# Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

Eric Thoma

(Date:  December 13, 2021 )

###### Abstract.

We characterize the behavior of a random discrete interface $\phi$ on
$[-L,L]^{d}\cap\mathbb{Z}^{d}$ with energy $\sum V(\Delta\phi(x))$ as
$L\to\infty$, where $\Delta$ is the discrete Laplacian and $V$ is a uniformly
convex, symmetric, and smooth potential. The interface $\phi$ is called the
non-Gaussian membrane model. By analyzing the Helffer-Sjöstrand representation
associated to $\Delta\phi$, we provide a unified approach to continuous
scaling limits of the rescaled and interpolated interface in dimensions
$d=2,3$, Gaussian approximation in negative regularity spaces for all $d\geq
2$, and the infinite volume limit in $d\geq 5$. Our results generalize some of
those of [7].

###### 2020 Mathematics Subject Classification:

82B20, 60F05, 35Q82

Eric Thoma, Courant Institute<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 The Helffer-Sjöstrand Equation
  3. 3 Minimization for Smooth Harmonic Perturbations
  4. 4 Marginals and the Simplified Energy
  5. 5 Minimization of the Helffer-Sjöstrand Energy
  6. 6 Proofs of the Main Theorems
  7. 7 Appendix: Bergman projections and Green’s function estimates

##  1\. Introduction

This article is concerned with the membrane model, which is a random field
$(\phi(x))_{x\in\mathbb{Z}^{d}}$ with distribution given by

$\mathrm{MM}_{L}(d\phi)=\frac{1}{\mathrm{Z}(\Box_{L})}\exp\left(-\sum_{x\in\mathbb{Z}^{d}}V(\Delta\phi(x))\right)\prod_{x\in\Box_{L}}d\phi(x)\prod_{x\in\mathbb{Z}^{d}\setminus\Box_{L}}\delta_{0}(d\phi(x)).$

where $\Box_{L}=[-L,L]^{d}\cap\mathbb{Z}^{d}$ is a $d$-dimensional discrete
cube, $V:\mathbb{R}\to\mathbb{R}$ is a potential satisfying certain
conditions, and $\Delta$ is the discrete Laplacian on $\mathbb{Z}^{d}$. The
constant $\mathrm{Z}(\Box_{L})$ is a normalizing factor, and we have enforced
$\phi(x)=0$ for $x\not\in\Box_{L}$ through the Dirac delta measure
$\delta_{0}$, though other boundary conditions and other domains besides cubes
are possible.

In particular, we will investigate limits of $\mathrm{MM}_{L}$ as $L\to\infty$
in $d\geq 2$. There are multiple senses in which we can take a limit, and we
will consider three types. The first type is the infinite-volume limit, also
known as the thermodynamic limit, which is the distributional limit of
$(\phi(x))_{x\in\Box_{K}}$ as $L\to\infty$ and $K$ stays fixed. This limit
only exists in the supercritical dimensions $d\geq 5$, and is generally non-
Gaussian. The second and third types concern the scaling limit of the rescaled
field $\overline{\phi}(x)=L^{d/2-2}\phi(Lx)$ defined on
$[-1,1]^{d}\cap\frac{1}{L}\mathbb{Z}^{d}$. We consider for $d\geq 2$ the
random quantity

$L^{-d}\sum_{x\in[-1,1]^{d}\cap\frac{1}{L}\mathbb{Z}^{d}}\overline{\phi}(x)f(x),$

where $f:[-1,1]^{d}\to\mathbb{R}$ is a smooth enough function, and approximate
its law by a Gaussian. In dimensions $d=2,3$, we consider an appropriate
interpolation of $\overline{\phi}(x)$ to a function with domain $[-1,1]^{d}$
and prove convergence of its distribution in the space of continuous
functions.

We prove the limits under the condition that $V$ is symmetric, uniformