-d/2}\sum_{e\in\mathcal{E}(Q_{R})}\nabla\phi(e)\cdot
f_{R}\left(e\right)]=-2\inf_{w\in
H^{1}_{0}(Q_{R},\mu_{R})}\mathsf{E}_{\mu_{R},Q_{R},f_{R}}\left[w\right],$

where we let $\mathsf{E}_{\mu,U,f}\left[\cdot\right]$ denote the energy
functional

$\displaystyle\mathsf{E}_{\mu,U,f}\left[w\right]$
$\displaystyle:=\frac{1}{2}\sum_{y\in Q}\sum_{x\in
U^{\circ}}\left\langle(\partial_{y}w(x,\cdot))^{2}\right\rangle_{\mu}+\frac{1}{2}\sum_{e\in\mathcal{E}(U)}\left\langle
V^{\prime\prime}(e)(\nabla w(e,\cdot))^{2}\right\rangle_{\mu}$
$\displaystyle\qquad-\sum_{x\in U^{\circ}}\left\langle
f(x,\cdot)w(x,\cdot)\right\rangle_{\mu}.$

The minimizer of (4.1) can be written as $R^{-\frac{d}{2}}u_{R}$, where
$u_{R}$ solves the Helffer-Sjöstrand PDE

(4.2) $\left\\{\begin{aligned} &-\mathcal{L}_{\mu}u_{R}+\nabla^{*}\cdot
V^{\prime\prime}\nabla u_{R}=\nabla^{*}\cdot f_{R}\quad&\mbox{in}&\
Q_{R}\times\Omega_{0}(Q_{R})\\\ &u_{R}=0&\mbox{on}&\ \partial
Q_{R}\times\Omega_{0}(Q_{R}),\end{aligned}\right.$

and by testing (4.2) with $u_{R}$ and integration by parts, we may rewrite the
energy functional $\mathsf{E}_{\mu_{R},Q_{R},f_{R}}\left[u_{R}\right]$, and
thus (4.1) as [22, 5]

$\operatorname{var}_{\mu_{R}}[R^{-d/2}\sum_{x\in Q_{R}}\nabla\phi(x)\cdot
f_{R}\left(x\right)]=\sum_{x\in Q_{R}}R^{-d}\left\langle
f_{R}\left(x\right)\nabla u_{R}(x)\right\rangle_{\mu_{R}}$

Therefore Theorem 4.1 follows from the quantitative homogenization of the
Hellfer-Sjöstrand equation (4.2), presented below.

######  Theorem 4.4.

Suppose that $f_{R},f$ satisfy the conditions in Theorem 4.1, and let
$\mathbf{a}$ be the diagonal matrix with
$\mathbf{a}(e,e)=\mathsf{V}^{\prime\prime}(\nabla\phi(e))$, where $\nabla\phi$
is sampled from the Gibbs measure $\mu_{R}$ (1.1). Let $u_{R},u$ denote
respectively the solution to the equations:

(4.3) $\left\\{\begin{aligned}
&-\mathcal{L}_{\mu}u_{R}+\nabla^{*}\cdot\mathbf{a}\nabla u_{R}=\nabla^{*}\cdot
f_{R}\quad&\mbox{in}&\ Q_{R}\times\Omega_{0}(Q_{R})\\\ &u_{R}=0&\mbox{on}&\
\partial Q_{R}\times\Omega_{0}(Q_{R}),\end{aligned}\right.$

and

(4.4) $\left\\{\begin{aligned} &-\nabla\cdot{\overline{\mathbf{a}}}\nabla
u=f\quad&\mbox{in}&\ [0,1]^{2}\\\ &u=0&\mbox{on}&\
\partial([0,1]^{2}).\end{aligned}\right.$

Then there exists $\beta=\beta(d,\lambda,\Lambda)>0$, such that

(4.5)
$\left\|u_{R}(R\cdot)-u(\cdot)\right\|_{L^{2}(\frac{1}{R}Q_{R},\mu_{R})}+\left\|\nabla
u_{R}(R\cdot)-\nabla u(\cdot)\right\|_{H^{-1}(\frac{1}{R}Q_{R},\mu_{R})}\leq
CR^{-\alpha}\|\nabla^{*}\cdot f_{R}\|_{L^{\infty}}.$

Applying Theorem 4.4 and rescale the domain by $R$, we have

$\left|\operatorname{var}_{\mu_{R}}[R^{-d/2}\sum_{x\in
Q_{R}}\nabla\phi(x)\cdot f_{R}\left(x\right)]-\sum_{x\in
Q_{R}}R^{-d}f_{R}\left(x\right)\nabla u\left(\frac{x}{R}\right)\right|\\\
\leq\|\nabla^{*}\cdot
f_{R}\|_{L^{\infty}}\|u_{R}-u\left(\frac{\cdot}{R}\right)\|_{\underline{L}^{2}(Q_{R},\mu_{R})}\leq
CR^{-\beta}\|\nabla^{*}\cdot f_{R}\|_{L^{\infty}}$

Moreover, the limit

$\mathbf{g}:=\lim_{R\to\infty}\sum_{x\in Q_{R}}R^{-d}f_{R}\left(x\right)\nabla
u\left(\frac{x}{R}\right)=\int_{[0,1]^{2}}f(x)u(x)\,dx=\int_{[0,1]^{2}\times[0,1]^{2}}f(x)(\nabla^{*}\cdot{\overline{\mathbf{a}}}\nabla)^{-1}(x,y)f(y)\,dxdy$

exists, by the convergence of Riemann sum to integral, with a rate of
convergence $O(R^{-\alpha})$. Combining these estimates we conclude Theorem
4.1.

###  4.1. Finite-volume energy quantities

In this section we recall the energy quantities and their quantitative
convergence results established in [5]. As can be seen from the variational
characterization, the convergence of the energy quantities will play an
essential role in the proof of Theorem 4.1. Define the subadditive energy
quantity

$\nu_{R}(Q_{R},f,p):=\inf_{w\in
H^{1}_{0}(Q_{R},\mu_{R,p})}\mathsf{E}_{\mu_{R,p},Q_{R},f}\left[w\right],$

where $\mu_{R,p}$ denotes the finite volume Gibbs measure in $Q_{R}$ with an
affine boundary condition $\ell_{p}(x)=p\cdot x$. In what follows we consider
$f=0$ and simply write it as $\nu_{R}(Q_{R},p)$. As was explained in (4.1),
the minimizer of $\nu_{R}(Q_{R},p)$, which we denote as $v(\cdot,Q_{R},p)$,
solves the Helffer-Sjöstrand equation with an affine boundary condition:

(4.6) $\left\\{\begin{aligned}
&\left(-\mathcal{L}_{\mu}+\nabla^{*}\mathbf{a}\nabla\right)v(\cdot,Q_{R},p)=0&\mbox{in}&\
Q^{\circ}_{R}\times\Omega_{0}(Q_{R}),\\\
&v(\cdot,Q_{R},p)-\ell_{p}=0&\mbox{on}&\ \partial
Q_{R}\times\Omega_{0}(Q_{R}),\end{aligned}\right.$

We recall the fact that $\nu_{R}$ are actually quadratic polynomials for all
$R\geq 1$, and one may compute the first and second variations of their
defining optimization problems. The following lemma is [5, Lemma 5.2].

######  Lemma 4.5 (Basic properties of $\nu_{R}$ ).

Fix a cube $Q\subseteq Q_{R}$. The quantities $\nu_{R}(Q,p)$ and its
optimizing functions $v(\cdot,Q,p)$ satisfies

  * •

_Quadratic representation._ There exist symmetric matrices
${\overline{\mathbf{a}}}(Q)\in\mathbb{R}^{d\times d}$, such that

(4.7)
$\nu_{R}(Q,p)=\frac{1}{2}p\cdot{\overline{\mathbf{a}}}(Q)p+\nu_{R}(Q,0)\quad\forall
p\in{\mathbb{R}^{d}}.$

where the matrix ${\overline{\mathbf{a}}}(Q)$ can be characterized such that
for all $p,p^{\prime}\in{\mathbb{R}^{d}}$,

(4.8)
$p^{\prime}\cdot{\overline{\mathbf{a}}}(Q)p=\mathsf{B}_{\mu_{R,p},Q}\left[\ell_{p^{\prime}},v(\cdot,Q,p)\right].$

  * •

_First variation._ The optimizing functions are characterized as follows:
$v(\cdot,Q,p)$ is the unique element of $\ell_{p}+H^{1}_{0}(Q,\mu_{R,p})$
satisfying

(4.9)
$\displaystyle\mathsf{B}_{\mu_{R,p},Q}\left[v(\cdot,Q,p),w\right]=0,\quad\forall
w\in H^{1}_{0}(Q,\mu_{R,p});$

  * •

_Second variation._ For every
$w\in\ell_{p}+H_{0}^{1}\left(Q,\mu_{R,p}\right)$,

(4.10)
$\mathsf{E}_{\mu_{R,p},Q}\left[w\right]-|Q|\nu_{R}(Q,p)=\frac{1}{2}\mathsf{B}_{\mu_{R,p},Q}\left[v(\cdot,Q,p)-w,v(\cdot,Q,p)-w\right]$

As $R\to\infty$, the subadditive quantity $\nu_{R}$ is proved to converge with
an algebraic rate of convergence. Define, for some positive definite matrix
${\overline{\mathbf{a}}}={\overline{\mathbf{a}}}(V)$ and ${\overline{c}}$,

(4.11)
$\overline{\nu}(p)=\frac{1}{2}p\cdot{\overline{\mathbf{a}}}p-{\overline{c}}.$

We have

######  Proposition 4.6 (Proposition 6.9 of [5]).

There exist $\beta(d,\lambda,\Lambda)\in\left(0,\frac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that, for every $L\in\mathbb{N}$ with
$L\leq R$, we have

(4.12) $\left|\nu_{R}(Q_{L},p)-\overline{\nu}(p)\right|\leq CL^{-\beta}.$

Combine with the quadratic representation (4.11), this implies

######  Corollary 4.7.

There exist $\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that, for every $R\in\mathbb{N}$,

(4.13)
$\left|{\overline{\mathbf{a}}}(Q_{R})-{\overline{\mathbf{a}}}\right|\leq
CR^{-\beta}$

###  4.2. Estimates on finite-volume correctors

A direct consequence of Proposition 4.6 and the quadratic response (4.10)
implies the quantitative convergence of the solution to the Dirichlet problem
(4.6) to the affine function.

######  Lemma 4.8.

There exist $\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that, for every $R\in\mathbb{N}$,

(4.14)
$\frac{1}{R}\left\|v(\cdot,Q_{R},p)-\ell_{p}\right\|_{\underline{L}^{2}(Q_{R},\mu_{R})}\leq
CR^{-\beta}.$

An application of the multiscale Poincaré inequality (Proposition A.1) implies
the quantitative convergence of the fluxes along the geometric scales
$R=3^{n}$. We define for every $m\in\mathbb{N}$,

(4.15)
${\scaleobj{1.2}{\square}}_{m}:=\left[-3^{m},3^{m}\right]^{d}\cap\mathbb{Z}^{d}.$

We also define, for $m\leq n$,
$\mathcal{Z}_{m}:=3^{m}\mathbb{Z}^{d}\cap{\scaleobj{1.2}{\square}}_{n}$, so
that $\\{y+{\scaleobj{1.2}{\square}}_{m}:y\in\mathcal{Z}_{m}\\}$ is a
partition of ${\scaleobj{1.2}{\square}}_{n}$. The next lemma shows the spatial
average of the flux is concentrated around its mean.

######  Lemma 4.9.

There exist $\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that, for every $m\in\mathbb{N}$,

(4.16)
$\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{m},p)\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]\leq
C3^{-m\beta}.$

###### Proof.

We define a localized solution $v_{loc}$, such that if
$x\in{\scaleobj{1.2}{\square}}_{m}$ is contained in
$z+{\scaleobj{1.2}{\square}}_{m/3}$ for some $z\in\mathcal{Z}_{m/3}$, set
$v_{loc}(x)=v(x,z+{\scaleobj{1.2}{\square}}_{m/3},p)$. We notice that

$\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{m},p)\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]\leq\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v_{loc}\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]+C3^{-m\beta/3}.$

Indeed, it follows from the second variation and triangle inequality that

$\left|\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{m},p)\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]-\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v_{loc}\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]\right|\\\ \leq
2\left\|\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{m},p)-\mathbf{a}\nabla
v_{loc}\right\|_{\underline{L}^{2}({\scaleobj{1.2}{\square}}_{m},\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C(\nu_{3^{m}}({\scaleobj{1.2}{\square}}_{m/3},p)-\nu_{3^{m}}({\scaleobj{1.2}{\square}}_{m},p)),$

and this is bounded by $3^{-m\beta/3}$ by the quantitative convergence of
energy (Proposition 4.6).

Therefore it suffices to estimate
$\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v_{loc}\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]$, which we do by using
the spectral gap (namely, the Brascamp-Lieb inequality) and that $v_{loc}$ is
a localized solution, to derive a correlation decay for $\mathbf{a}\nabla
v_{loc}$. For simplicity, denote by $\overline{\mathbf{a}\nabla
v_{loc}}:=\left(\mathbf{a}\nabla
v_{loc}\right)_{{\scaleobj{1.2}{\square}}_{m}}$. Applying the Brascamp-Lieb
inequality (Proposition 3.1) then yields

(4.17)
$\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v_{loc}\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]\leq\frac{1}{\lambda}\sum_{x,y\in{\scaleobj{1.2}{\square}}_{m}^{\circ}}G_{{\scaleobj{1.2}{\square}}_{m}}(x,y)\left\langle\left(\partial_{x}\overline{\mathbf{a}\nabla
v_{loc}}\right)\left(\partial_{y}\overline{\mathbf{a}\nabla
v_{loc}}\right)\right\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}$

Notice that by definition (3.2)

$\partial_{x}\overline{\mathbf{a}\nabla v_{loc}}=\lim_{h\to 0}\frac{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}{h}\frac{\overline{\mathbf{a}\nabla
v_{loc}}(\phi+h\omega_{x})-\overline{\mathbf{a}\nabla
v_{loc}}(\phi)}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}$

It follows from the regularity assuption that
$\mathbf{a}_{e}=V^{\prime\prime}$ is uniformly Lipshitz, and since
$\mathbf{a}_{e}$ is a function of $\nabla\phi$, we may write $\frac{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}{h}=\nabla^{*}\cdot\mathsf{b}^{h}$,
where $\mathsf{b}^{h}$ is bounded by a constant independent of $h$. We claim
that there exists $C(d,\lambda,\Lambda)<\infty$, such that

(4.18) $\left\|\frac{\overline{\mathbf{a}\nabla
v_{loc}}(\phi+h\omega_{x})-\overline{\mathbf{a}\nabla
v_{loc}}(\phi)}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C3^{-4m/3}.$

This implies, by (4.17), and the estimate that $|\nabla_{x}\nabla_{y}\cdot
G_{{\scaleobj{1.2}{\square}}_{m}}(x,y)|\leq\frac{C}{\max\\{1,|x-y|^{2}\\}}$,
that

$\operatorname{var}_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\left[\left(\mathbf{a}\nabla
v_{loc}\right)_{{\scaleobj{1.2}{\square}}_{m}}\right]\\\
\leq\frac{1}{\lambda}\sum_{x,y\in{\scaleobj{1.2}{\square}}_{m}^{\circ}}\left|\mathsf{b}(x)\nabla_{x}^{\cdot}G_{{\scaleobj{1.2}{\square}}_{m}}(x,y)\nabla_{y}^{*}\cdot\mathsf{b}(y)\right|\left\|\partial_{x}\overline{\mathbf{a}\nabla
v_{loc}}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\left\|\partial_{y}\overline{\mathbf{a}\nabla
v_{loc}}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\\\ \leq
C3^{-8m/3}\sum_{x,y\in{\scaleobj{1.2}{\square}}_{m}^{\circ}}\left|\mathsf{b}(x)\nabla_{x}^{*}\cdot
G_{{\scaleobj{1.2}{\square}}_{m}}(x,y)\nabla_{y}\mathsf{b}(y)\right|\\\ \leq
C3^{-8m/3}\sum_{x,y\in{\scaleobj{1.2}{\square}}_{m}^{\circ}}\frac{C}{\max\\{1,|x-y|^{2}\\}}\leq
Cm3^{-2m/3}$

Thus we conclude the lemma. To prove (4.18), notice that

$\left\|\frac{\overline{\mathbf{a}\nabla
v_{loc}}(\phi+h\omega_{x})-\overline{\mathbf{a}\nabla
v_{loc}}(\phi)}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
2\left\|\frac{(\mathbf{a}(\phi+h\omega_{x})(\nabla
v_{loc}(\phi+h\omega_{x})-\nabla
v_{loc}(\phi)))_{{\scaleobj{1.2}{\square}}_{m}}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\\\
+2\left\|\frac{((\mathbf{a}(\phi+h\omega_{x})-\mathbf{a}(\phi))\nabla
v_{loc}(\phi)))_{{\scaleobj{1.2}{\square}}_{m}}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}$

And since

(4.19) $((\mathbf{a}(\phi+h\omega_{x})-\mathbf{a}(\phi))\nabla
v_{loc}(\phi)))_{{\scaleobj{1.2}{\square}}_{m}}=3^{-2m}\sum_{e\ni
x}(\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi))\nabla v_{loc}(e)$

Therefore

$\left\|\frac{((\mathbf{a}(\phi+h\omega_{x})-\mathbf{a}(\phi))\nabla
v_{loc}(\phi)))_{{\scaleobj{1.2}{\square}}_{m}}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C3^{-2m}\sum_{e\ni x}\|\nabla
v_{loc}(e)\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}$

where the right side is bounded by

$C3^{-2m}\sum_{e\in\mathcal{E}({\scaleobj{1.2}{\square}}_{m/3})}\|\nabla
v_{loc}(e)\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C3^{-2m}3^{2m/3}\nu_{3^{m}}({\scaleobj{1.2}{\square}}_{m/3},p)\leq C3^{-4m/3}$

To estimate the other term, we claim that for the unique
$z\in\mathcal{Z}_{m/3}$ such that $x\in z+{\scaleobj{1.2}{\square}}_{m/3}$,

(4.20) $\left\|\frac{\nabla v_{loc}(\phi+h\omega_{x})-\nabla
v_{loc}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(z+{\scaleobj{1.2}{\square}}_{m/3},\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C3^{2m/3}$

Indeed, since $\widetilde{v}_{loc}:=v_{loc}(\phi+h\omega_{x})$ is a solution
to the Dirichlet problem:

(4.21) $\left\\{\begin{aligned}
&\left(-\mathcal{L}_{\mu}+\nabla^{*}\mathbf{a}(\phi+h\omega_{x})\nabla\right)\widetilde{v}_{loc}=0&\mbox{in}&\
z+{\scaleobj{1.2}{\square}}_{m/3}\times\Omega_{0}(z+{\scaleobj{1.2}{\square}}_{m/3}),\\\
&\widetilde{v}_{loc}-\ell_{p}=0&\mbox{on}&\ \partial
z+{\scaleobj{1.2}{\square}}_{m/3}\times\Omega_{0}(z+{\scaleobj{1.2}{\square}}_{m/3}),\end{aligned}\right.$

Testing the equation of $\widetilde{v}_{loc}$ and $v_{loc}$ and subtract them,
we obtain

$\left\langle\sum_{e\in\mathcal{E}(z+{\scaleobj{1.2}{\square}}_{m/3})}\mathbf{a}_{e}(\phi)\nabla
v_{loc}(e)\nabla(v_{loc}(e)-\widetilde{v}_{loc}(e))\right\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}\\\
\leq\left\langle\sum_{e\in\mathcal{E}(z+{\scaleobj{1.2}{\square}}_{m/3})}\mathbf{a}_{e}(\phi+h\omega_{x})\nabla\widetilde{v}_{loc}(e)\nabla(v_{loc}(e)-\widetilde{v}_{loc}(e))\right\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}$

Therefore

$\left\|\nabla\widetilde{v}_{loc}-\nabla
v_{loc}\right\|_{L^{2}(z+{\scaleobj{1.2}{\square}}_{m/3},\mu_{{\scaleobj{1.2}{\square}}_{m}})}^{2}\leq\left\langle\sum_{e\ni
x}|\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)|\nabla
v_{loc}(e)\nabla(v_{loc}(e)-\widetilde{v}_{loc}(e))\right\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}$

thus by Cauchy-Schwarz

$\left\|\frac{\nabla\widetilde{v}_{loc}-\nabla v_{loc}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(z+{\scaleobj{1.2}{\square}}_{m/3},\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq\sum_{e\ni
x}\|\nabla v_{loc}(e)\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C3^{2m/3}\nu_{3^{m}}({\scaleobj{1.2}{\square}}_{m/3},p)$

this is (4.20). Therefore

(4.22) $\left\|\frac{(\mathbf{a}(\phi+h\omega_{x})(\nabla
v_{loc}(\phi+h\omega_{x})-\nabla
v_{loc}(\phi)))_{{\scaleobj{1.2}{\square}}_{m}}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(\mu_{{\scaleobj{1.2}{\square}}_{m}})}\\\
\leq C3^{-2m}\left\|\frac{\nabla v_{loc}(\phi+h\omega_{x})-\nabla
v_{loc}}{\sum_{e\ni
x}\mathbf{a}_{e}(\phi+h\omega_{x})-\mathbf{a}_{e}(\phi)}\right\|_{L^{2}(z+{\scaleobj{1.2}{\square}}_{m/3},\mu_{{\scaleobj{1.2}{\square}}_{m}})}\leq
C3^{-4m/3}$

Combine the estimates (4.19) and (4.22) above we conclude (4.18), and thus
finish the proof of the lemma. ∎

We are ready to prove the convergence of the fluxes with an algebraic rate.

######  Lemma 4.10.

There exist $\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that, for every $n\in\mathbb{N}$,

(4.23) $3^{-2n}\left\|\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-{\overline{\mathbf{a}}}p\right\|_{\underline{H}^{-1}({\scaleobj{1.2}{\square}}_{n},\mu_{{\scaleobj{1.2}{\square}}_{n}})}\leq
C3^{-n\beta}.$

###### Proof.

We first notice, that by the representation of ${\overline{\mathbf{a}}}(Q)$ in
(4.8), and the definition of $\mathsf{B}$ norm in (3.4), for all $m\geq 1$ we
may write the spatial average of the flux as

(4.24) $\langle\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{m},p)\right)_{{\scaleobj{1.2}{\square}}_{m}}\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{m}}}={\overline{\mathbf{a}}}({\scaleobj{1.2}{\square}}_{m})p.$

Recall that for $m\leq n$,
$\mathcal{Z}_{m}:=3^{m}\mathbb{Z}^{d}\cap{\scaleobj{1.2}{\square}}_{n}$. We
apply the multiscale Poincaré inequality ( Proposition A.1) to obtain

(4.25) $\displaystyle 3^{-2n}\left\|\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-{\overline{\mathbf{a}}}p\right\|_{\underline{H}^{-1}({\scaleobj{1.2}{\square}}_{n},\mu_{{\scaleobj{1.2}{\square}}_{n}})}\leq
C3^{-n}\left\|\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-{\overline{\mathbf{a}}}p\right\|_{\underline{L}^{2}({\scaleobj{1.2}{\square}}_{n},\mu_{{\scaleobj{1.2}{\square}}_{n}})}$
$\displaystyle+C\sum_{m=0}^{n-1}3^{m-n}\left\langle\left(\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-{\overline{\mathbf{a}}}p\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\right)^{1/2}\right\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{n}}}.$

The first term on the right side above is bounded by

$C3^{-n}\left(\left\|\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)\right\|_{\underline{L}^{2}({\scaleobj{1.2}{\square}}_{n},\mu_{{\scaleobj{1.2}{\square}}_{n}})}+|p|^{2}\right)\leq
C^{\prime}3^{-n}$

For the second term, triangle inequality implies that

$\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-{\overline{\mathbf{a}}}p\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\\\
\leq\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\langle\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\rangle_{\mu_{y+{\scaleobj{1.2}{\square}}_{m}}}-{\overline{\mathbf{a}}}p\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\\\
+\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\langle\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\rangle_{\mu_{y+{\scaleobj{1.2}{\square}}_{m}}}-\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\\\
+\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}$

By (4.24) and Proposition 4.6, we have

(4.26) $\left|\left(\langle\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\rangle_{\mu_{y+{\scaleobj{1.2}{\square}}_{m}}}-{\overline{\mathbf{a}}}p\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\leq\left|{\overline{\mathbf{a}}}({\scaleobj{1.2}{\square}}_{m})-{\overline{\mathbf{a}}}\right|^{2}|p|^{2}\leq
C\left(|p|+|q|+\mathsf{K}_{0}\right)^{2}3^{-2\beta m}$

By the second variation, we have

(4.27)
$\left\langle\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\right\rangle_{\mu_{{\scaleobj{1.2}{\square}}_{n}}}\leq
C(\nu_{3^{n}}({\scaleobj{1.2}{\square}}_{m},p)-\nu_{3^{n}}({\scaleobj{1.2}{\square}}_{n},p))\leq
C3^{-m\beta}$

where the last inequality follows from the quantitative convergence of
$\nu_{3^{n}}$ (Proposition 4.6) and triangle inequality.

We also apply the variance estimate Lemma 4.9 to conclude that for every
$y\in\mathcal{Z}_{m}$,

(4.28) $\left|\left(\langle\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\rangle_{\mu_{y+{\scaleobj{1.2}{\square}}_{m}}}-\mathbf{a}\nabla
v(\cdot,y+{\scaleobj{1.2}{\square}}_{m},p)\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\leq
C3^{-m\beta}$

Combining (4.26), (4.27) and (4.28), we conclude there exists
$\beta=\beta(d,\lambda,\Lambda)>0$ and $C=C(d,\lambda,\Lambda)<\infty$ , such
that

(4.29)
$\frac{1}{\left|\mathcal{Z}_{m}\right|}\sum_{y\in\mathcal{Z}_{m}}\left|\left(\mathbf{a}\nabla
v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)-{\overline{\mathbf{a}}}p\right)_{y+{\scaleobj{1.2}{\square}}_{m}}\right|^{2}\leq
C3^{-m\beta}$

Substitute the above estimates into (4.25), and summing over $m$, we conclude
the Lemma. ∎

###  4.3. Proof of Theorem 4.4

In the previous subsection, we established the convergence of the solution to
a Dirichlet problem with an affine boundary condition, with an algebraic rate
of convergence. The equation (4.2) we would like to homogenize is more
general, but if we localize it on a mesoscale, the boundary condition becomes
approximately affine. In this section we prove Theorem 4.4 by estimating the
homogenization error in terms of the error in the convergence of the
correctors and fluxes defined in the previous subsection. The proof goes
through a standard, deterministic argument known as the two-scale expansions,
that follows closely along the argument of [3, 4].

Given $R\geq 1$, let $m=\inf\\{k\in\mathbb{N}:3^{k}\geq R\\}$. We may view the
solution $u_{R}$ (and respectively, $u$) to the the equation (4.2) (resp.
(4.4)) as elements in a slightly larger domain
${\scaleobj{1.2}{\square}}_{m}\times\Omega_{0}({\scaleobj{1.2}{\square}}_{m})$
(resp. in $U:=\frac{1}{R}{\scaleobj{1.2}{\square}}_{m}$). Specifically, we set
the value of $u_{R}$ to be $0$ and $v(\cdot,{\scaleobj{1.2}{\square}}_{n},p)$
(defined in (4.6)) to be $\ell_{p}$ outside $Q_{R}$, and set $u=0$ outside
$[0,1]^{2}$.

We now rescale the problem and study the Dirichlet problem in a fixed domain
with mesh size goes to zero. Denote by $\varepsilon=R^{-1}$ and for
$i=1,\cdots,d$ define the finite volume corrector

$\chi_{e_{i}}^{\varepsilon}(\cdot):=v(\cdot,{\scaleobj{1.2}{\square}}_{m},e_{i})-\ell_{e_{i}}.$

For $u$ that solves (4.4), we construct the modified two scale expansion

$w^{\varepsilon}\left(x\right)=u\left(x\right)+\varepsilon\nabla
u\left(x\right)\cdot\chi^{\varepsilon}(\frac{x}{\varepsilon}),\quad\forall
x\in U$

Step 1.  Substitute $w^{\varepsilon}$ into the Eq. (4.2) to obtain there exist
$\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that,

$\left\|\nabla^{*}\cdot\left(\mathbf{a}\nabla
w^{\varepsilon}\right)-\mathcal{L}_{\mu}w^{\varepsilon}-\nabla\cdot
f^{\varepsilon}\right\|_{\underline{H}^{-1}\left(U,\mu_{R}\right)}\leq
C\|f\|_{L^{\infty}(U)}\varepsilon^{\beta}.$

We begin by computing

(4.30) $\nabla w^{\varepsilon}=\nabla u+\nabla
u\left(\cdot\right)\cdot\nabla\chi^{\varepsilon}+\varepsilon\sum_{j=1}^{d}\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\nabla\nabla_{j}u\\\
=\sum_{j=1}^{d}\nabla_{j}u\nabla
v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})+\varepsilon\sum_{j=1}^{d}\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\nabla\nabla_{j}u$

where we used the definition of $\chi^{\varepsilon}_{e_{j}}$ in the last line.
This yields

(4.31) $\nabla^{*}\cdot\mathbf{a}\nabla
w^{\varepsilon}=\sum_{j=1}^{d}\nabla_{j}u\nabla^{*}\cdot\mathbf{a}\nabla
v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})+\sum_{j=1}^{d}\nabla^{*}\cdot\nabla_{j}u\left(\mathbf{a}\nabla
v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})\right)+\varepsilon\nabla^{*}\cdot\mathbf{a}\left(\sum_{j=1}^{d}\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\nabla\nabla_{j}u\right)$

We also obtain, using that $u(x)$ has no $\phi$-dependence,

$\mathcal{L}_{\mu}w^{\varepsilon}=\sum_{j=1}^{d}\nabla_{j}u\cdot\mathcal{L}_{\mu}v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})$

Therefore

(4.32) $\nabla^{*}\cdot\mathbf{a}\nabla
w^{\varepsilon}-\mathcal{L}_{\mu}w^{\varepsilon}=\sum_{j=1}^{d}\nabla_{j}u\left(\nabla^{*}\cdot\mathbf{a}\nabla
v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})-\mathcal{L}_{\mu}v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})\right)\\\
+\sum_{j=1}^{d}\nabla^{*}\cdot\nabla_{j}u\left(\mathbf{a}\nabla
v(\frac{\cdot}{\varepsilon},{\scaleobj{1.2}{\square}}_{m},e_{j})-{\overline{\mathbf{a}}}e_{j}\right)+\sum_{j=1}^{d}\nabla^{*}\cdot\nabla_{j}u\cdot{\overline{\mathbf{a}}}e_{j}+\varepsilon\nabla^{*}\cdot\mathbf{a}\left(\sum_{j=1}^{d}\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\nabla\nabla_{j}u\right)$

The first term on the right side above vanishes since $v$ solves the equation
(4.6). The third term is

$\sum_{j=1}^{d}\nabla^{*}\cdot\nabla_{j}u\cdot{\overline{\mathbf{a}}}e_{j}=\nabla^{*}\cdot{\overline{\mathbf{a}}}\nabla
u=f$

To estimate the rest two terms, notice that $u$ solves the constant
coefficient equation (4.4), and therefore

$\left\|\nabla^{k}\nabla_{j}u\right\|_{L^{\infty}(U)}\leq\left\|\int_{[0,1]^{2}}\nabla^{k+1}\nabla_{j}G_{{\overline{\mathbf{a}}}}(x,y)f(y)\,dy\right\|_{L^{\infty}([0,1]^{2})}\leq
C\|f\|_{L^{\infty}([0,1]^{2})},$

where $G_{{\overline{\mathbf{a}}}}(x,y)$ is the Green’s function for the
operator $\nabla^{*}\cdot{\overline{\mathbf{a}}}\nabla$ in $[0,1]^{2}$ with
zero boundary condition.

We apply the quantitative convergence of the corrector (Lemma 4.8) and the
fluxes (Lemma 4.10) in the rescaled setting, which implies there there exist
$\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that,

$\sum_{j=1}^{d}\left\|\nabla^{*}\cdot\nabla_{j}u\left(\mathbf{a}\nabla
v(\cdot,e_{j})-{\overline{\mathbf{a}}}e_{j}\right)\right\|_{\underline{H}^{-1}(U,\mu_{R})}\\\
\leq
C\left\|\nabla_{j}u\right\|_{L^{\infty}}\sum_{j=1}^{d}\left\|\mathbf{a}\nabla
v(\cdot,e_{j})-{\overline{\mathbf{a}}}e_{j}\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq
C\|f\|_{L^{\infty}([0,1]^{2})}\varepsilon^{\beta}$

and

$\left\|\varepsilon\nabla^{*}\cdot\mathbf{a}\left(\sum_{j=1}^{d}\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\nabla\nabla_{j}u\right)\right\|_{L^{2}(U,\mu_{R})}\leq
C\left\|\nabla\nabla_{j}u\right\|_{L^{\infty}}\varepsilon\sum_{j=1}^{d}\|\chi^{\varepsilon}_{e_{j}}\|_{\underline{L}^{2}(Q_{R},\mu_{R})}\leq
C\|f\|_{L^{\infty}([0,1]^{2})}\varepsilon^{\beta}$

Thus we conclude

$\left\|\nabla^{*}\cdot\left(\mathbf{a}\nabla
w^{\varepsilon}\right)+\mathcal{L}_{\mu}w^{\varepsilon}-f\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq
C\varepsilon^{\beta}.$

And Step 1 follows from the fact that $\|\nabla^{*}\cdot
f^{\varepsilon}(\frac{\cdot}{\varepsilon})-f(\cdot)\|_{L^{\infty}([0,1]^{2})}\leq\varepsilon^{\alpha}$.

Step 2 We deduce that there exist
$\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that,

$\left\|u^{\varepsilon}-w^{\varepsilon}\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq
C\|f\|_{L^{\infty}(U)}\varepsilon^{\beta}.$

Since $u$ solves (4.2), we use the coercivity of the $\mathsf{B}_{\mu,Q}$ norm
defined in (3.4) to obtain

$\left\|u^{\varepsilon}-w^{\varepsilon}\right\|_{\underline{H}^{-1}(U,\mu_{R})}^{2}\leq
C\mathsf{B}_{\mu_{R},Q_{R}}\left[u^{\varepsilon}-w^{\varepsilon},u^{\varepsilon}-w^{\varepsilon}\right]$

Testing (4.2) by $u^{\varepsilon}-w^{\varepsilon}$ then implies

$\mathsf{B}_{\mu_{R},Q_{R}}\left[u^{\varepsilon}-w^{\varepsilon},u^{\varepsilon}\right]=\frac{1}{|Q_{R}|}\sum_{e\in\mathcal{E}(\varepsilon
Q_{R})}\left\langle(\nabla u^{\varepsilon}(\frac{e}{\varepsilon},\cdot)-\nabla
w^{\varepsilon}(e,\cdot)f^{\varepsilon}(\frac{e}{\varepsilon}))\right\rangle_{\mu_{R}},$

Therefore by the definition of $\mathsf{B}_{\mu,Q}$ and Poincaré inequality,

$\left\|\nabla u^{\varepsilon}-\nabla
w^{\varepsilon}\right\|_{L^{2}\left(U,\mu_{R}\right)}^{2}\leq\frac{1}{|Q_{R}|}\sum_{e\in\mathcal{E}(\varepsilon
Q_{R})}\left\langle(\nabla u^{\varepsilon}(\frac{e}{\varepsilon},\cdot)-\nabla
w^{\varepsilon}(e,\cdot)f^{\varepsilon}(\frac{e}{\varepsilon}))\right\rangle_{\mu_{R}}-C\mathsf{B}_{\mu_{R},Q_{R}}\left[w^{\varepsilon},u^{\varepsilon}-w^{\varepsilon}\right]\\\
\leq C\left\|u^{\varepsilon}-w^{\varepsilon}\right\|_{H^{1}\left(\varepsilon
Q_{R},\mu_{R}\right)}\left\|\nabla^{*}\cdot\left(\mathbf{a}\nabla
w^{\varepsilon}\right)-\mathcal{L}_{\mu}w^{\varepsilon}-\nabla^{*}\cdot
f^{\varepsilon}\right\|_{H^{-1}\left(\varepsilon Q_{R},\mu_{R}\right)}\\\ \leq
C\left\|\nabla u^{\varepsilon}-\nabla
w^{\varepsilon}\right\|_{L^{2}\left(\varepsilon
Q_{R},\mu_{R}\right)}\left\|\nabla^{*}\cdot\left(\mathbf{a}\nabla
w^{\varepsilon}\right)-\mathcal{L}_{\mu}w^{\varepsilon}-\nabla^{*}\cdot
f^{\varepsilon}\right\|_{\underline{H}^{-1}(U,\mu_{R})}$

Absorbing $\left\|\nabla u^{\varepsilon}-\nabla
w^{\varepsilon}\right\|_{L^{2}\left(\varepsilon Q_{R},\mu_{R}\right)}$ to the
left side and combining with Step 1 we conclude Step 2.

Step 3 We show that there exist
$\beta(d,\lambda,\Lambda)\in\left(0,\tfrac{1}{2}\right]$ and
$C(d,\lambda,\Lambda)<\infty$ such that,

$\left\|w^{\varepsilon}-u\right\|_{L^{2}\left(\varepsilon
Q_{R},\mu_{R}\right)}+\left\|\nabla w^{\varepsilon}-\nabla
u\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq C\varepsilon^{\beta},$

When $f^{\varepsilon}$ is a constant in $Q_{R}$, this just follows from the
quantitative convergence of the corrector (Lemma 4.8). Here we simply compute
the derivative of the second term in the two-scale expansion.

Indeed, we have

$\left\|\nabla w^{\varepsilon}-\nabla
u\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq
C\left\|\varepsilon\sum_{j=1}^{d}\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\nabla_{j}u\right\|_{L^{2}(U,\mu_{R})}\leq
C\|\nabla_{j}u\|_{L^{\infty}([0,1]^{2})}\sum_{j=1}^{d}\varepsilon\|\chi^{\varepsilon}_{e_{j}}(\frac{\cdot}{\varepsilon})\|_{L^{2}(\varepsilon
Q_{R},\mu_{R})}$

Applying Lemma 4.8 we conclude

$\left\|\nabla w^{\varepsilon}-\nabla
u\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq C\varepsilon^{\beta},$

Since $w^{\varepsilon}-u\in H_{0}^{1}(U,\mu_{R})$, we apply Lemma A.2 to
conclude

$\left\|w^{\varepsilon}-u\right\|_{L^{2}\left(\varepsilon
Q_{R},\mu_{R}\right)}\leq C\left\|\nabla w^{\varepsilon}-\nabla
u\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq C\varepsilon^{\beta}.$

Step 4 Finally, we conclude by combining Steps 2 and 3 which yields

$\left\|\nabla u^{\varepsilon}-\nabla
u\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq\left\|\nabla
u^{\varepsilon}-\nabla
w^{\varepsilon}\right\|_{\underline{H}^{-1}(U,\mu_{R})}+\left\|\nabla
w^{\varepsilon}-\nabla u\right\|_{\underline{H}^{-1}(U,\mu_{R})}\leq
C(1+\|f\|_{L^{\infty}(U)})\varepsilon^{\beta}.$

Finally, the bound for $\|u^{\varepsilon}-u\|_{L^{2}(\varepsilon
Q_{R},\mu_{R})}$ follows from the above $\underline{H}^{-1}$ bound for the
gradients and the fact that since $u^{\varepsilon}-u\in H_{0}^{1}(U,\mu_{R})$,
we have by Lemma A.2

$\|u^{\varepsilon}-u\|_{\underline{L}^{2}(\varepsilon Q_{R},\mu_{R})}\leq
C\left\|\nabla u^{\varepsilon}-\nabla
u\right\|_{\underline{H}^{-1}(U,\mu_{R})}.$

##  5\. Decoupling of the $\nabla\phi$-field

###  5.1. Approximate harmonic coupling

For the discrete Gaussian Free Field(GFF), there is a nice orthogonal
decomposition. More precisely, the conditioned field inside the domain is the
discrete harmonic extension of the boundary value to the whole domain plus an
independent copy of a zero boundary discrete GFF.

While this exact decomposition does not carry over to general gradient Gibbs
measures, the next result due to Jason Miller, see [20], provides an
approximate version.

######  Theorem 5.1 ([20]).

Let $D\subseteq\mathbb{Z}^{2}$ be a simply connected domain of diameter $R$,
and denote $D^{r}=\left\\{x\in D:\text{dist}(x,\partial D)>r\right\\}$.
Suppose that $f:\partial D\rightarrow\mathbb{R}$ satisfies $\max_{x\in\partial
D}\left|f\left(x\right)\right|\leq 2\left|\log R\right|^{2}$. Let $\phi$ be
sampled from the Gibbs measure (3.5) on $D$ with zero boundary condition, and
let $\phi^{f}$ be sampled from Gibbs measure on $D$ with boundary condition
$f$. Then there exist constants $c,\gamma,\delta^{\prime}\in\left(0,1\right)$,
that only depend on $V$, so that if $r>cR^{\gamma}$ then the following holds.
There exists a coupling $\left(\phi,\phi^{f}\right)$, such that if
$\widehat{\phi}:D^{r}\rightarrow\mathbb{R}$ is discrete harmonic with
$\widehat{\phi}|_{\partial D^{r}}=\phi^{f}-\phi|_{\partial D^{r}}$, then

$\mathbb{P}\left(\phi^{f}=\phi+\widehat{\phi}\text{ in }D^{r}\right)\geq
1-cR^{-\delta^{\prime}}.$

Here and in the sequel of the paper, for a set $A\subseteq\mathbb{Z}^{2}$ and
a point $x\in\mathbb{Z}^{2}$, we use $\text{dist}(x,A)$ to denote the
(lattice) distance from $x$ to $A$. Since the above theorem requires that the
boundary condition $f$ is not too large, we introduce the “good” event

$\mathcal{M}=\left\\{\phi:\max_{v\in
D}\left|\phi\left(v\right)\right|<\left(\log R\right)^{2}\right\\},$

which is typical since the Brascamp-Lieb inequality implies that $\max_{v\in
D}|\phi(v)|\leq O\left(\log R\right)$ with high probability. Indeed, by
applying the exponential Brascamp-Lieb inequality (3.7) and a union bound we
immediately obtain

######  Lemma 5.2.

There is some $c_{1}>0$, such that
$\mathbb{P}^{D,0}\left(\mathcal{M}^{c}\right)\leq\exp\left(-c_{1}\left(\log
R\right)^{3}\right)$.

We will use repeatedly the following consequence of Theorem 5.1. It applies to
functions $\rho$ such that the integral of $\rho$ against a harmonic function
is always zero.

######  Lemma 5.3 ([8], Lemma 2.7).

There exists constants $\delta=\delta(V),\gamma=\gamma(V)>0$ and $C<\infty$,
such that for any simply connected $D\subseteq\mathbb{Z}^{2}$ of diameter $R$,
any $r>R^{\gamma}$ and any $\rho:D\rightarrow\mathbb{R}$ supported on $D^{r}$
that satisfies $\sum_{x\in D^{r}}\rho\left(x\right)f\left(x\right)=0$ for all
functions $f$ harmonic in $D^{r}$, and $\sum_{y\in
D}\left|\rho\left(y\right)\right|<\infty$, we have for $R$ large enough,

$\displaystyle\left|\mathbb{E}^{D,f}\left[\exp\left(\sum_{x\in
D}\rho\left(x\right)\phi^{f}\left(x\right)\right)1_{\mathcal{M}}\right]-\mathbb{E}^{D,0}\left[\exp\left(\sum_{x\in
D}\rho\left(x\right)\phi\left(x\right)\right)1_{\mathcal{M}}\right]\right|$
$\displaystyle\leq$ $\displaystyle
2\exp\left(C\operatorname{var}_{G,D}\left(\sum_{x\in
D}\rho\left(x\right)\phi\left(x\right)\right)\right)R^{-\delta}\text{.}$

We will apply Lemma 5.3 to the increment of harmonic averages of the field in
$D$, defined in the section below. The increments have finite variances and
thus changing the boundary to zero only gives an error of order $R^{-\delta}$.

###  5.2. Harmonic averages

We will apply Theorem 5.1 to study the harmonic average of the $\nabla\phi$
field. Given $B\subseteq\mathbb{Z}^{2}$, $v\in B$ and $y\in\partial B$, we
denote by $a_{B}\left(v,\cdot\right)$ the harmonic measure on $\partial B$
seen from $v$. In other words, let $S^{x}$ denote the simple random walk
starting at $x$, and $\tau_{\partial
B}=\inf\left\\{t>0:S\left[t\right]\in\partial B\right\\}$, we have

$a_{B}\left(x,y\right)=\mathbb{P}\left(S^{x}\left[\tau_{\partial
B}\right]=y\right).$

Given $v\in\mathbb{Z}^{2}$ and $R>0$, let
$B_{R}\left(v\right)=\left\\{y\in\mathbb{Z}^{2}:\left|v_{1}-y_{1}\right|^{2}+\left|v_{2}-y_{2}\right|^{2}<R^{2}\right\\}$.
When $v=0$ we simply write $B_{R}\left(v\right)$ as $B_{R}$. Define the circle
average of the field with radius $R$ at $v$ by

(5.1) $C_{R}\left(v,\phi\right)=\sum_{y\in\partial
B_{R}\left(v\right)}a_{B_{R}\left(v\right)}\left(v,y\right)\phi\left(y\right).$

We introduce the geometric scales in order to carry out the multiscale
argument to prove Lemma 2.1 and 2.3. Let $\gamma=\gamma(V)$ be the constant in
Theorem 5.1, define the sequence of numbers
$\left\\{r_{k}\right\\}_{k=1}^{\infty}$,
$\left\\{r_{k,+}\right\\}_{k=0}^{\infty}$ and
$\left\\{r_{k,-}\right\\}_{k=0}^{\infty}$ by

(5.2) $\displaystyle r_{k}$ $\displaystyle=$ $\displaystyle e^{-k}N,$
$\displaystyle r_{k,+}$ $\displaystyle=$
$\displaystyle\left(1+r_{k}^{-\gamma}\right)r_{k},$ $\displaystyle r_{k,-}$
$\displaystyle=$ $\displaystyle\left(1-r_{k}^{-\gamma}\right)r_{k}.$

We also define

$\displaystyle X_{r_{k,+}}\left(v\right)$ $\displaystyle=$
$\displaystyle\sum_{r=\left(1-\frac{1}{4}r_{k}^{-\gamma}\right)r_{k,+}}^{\left(1+\frac{1}{4}r_{k}^{-\gamma}\right)r_{k,+}}\left(\frac{1}{2}r_{k}^{1-\gamma}\right)^{-1}C_{r}\left(v,\phi\right),$
$\displaystyle X_{r_{k,-}}\left(v\right)$ $\displaystyle=$
$\displaystyle\sum_{r=\left(1-\frac{1}{4}r_{k}^{-\gamma}\right)r_{k,-}}^{\left(1+\frac{1}{4}r_{k}^{-\gamma}\right)r_{k,-}}\left(\frac{1}{2}r_{k}^{1-\gamma}\right)^{-1}C_{r}\left(v,\phi\right),$
$\displaystyle X_{r_{k}}\left(v\right)$ $\displaystyle=$
$\displaystyle\sum_{r=\left(1-\frac{1}{4}r_{k}^{-\gamma}\right)r_{k}}^{\left(1+\frac{1}{4}r_{k}^{-\gamma}\right)r_{k}}\left(\frac{1}{2}r_{k}^{1-\gamma}\right)^{-1}C_{r}\left(v,\phi\right),$

where $C_{r}$ is defined in (5.1).

We also set the increment process $A_{k}:=X_{r_{k+1}}-X_{r_{k-}}$ and the
boundary layer error $\mathcal{E}_{k}:=X_{r_{k}}-X_{r_{k-}}$.

Notice that the harmonic average process $X$ defined above is slightly
different from the one in [8], Section 3. Namely, the width of the harmonic
average is thinner (of the order $r_{k}^{1-\gamma}$ instead of $r_{k}$). The
increment of the harmonic average process, $A_{k}$, is crucial to the
multiscale argument below. We may write

(5.3)
$A_{k}\left(v,\phi\right)=\left(\sum_{r=\left(1-\frac{1}{4}r_{k+1}^{-\gamma}\right)r_{k+1}}^{\left(1+\frac{1}{4}r_{k+1}^{-\gamma}\right)r_{k+1}}\left(\frac{1}{2}r_{k+1}^{1-\gamma}\right)^{-1}-\sum_{r=\left(1-\frac{1}{4}r_{k}^{-\gamma}\right)r_{k,-}}^{\left(1+\frac{1}{4}r_{k}^{-\gamma}\right)r_{k,-}}\left(\frac{1}{2}r_{k}^{1-\gamma}\right)^{-1}\right)\sum_{y\in\partial
B_{r}\left(v\right)}a_{B_{r}\left(v\right)}\left(v,y\right)\phi\left(y\right).$

This can be written as $\sum_{y\in
B_{r_{k}}(v)}\rho_{r_{k}}\left(v,y\right)\phi\left(y\right)$, where we define

(5.4)
$\rho_{r_{k}}\left(v,y\right)=\left[\left(\frac{1}{2}r_{k+1}^{1-\gamma}\right)^{-1}\mathds{1}_{|r-r_{k+1}|\leq\frac{1}{4}r_{k+1}^{1-\gamma}}-\left(\frac{1}{2}r_{k}^{1-\gamma}\right)^{-1}\mathds{1}_{|r-r_{k,-}|\leq\frac{1}{4}r_{k}^{1-\gamma}}\right]a_{B_{\left|v-y\right|}\left(v\right)}\left(v,y\right).$

Here we omit the dependence of $r$ in the definition of $\rho_{r_{k}}$.

We now state two important properties of the increment process $A_{k}$. They
are consequences of Theorem 5.1, and the proof are essentially the same as
Lemma 3.1 and 3.2 of [8].

######  Lemma 5.4.

For any $k\in\mathbb{N}$, and any discrete harmonic function $h$ in
$B_{r_{k}}$, we have $\sum_{y\in
B_{r_{k}}(v)}\rho_{r}\left(v,y\right)h\left(y\right)=0.$

###### Proof.

Denote $B_{r_{k}}$ by $D$. Suppose $h$ takes the boundary value $h|_{\partial
D}=H$. We conclude the proof by showing for any $r\geq 1$

$\sum_{y\in\partial
B_{r}\left(v\right)}a_{B_{r}\left(v\right)}\left(v,y\right)h\left(y\right)=h\left(v\right).$

Indeed, since $h$ is harmonic,

$h\left(y\right)=\sum_{z\in\partial D}a_{D}\left(y,z\right)H\left(z\right).$

Using the fact that

$\sum_{y\in\partial
B_{r}\left(v\right)}a_{B_{r}\left(v\right)}\left(v,y\right)a_{D}\left(y,z\right)=a_{D}\left(v,z\right),$

we obtain

$\sum_{y\in\partial
B_{r}\left(v\right)}a_{B_{r}\left(v\right)}\left(v,y\right)h\left(y\right)=\sum_{z\in\partial
D}a_{D}\left(v,z\right)H\left(z\right)=h\left(v\right).$

∎

The following lemma is a consequence of Theorem 5.1 and the lemma above.

######  Lemma 5.5.

Suppose the same conditions in Theorem 5.1 holds. Let $\delta$ be the constant
from Theorem 5.1. Let $\phi^{f}$ be sampled from gradient field (3.5) on
$B_{r_{k}}$ with boundary condition $f$, and $\phi^{0}$ be sampled from the
zero boundary gradient field on $B_{r_{k}}$. Then, on an event with
probability $1-O\left(r_{k}^{-\delta}\right)$, we have

$A_{k}\left(v,\phi^{f}\right)=A_{k}\left(v,\phi^{0}\right).$

###  5.3. Proof of Lemma 2.1 and Lemma 2.3

A key ingredient for the characteristic function estimates (Lemma 2.1 and
Lemma 2.3) is the decoupling estimates Proposition 5.6 below.

For $r>0$ denote by $\mathbb{P}^{r,0}$ the law of the Ginzburg-Landau field in
$B_{r}\left(v\right)$ with zero boundary condition (and denote by
$\mathbb{E}^{r,0}$ the corresponding expectation). The basic building block of
Lemma 2.1 and 2.3 is the following.

######  Proposition 5.6.

There exists $\delta=\delta(V)>0$ and $C=C(d,\lambda,\Lambda)<\infty$, such
that for all $|s|^{2}<\min\\{\frac{\delta(\log N-k-1)}{Ck},\frac{1}{2}\\}$ and
$k\leq\log N-\sqrt{\log N}$,

(5.5)
$\log\left\langle\exp\left(isX_{r_{k}}\right)\right\rangle_{\mu_{N}}=\sum_{j=1}^{k}\log\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]+\log\left\langle\exp\left(isX_{r_{0}}\right)\right\rangle_{\mu_{N}}+O\left(\sum_{j=1}^{k}e^{Cs^{2}k}r_{j}^{-\delta}\right).$

###### Proof.

We gave an inductive proof, by running an induction jointly with (5.5) and

(5.6) $\left\langle\exp\left(isX_{r_{k}}\right)\right\rangle_{\mu_{N}}\geq
e^{-Cs^{2}k}$

for some $C=C(d,\lambda,\Lambda)<\infty$. For the base case $k=0$, (5.5)
trivially holds. To see (5.6) holds for $k=0$, we Taylor expand the
exponential and apply the Brascamp-Lieb inequality to obtain

$\left\langle\exp\left(isX_{r_{0}}\right)\right\rangle_{\mu_{N}}\geq
1-\frac{s^{2}}{2}\operatorname{var}_{\mu_{N}}X_{r_{0}}\geq
1-Cs^{2}\operatorname{var}_{\mu_{G,Q_{N}}}X_{r_{0}}\geq e^{-C^{\prime}s^{2}}$

To keep iterating, we denote by $\mathcal{M}_{k}:=\left\\{\max_{x\in
B_{r_{k-1}}}|\phi(x)|\leq(\log r_{k-1})^{2}\right\\}$. By Lemma 5.7,
$\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]\leq N^{-c_{1}}$. On the event
$\mathcal{M}_{k}$ , we may write

(5.7)
$\left\langle\exp\left(isX_{r_{k}}\right)\mathds{1}_{\mathcal{M}_{k}}\right\rangle_{\mu_{N}}=\left\langle\exp\left(isX_{r_{k-1}}\right)\mathbb{E}\left[\exp\left(isA_{k-1}\right)\exp(is\mathcal{E}_{k-1})|\mathcal{F}_{k-1}\right]\mathds{1}_{\mathcal{M}_{k}}\right\rangle_{\mu_{N}}\\\
=\left\langle\exp\left(isX_{r_{k-1}}\right)\mathbb{E}\left[\exp\left(isA_{k-1}\right)|\mathcal{F}_{k-1}\right]\mathds{1}_{\mathcal{M}_{k}}\right\rangle_{\mu_{N}}\\\
+\left\langle\exp\left(isX_{r_{k-1}}\right)\exp\left(isA_{k-1}\right)\left(\exp(is\mathcal{E}_{k-1})-1\right)\mathds{1}_{\mathcal{M}_{k}}\right\rangle_{\mu_{N}}$

We conclude using Lemma 5.3 that

$\mathbb{E}\left[\exp\left(isA_{k-1}\right)|\mathcal{F}_{k-1}\right]=\mathbb{E}^{r_{k-1},0}\left[\exp\left(isA_{k-1}\right)\right]+R_{k-1}$

such that with probability $1-\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]$ , there
is some $C<\infty$ and $\delta_{1}>0$ such that $|R_{k-1}|\leq
2\exp\left(C_{1}\operatorname{var}_{G,B_{r_{k-1}}}A_{k-1}\right)r_{k-1}^{-\delta_{1}}\leq
Cr_{k-1}^{-\delta_{1}}$, and on an event with probability
$\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]$, $|R_{k-1}|$ is bounded by $1$. This
implies, in particular, $\left\langle|R_{k-1}|^{2}\right\rangle_{\mu_{N}}\leq
Cr_{k-1}^{-\delta_{1}}$. For the last term in (5.7), applying the Brascamp-
Lieb inequality yields

$\left\langle\left|\exp(is\mathcal{E}_{k-1})-1\right|\right\rangle_{\mu_{N}}\leq|s|\left\langle\left|\mathcal{E}_{k-1}\right|\right\rangle_{\mu_{N}}\leq\left\langle\left|\mathcal{E}_{k-1}^{2}\right|^{\frac{1}{2}}\right\rangle_{\mu_{N}}\leq
Cr_{k-1}^{-\gamma/2}.$

The two estimates above implies that for every $k\in\mathbb{N}$, we have

(5.8)
$\left\langle\exp\left(isX_{r_{k}}\right)\mathds{1}_{\mathcal{M}_{k}}\right\rangle_{\mu_{N}}=\left\langle\exp\left(isX_{r_{k-1}}\right)\right\rangle_{\mu_{N}}\mathbb{E}^{r_{k-1},0}\left[\exp\left(isA_{k-1}\right)\right]+O(r_{k-1}^{-\delta_{1}/2})+O(r_{k-1}^{-\gamma/2}).$

Denote $\min\\{\delta_{1}/2,\gamma/2\\}$ as $\delta$. We now apply the
induction hypothesis and the fact that

$\mathbb{E}^{r_{k-1},0}\left[\exp\left(isA_{k-1}\right)\right]\geq
1-\frac{s^{2}}{2}\operatorname{var}_{\mu_{B_{r_{k-1}}}}A_{k-1}\geq
1-Cs^{2}\operatorname{var}_{\mu_{G,B_{r_{k-1}}}}A_{k-1}\geq
e^{-C^{\prime}s^{2}}$

to obtain for some $C<\infty$,

$\left\langle\exp\left(isX_{r_{k}}\right)\mathds{1}_{\mathcal{M}_{k}}\right\rangle_{\mu_{N}}\geq
e^{-Cs^{2}(k+1)}+O(r_{k-1}^{-\delta})$

therefore (5.6) follows from the fact that
$e^{-Cs^{2}(k+1)}>r_{k-1}^{-\delta/2}$, which is implied by the condition
$|s|^{2}<\min\\{\frac{\delta(\log N-k-1)}{Ck},\frac{1}{2}\\}$, and
$\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]\leq N^{-c_{1}}$. Finally, by taking the
logarithm of (5.8) we obtain (5.5) for the case $k+1$. ∎

######  Lemma 5.7.

There exists $c_{1}>0$, such that for all $k\leq\log N-\sqrt{\log N}$,
$\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]\leq N^{-c_{1}}$.

###### Proof.

By making a union bound and applying the exponential Brascamp-Lieb (3.7) and
the Chebyshev inequality, we have for all $t\in\mathbb{R}$,

$\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]\leq\sum_{x\in
B_{r_{k}}}\mathbb{P}^{N,0}[\phi(x)>(\log r_{k})^{2}]\leq r_{k}^{2}e^{-t(\log
r_{k})^{2}}e^{Ct^{2}\log N}.$

Optimize over $t$, and use the fact that $\log r_{k}\geq(\log
N)^{\frac{1}{2}}$, we see that there exists $c_{1}>0$, such that for $N$
sufficiently large,

$\mathbb{P}^{N,0}[\mathcal{M}_{k}^{c}]\leq r_{k}^{2}e^{-\frac{2c_{1}(\log
r_{k})^{4}}{\log N}}\leq N^{-c_{1}}$

∎

We also need an algebraic rate convergence of the variance of $A_{k}$ stated
below.

######  Lemma 5.8.

There exists $\mathbf{g}=\mathbf{g}(V)>0$ and
$\beta=\beta(d,\lambda,\Lambda)>0$, such that for all $j=1,\cdots\log N$,

$\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]=\mathbf{g}+O(r_{j}^{-\beta}).$

###### Proof.

Recall from (5.3) that, we may write $A_{j}=\sum_{x\in
B_{r_{j}}}\phi(x)\rho_{r_{j}}(x)$, where $\rho_{r_{j}}$ is defined in (5.4).
Denote by $G_{r_{j}}(x,\cdot)$ the Dirichlet Green’s function in $B_{r_{j}}$.
We may use the integration by parts
$\phi(x)=\sum_{e\in\mathcal{E}(B_{r_{j}})}\nabla\phi(e)\nabla G_{r_{j}}(x,e)$
to write $A_{j}$ as

$A_{j}=\sum_{e\in\mathcal{E}(B_{r_{j}})}\nabla\phi(e)\sum_{x\in
B_{r_{j}}}\nabla G_{r_{j}}(x,e)\rho_{r_{j}}(x)$

In order to apply Theorem 4.1, define

$f_{r_{j}}(e):=r_{j}\sum_{x\in B_{r_{j}}}\nabla G_{r_{j}}(x,e)\rho_{r_{j}}(x)$

Thus $\nabla^{*}\cdot f_{r_{j}}(x)=r_{j}\rho_{r_{j}}(x)$ and
$A_{j}=r_{j}^{-1}\sum_{e\in\mathcal{E}(B_{r_{j}})}\nabla\phi(e)f_{r_{j}}(e)$.
Notice that, as $r\rightarrow\infty$, the rescaled harmonic measure

$ra_{B_{r}\left(0\right)}\left(0,\cdot\right)\rightarrow 1/2\pi.$

Thus as $r\rightarrow\infty$, $r\rho_{r}$ converges to

$f(x):=\frac{1}{2\pi r_{j+1}}\mathds{1}_{|x|=r_{j+1}}-\frac{1}{2\pi
r_{j,-}}\mathds{1}_{|x|=r_{j,-}}$

and that $\|r\rho_{r}-f\|_{L^{\infty}(r^{-1}B_{r})}=O(r^{-1})$.

Applying Theorem 4.1 using that $sup_{j\geq
1}\|r_{j}\rho_{r_{j}}\|_{L^{\infty}}<\infty$, there exists
$\mathbf{g}=\mathbf{g}(V)>0$ and $\beta=\beta(d,\lambda,\Lambda)>0$, such that

$\left|\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]-\mathbf{g}\right|\leq
Cr_{j}^{-\beta}.$

And we conclude the lemma.

∎

We are now ready to finish the proof of Lemma 2.1 and 2.3.

###### Proof of Lemma 2.1 .

Denote by $s=\frac{t}{\sqrt{\log N}}$. We apply Proposition 5.6 and stop at
$k_{2}:=\log N-\sqrt{\log N}$. It follows that

(5.9)
$\left\langle\exp(is\phi(0))\right\rangle_{\mu_{N}}=\left\langle\mathbb{E}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})|\mathcal{F}_{k_{2}}\right]\exp(isX_{r_{k_{2}}})\exp(is\mathcal{E}_{k_{2}})\right\rangle_{\mu_{N}}$

We apply Lemma 5.3 and Lemma 5.7 to conclude

$\mathbb{E}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})|\mathcal{F}_{k_{2}}\right]=\mathbb{E}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})\mathds{1}_{\mathcal{M}_{k_{2}}}|\mathcal{F}_{k_{2}}\right]\\\
+\mathbb{E}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})\mathds{1}_{\mathcal{M}_{k_{2}}^{c}}|\mathcal{F}_{k_{2}}\right]=\mathbb{E}^{r_{k_{2}},0}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})\right]+R_{k_{2}},$

where, for some $\delta=\delta(V)\in(0,\frac{1}{2}]$,
$\left\langle|R_{k_{2}}|^{2}\right\rangle_{\mu_{N}}\leq
r_{k_{2}}^{-\delta}\leq e^{-\delta\sqrt{\log N}}$. Thus

(5.10)
$\left\langle\exp(is\phi(0))\right\rangle_{\mu_{N}}=\mathbb{E}^{r_{k_{2}},0}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})\right]\left\langle\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}+\left\langle
R_{k_{2}}\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}\\\
+\left\langle(\exp(is\mathcal{E}_{k_{2}})-1)\exp(is\phi(0)-isX_{r_{k_{2},-}})\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}$

The first term on the right side gives the main contribution. We claim that
there exists $\mathbf{g}=\mathbf{g}(V)>0$ and $\beta=\beta(V)>0$, such that

(5.11)
$\left\langle\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}=e^{-\frac{t^{2}}{2\mathbf{g}}}\left(1+O\left(\frac{t^{2}}{(\log
N)^{\frac{1}{2}}}\right)\right)$

together with the Brascamp-Lieb inequality, which implies

$1\geq\mathbb{E}^{r_{k_{2}},0}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})\right]\geq
1-\frac{s^{2}}{2}\operatorname{var}_{\mu_{B_{r_{k_{2}}}}}[\phi(0)-X_{r_{k_{2},-}}]\geq
1-\frac{C_{1}s^{2}}{2}\operatorname{var}_{\mu_{G,B_{r_{k_{2}}}}}[\phi(0)-X_{r_{k_{2},-}}]\\\
\geq 1-C_{2}\frac{t^{2}}{(\log N)^{\frac{1}{2}}}$

Therefore

(5.12)
$\mathbb{E}^{r_{k_{2}},0}\left[\exp(is\phi(0)-isX_{r_{k_{2},-}})\right]\left\langle\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}=e^{-\frac{t^{2}}{2\mathbf{g}}}\left(1+O(\frac{t^{2}}{(\log
N)^{\frac{1}{2}}})\right).$

To show (5.11), we apply Proposition 5.6 with $s=\frac{t}{\sqrt{\log N}}$
(thus $s^{2}\ll\frac{\delta(\log N-k-1)}{Ck}=O((\log N)^{-1/2})$), which
yields

$\displaystyle\log\left\langle\exp\left(isX_{r_{k_{2}}}\right)\right\rangle_{\mu_{N}}=\sum_{j=1}^{k_{2}}\log\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]+\log\left\langle\exp\left(isX_{r_{0}}\right)\right\rangle_{\mu_{N}}+O\left(\sum_{j=1}^{k_{2}}e^{Cs^{2}k}r_{j}^{-\delta}\right).$

Note that $\sum_{j=1}^{k_{2}}e^{Cs^{2}k}r_{j}^{-\delta}\leq
Cr_{k_{2}}^{-\delta}\leq Ce^{-\delta\sqrt{\log N}}$. We also apply the
Brascamp-Lieb inequality to see that

$\left|\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]-1-\frac{s^{2}}{2}\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]\right|\leq
Cs^{4}\mathbb{E}^{r_{j},0}\left[A_{j}^{4}\right]\leq\frac{Ct^{4}}{(\log
N)^{2}}\mathbb{E}^{G,B_{r_{j}}}\left[A_{j}^{4}\right]=O(\frac{t^{4}}{(\log
N)^{2}})$

Thus

$\sum_{j=1}^{k_{2}}\log\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]=\sum_{j=1}^{k_{2}}-\frac{s^{2}}{2}\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]+O(\frac{t^{4}}{\log
N})$

We apply Lemma 5.8 to conclude that there exists $\mathbf{g}=\mathbf{g}(V)>0$
and $\beta=\beta(d,\lambda,\Lambda)>0$, such that

$\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]=\mathbf{g}+O(r_{j}^{-\beta}),$

this yields

$\sum_{j=1}^{k_{2}}-\frac{s^{2}}{2}\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]=-\frac{t^{2}}{2\mathbf{g}}\frac{\log
N-(\log N)^{\frac{1}{2}}}{\log N}+O((e^{-\beta\sqrt{\log N}}),$

together with the variance estimate that follows from the Brascamp-Lieb
inequality, which gives
$\log\left\langle\exp\left(isX_{r_{0}}\right)\right\rangle_{\mu_{N}}=O(\frac{1}{\log
N})$, we conclude (5.11).

The other terms on the right side of (5.10) can be estimated by

$\left|\left\langle
R_{k_{2}}\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}\right|\leq
e^{-\delta\sqrt{\log N}},$

and

$\left\langle(\exp(is\mathcal{E}_{k_{2}})-1)\exp(is\phi(0)-isX_{r_{k_{2},-}})\exp(isX_{r_{k_{2}}})\right\rangle_{\mu_{N}}\leq\frac{s^{2}}{2}\left\langle\mathcal{E}_{k_{2}}^{2}\right\rangle_{\mu_{N}}\\\
\leq\frac{Ct^{2}}{2\log
N}\left\langle\mathcal{E}_{k_{2}}^{2}\right\rangle_{\mu_{N}}=O(\frac{1}{\log
N})$

Substitutes the estimates above into (5.10) we conclude Lemma 2.1.

∎

###### Proof of Lemma 2.3.

The proof is very similar to Lemma 2.1 and thus we give a sketch here. We
apply Proposition 5.6 and stop at $k_{1}:=\frac{1}{2}\log N$. By conditioning,

(5.13)
$\left\langle\exp(is\phi(0))\right\rangle_{\mu_{N}}=\left\langle\mathbb{E}\left[\exp(is\phi(0)-isX_{r_{k_{1},-}})|\mathcal{F}_{k_{1}}\right]\exp(isX_{r_{k_{1}}})\exp(is\mathcal{E}_{k_{1}})\right\rangle_{\mu_{N}}$

We then apply Lemma 5.3 and Lemma 5.7 to obtain

(5.14)
$\left\langle\exp(is\phi(0))\right\rangle_{\mu_{N}}=\mathbb{E}^{r_{k_{1}},0}\left[\exp(is\phi(0)-isX_{r_{k_{1},-}})\right]\left\langle\exp(isX_{r_{k_{1}}})\right\rangle_{\mu_{N}}+\left\langle
R_{k_{1}}\exp(isX_{r_{k_{1}}})\right\rangle_{\mu_{N}}\\\
+\left\langle(\exp(is\mathcal{E}_{k_{1}})-1)\exp(is\phi(0)-isX_{r_{k_{1},-}})\exp(isX_{r_{k_{1}}})\right\rangle_{\mu_{N}},$

where, for some $\delta=\delta(V)\in(0,\frac{1}{2}]$,
$\left\langle|R_{k_{1}}|^{2}\right\rangle_{\mu_{N}}\leq
r_{k_{1}}^{-\delta}\leq N^{-\delta/2}$. Thus,

$\left|\left\langle
R_{k_{1}}\exp(isX_{r_{k_{1}}})\right\rangle_{\mu_{N}}\right|\leq
N^{-\delta/2},$

and by the Brascamp-Lieb inequality,

$\left\langle(\exp(is\mathcal{E}_{k_{1}})-1)\exp(is\phi(0)-isX_{r_{k_{1},-}})\exp(isX_{r_{k_{1}}})\right\rangle_{\mu_{N}}\leq\frac{s^{2}}{2}\left\langle\mathcal{E}_{k_{1}}^{2}\right\rangle_{\mu_{N}}\\\
=O(N^{-\gamma})$

Let $\delta>0$ and $C<\infty$ be the constants from Proposition 5.6, and we
take $\varepsilon$ sufficiently small so that
$\varepsilon^{2}\leq\frac{\delta}{2C}$. For the first term on the right side
of (5.14), which is absolutely bounded by
$\left|\left\langle\exp(isX_{r_{k_{1}}})\right\rangle_{\mu_{N}}\right|$, we
apply Proposition 5.6 (with $|s|<\varepsilon$) to obtain

$\displaystyle\log\left\langle\exp\left(isX_{r_{k_{1}}}\right)\right\rangle_{\mu_{N}}=\sum_{j=1}^{k_{1}}\log\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]+\log\left\langle\exp\left(isX_{r_{0}}\right)\right\rangle_{\mu_{N}}+O\left(\sum_{j=1}^{k_{1}}e^{Cs^{2}k_{1}}r_{j}^{-\delta}\right).$

Since $s^{2}\leq\frac{\delta}{2C}$, we have

$\sum_{j=1}^{k_{1}}e^{Cs^{2}k_{1}}r_{j}^{-\delta}\leq
r_{k_{1}}^{-\delta/4}\leq N^{-\delta/8}.$

Notice that, since the distribution of $A_{j}$ is symmetric for the zero
boundary field,

$\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]\leq
1-\frac{s^{2}}{2}\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]+\frac{s^{4}}{24}\mathbb{E}^{r_{j},0}\left[A_{j}^{4}\right]$

We apply the Brascamp-Lieb inequality to conclude, there exists an absolute
constant $M<\infty$, such that

$\max_{j=1,\cdots,k_{1}}\mathbb{E}^{r_{j},0}\left[A_{j}^{4}\right]\leq
C\max_{j=1,\cdots,k_{1}}\mathbb{E}^{G,B_{r_{j}}}\left[A_{j}^{4}\right]\leq M$

On the other hand, applying Lemma 5.8 to conclude that there exists
$\mathbf{g}=\mathbf{g}(V)>0$ and $\beta=\beta(d,\lambda,\Lambda)>0$, such that
for each $j=1,\cdots,k_{1}$,

$\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]=\mathbf{g}+O(N^{-\beta}),$

Thus, by choosing $\varepsilon>0$ such that
$\varepsilon^{2}=\min\\{\frac{\mathbf{g}}{2M},\frac{\delta}{2C},\frac{3\gamma}{2\mathbf{g}},\frac{3\delta}{4\mathbf{g}}\\}$,
we see that for all $|s|<\varepsilon$,

$\mathbb{E}^{r_{j},0}\left[\exp(isA_{j})\right]\leq
1-\frac{s^{2}}{3}\mathbf{g}$

Thus for $N$ sufficiently large,

$\left\langle\exp\left(isX_{r_{k_{1}}}\right)\right\rangle_{\mu_{N}}\leq(1+O(N^{-\frac{\delta}{8}}))\prod_{j=1}^{k_{1}}\left(1-\frac{s^{2}}{4}\mathbf{g}\right)+O(N^{-\frac{\delta}{2}}+N^{-\gamma})\leq
2e^{-\frac{s^{2}}{3}\mathbf{g}\log N}$

Substitutes these estimates into (5.14) we conclude the Lemma.

∎

######  Remark 5.9.

Notice that if we aims for a weaker bound (2.4), then instead of applying 5.8
in the proof above, we only need a uniform lower bound, namely,
$\mathbb{E}^{r_{j},0}\left[A_{j}^{2}\right]\geq c_{1}$ for some $c_{1}>0$.
This can be proved, for example, by using the Mermin-Wagner argument as we did
in the next section.

##  6\.  A Mermin-Wagner bound

In this section we prove Lemma 2.5. The upper bound (2.5) is obtained by using
the following Lemma.

######  Lemma 6.1.

Let $X$ be a random variable taking values on a unit circle, and
$f:[0,2\pi)\to[0,1]$ be its density function. Suppose that $\forall
a\in[0,2\pi)$, we have

(6.1) $f(a+\frac{1}{2})f(a-\frac{1}{2})\geq cf(a)^{2}$

Then the random variable has a bounded density on the circle, and
$\int_{0}^{2\pi}e^{i\theta}f(\theta)\,d\theta\leq 1-\varepsilon$. Moreover, if
there exist $t>1$ and $C<\infty$, such that $\forall a,b\in[0,2\pi)$,

(6.2) $f(a+b)f(a-b)\geq e^{-\frac{Cb^{2}}{t^{2}}}f(a)^{2}$

then the characteristic function bound can be improved to
$\int_{0}^{2\pi}e^{i\theta}f(\theta)\,d\theta\leq\frac{C}{t^{2}}$.

We first prove Lemma 2.5 based on Lemma 6.1. The argument presented below
follows closely a Mermin-Wagner type estimate which has been done in e.g.,
[19, 24, 23]. Denote by $x_{k}=(2^{k},0)$ for $k=0,\cdots,\log N$. And

(6.3) $\mathcal{F}_{k}:=\sigma(\phi(x)-\phi(y):|x_{1}|+|x_{2}|=2^{k}\text{ and
}|y_{1}|+|y_{2}|=2^{k}).$

In other words, $\mathcal{F}_{k}$ specifies all the gradients of the field on
the boundary of a diamond of radius $2^{k}$. A key observation is that
conditioned on $\mathcal{F}_{k}$, the gradients of the field inside the region
$|x_{1}|+|x_{2}|<2^{k}$ and the gradients in the region
$|x_{1}|+|x_{2}|>2^{k}$ are independent. By progressively conditioning the
gradients on the layers $\mathcal{F}_{k}$, we have

(6.4)
$\left\langle\exp\left(is\phi(0)\right)\right\rangle_{\mu_{N}}=\left\langle\prod_{k=0}^{\log
N}\mathbb{E}\left[\exp\left(is(\phi(x_{k})-\phi(x_{k-1})\right)|\left(\mathcal{F}_{k}\right)_{0\leq
k\leq\log N}\right]\right\rangle_{\mu_{N}}$

Thus (2.5) follows if we can show there exist $\varepsilon>0$ and $C<\infty$,
such that $\forall k=1,\cdots,\log N$, and $\forall t>1$,

(6.5)
$E\left[\exp\left(it(\phi(x_{k})-\phi(x_{k-1})\right)|\left(\mathcal{F}_{k}\right)_{0\leq
k\leq\log N}\right]\leq\min\\{1-\varepsilon,\frac{C}{t^{2}}\\}.$

We show (6.5) using a Mermin-Wagner type argument. Define the deformation
$\forall b\in(0,2\pi]$,

(6.6) $\tau(x):=\left\\{\begin{aligned} &b,&&|x_{1}|+|x_{2}|\leq 2^{k-1},\\\
&b\left(1-\frac{|x_{1}|+|x_{2}|}{2^{k}}\right),&&2^{k-1}\leq|x_{1}|+|x_{2}|\leq
2^{k},\\\ &0,&&|x_{1}|+|x_{2}|\geq 2^{k}\end{aligned}\right.$

which interpolates between $b$ and $0$, and does not change the gradients for
all $\left(\mathcal{F}_{k}\right)_{0\leq k\leq\log N}$. We also notice that a
straightforward calculation yields
$\sum_{e\in\mathcal{E}(Q_{N})}(\nabla\tau(e))^{2}\leq Cb^{2}$ for some
$C<\infty$.

Let $\phi^{+}:=\phi+\tau$ and $\phi^{-}:=\phi-\tau$. We see the densities of
$\phi^{+}$ and $\phi^{-}$ (conditioning on all the gradients
$\left(\mathcal{F}_{k}\right)_{0\leq k\leq\log N}$) satisfies

(6.7)
$g(\phi^{+})g(\phi^{-})=\frac{1}{Z_{N}^{2}}\exp\left(-\sum_{(x,y)\in\mathcal{E}(Q_{N})}V(\phi(x)-\phi(y)+\tau(x)-\tau(y))\right)\\\
\cdot\exp\left(-\sum_{(x,y)\in\mathcal{E}(Q_{N})}V(\phi(x)-\phi(y)-\tau(x)+\tau(y))\right)\\\
\geq\frac{1}{Z_{N}^{2}}\exp\left(-2\sum_{(x,y)\in\mathcal{E}(Q_{N})}V(\phi(x)-\phi(y))-\frac{1}{2}\sup_{x\in\mathbb{R}}V^{\prime\prime}(x)\sum_{e\in\mathcal{E}(Q_{N})}(\nabla\tau(e))^{2}\right)\geq
e^{-Cb^{2}}g(\phi)^{2}$

for some $c>0$.

For any set $E\subseteq\Omega_{0}(Q_{N})$ of field configurations, integrate
the above inequality of densities over $E$ and applying the Cauchy-Schwarz
inequality, we obtain

(6.8) $\mathbb{P}\left(\phi^{+}\in E|\left(\mathcal{F}_{k}\right)_{0\leq
k\leq\log N}\right)\cdot\mathbb{P}\left(\phi^{-}\in
E|\left(\mathcal{F}_{k}\right)_{0\leq k\leq\log N}\right)\geq
e^{-Cb^{2}}\mathbb{P}\left(\phi\in E|\left(\mathcal{F}_{k}\right)_{0\leq
k\leq\log N}\right)^{2}.$

Set $t=1$ and $b=\frac{1}{2}$. If we denote by $f$ the conditional density of
$\exp\left(i(\phi(x_{k})-\phi(x_{k-1})\right)$ on the unit circle
(conditioning on all the gradients $\left(\mathcal{F}_{k}\right)_{0\leq
k\leq\log N}$) , then it follows that (6.1) is satisfied (for $t=1$).

To see that (6.2) holds for $t>1$, notice the fact that for $t$ large, we may
rescale $\phi\to t\phi$, so that the corresponding potential is given by
$V(\frac{\cdot}{t})$, which has second derivative of order $t^{-2}$. Thus
following the same argument as above, one obtains an improved bound for the
conditional densities (for large $t$),

(6.9) $g(\phi^{+})g(\phi^{-})\geq e^{-\frac{Cb^{2}}{t^{2}}}g(\phi)^{2}$

Therefore by applying Lemma 6.1 we conclude that

$\mathbb{E}\left[\exp\left(it(\phi(x_{k})-\phi(x_{k-1})\right)|\left(\mathcal{F}_{k}\right)_{0\leq
k\leq\log N}\right]\leq\min\\{1-\varepsilon,\frac{C}{t^{2}}\\}$

And by the conditioning (6.4) we conclude Lemma 2.5.

Finally we give a proof of Lemma 6.1 .

###### Proof of Lemma 6.1 .

We focus on the proof of
$\int_{0}^{2\pi}e^{i\theta}f(\theta)\,d\theta\leq\frac{C}{t^{2}}$ as the
$1-\varepsilon$ bound is a classical result and its proof can be found in [19,
24, 23]. Using (6.2), we see that the probability density has a ratio bounded
by $e^{C/t^{2}}$ uniformly over the circle .

Thus for all $x\in[0,2\pi)$ and any interval $I\subseteq[0,2\pi)$ of length at
most $\pi$,

$\left(\strokedint_{I}f(\theta)\,d\theta\right)^{2}e^{-\frac{C}{t^{2}}}\leq\strokedint_{I}f(\theta+x)f(\theta-x)\,d\theta\leq\strokedint_{I}f(\theta+x)e^{\frac{C}{t^{2}}}\strokedint_{I}f(\phi-x)\,d\phi\,d\theta\\\
\leq
e^{\frac{C}{t^{2}}}\left(\strokedint_{I+x}f(\theta)\,d\theta\right)\left(\strokedint_{I-x}f(\theta)\,d\theta\right),$

thus

$\left(\int_{I}f(\theta)\,d\theta\right)^{2}e^{-\frac{2C}{t^{2}}}\leq\left(\int_{I+x}f(\theta)\,d\theta\right)\left(\int_{I-x}f(\theta)\,d\theta\right)$

In particular, fix $t\in\mathbb{R}$, by taking $I=[(k-1)\pi/m,k\pi/m)$, for
some $m\in\mathbb{N}$, and $k=1,\cdots 2m$ we have

$\left(\max_{k=1,\cdots
2m}\int_{(k-1)\pi/m}^{k\pi/m}f(\theta)\,d\theta\right)^{2}\leq
e^{\frac{2C}{t^{2}}}\left(\min_{k=1,\cdots
2m}\int_{(k-1)\pi/m}^{k\pi/m}f(\theta)\,d\theta\right)\left(\max_{k=1,\cdots
2m}\int_{(k-1)\pi/m}^{k\pi/m}f(\theta)\,d\theta\right)$

Thus

$\left(\max_{k=1,\cdots
2m}\int_{(k-1)\pi/m}^{k\pi/m}f(\theta)\,d\theta\right)\leq
e^{\frac{2C}{t^{2}}}\left(\min_{k=1,\cdots
2m}\int_{(k-1)\pi/m}^{k\pi/m}f(\theta)\,d\theta\right)$

Therefore for each $k$, we have

$\left|\int_{\frac{(k-1)\pi}{m}}^{\frac{k\pi}{m}}e^{i\theta}f(\theta)\,d\theta+\int_{\pi+\frac{(k-1)\pi}{m}}^{\pi+\frac{k\pi}{m}}e^{i\theta}f(\theta)\,d\theta\right|\leq\left|e^{i\frac{k\pi}{m}}\int_{\frac{(k-1)\pi}{m}}^{\frac{k\pi}{m}}f(\theta)\,d\theta-e^{i\frac{k\pi}{m}}\int_{\pi+\frac{(k-1)\pi}{m}}^{\pi+\frac{k\pi}{m}}f(\theta)\,d\theta\right|\\\
+\max_{\theta\in[\frac{(k-1)\pi}{m},\frac{k\pi}{m}]}|e^{i\frac{(k-1)\pi}{m}}-e^{i\frac{k\pi}{m}}|\left|\int_{(\frac{(k-1)\pi}{m}}^{\frac{k\pi}{m}}f(\theta)\,d\theta+\int_{\pi+\frac{(k-1)\pi}{m}}^{\pi+\frac{k\pi}{m}}f(\theta)\,d\theta\right|\\\
\leq\left(1-e^{-\frac{2C}{t^{2}}}+O(\frac{1}{m})\right)\left|\int_{\frac{(k-1)\pi}{m}}^{\frac{k\pi}{m}}f(\theta)\,d\theta+\int_{\pi+\frac{(k-1)\pi}{m}}^{\pi+\frac{k\pi}{m}}f(\theta)\,d\theta\right|+O(\frac{1}{m^{4}t^{2}})$

Summing over $k$ yields

$\left|\int_{0}^{2\pi}e^{i\theta}f(\theta)\,d\theta\right|\leq
1-e^{-\frac{2C}{t^{2}}}+O(\frac{1}{m})$

by taking $m>\frac{1}{C}t^{2}$ we conclude the lemma. ∎

##  Appendix A Multiscale Poincare inequality,

We state a discrete version of the multiscale Poincare inequality, which
provides an estimate of the $H^{-1}\left({\scaleobj{1.2}{\square}}_{n}\right)$
norm of a function in terms of its spatial averages in triadic subcubes.

######  Proposition A.1 ([3, 5]).

(Multiscale Poincare inequality) Let
$\mathcal{Z}_{n}=3^{n}\mathbb{Z}^{2}\cap{\scaleobj{1.2}{\square}}_{m}$. Then

$\left\|f\right\|_{H^{-1}({\scaleobj{1.2}{\square}}_{m})}\leq
C\left\|f\right\|_{\underline{L}^{2}({\scaleobj{1.2}{\square}}_{m})}+C\sum_{n=0}^{m-1}3^{n}\left(\frac{1}{\left|\mathcal{Z}_{n}\right|}\sum_{y\in\mathcal{Z}_{n}}\left|\left(f\right)_{y+{\scaleobj{1.2}{\square}}_{n}}\right|^{2}\right)^{1/2}.$

We also record a lemma here that the $\underline{H}^{-1}$ norm of $\nabla u$
can bound the $L^{2}$ oscillation of $u$.

######  Lemma A.2.

There exists $C(d)<\infty$ such that for every $m\in\mathbb{N}$ and $u\in
H^{1}({\scaleobj{1.2}{\square}}_{m},\mu_{N})$,

$\left\|u-(u)_{{\scaleobj{1.2}{\square}}_{m}}\right\|_{\underline{L}^{2}({\scaleobj{1.2}{\square}}_{m},\mu_{N})}\leq
C\left\|\nabla
u\right\|_{\underline{H}^{-1}({\scaleobj{1.2}{\square}}_{m},\mu_{N})}$

And, for every $u\in\in H_{0}^{1}({\scaleobj{1.2}{\square}}_{m},\mu_{N})$,

$\left\|u\right\|_{\underline{L}^{2}({\scaleobj{1.2}{\square}}_{m},\mu_{N})}\leq
C\left\|\nabla
u\right\|_{\underline{H}^{-1}({\scaleobj{1.2}{\square}}_{m},\mu_{N})}$

### Acknowledgments

We thank Scott Armstrong, Ron Peled, Tom Spencer and Ofer Zeitouni for helpful
discussions, and Ofer Zeitouni for helpful comments on a previous draft of
this manuscript. The research was partially supported by the National Key R&D
Program of China and a grant from Shanghai Ministry Education Commission.

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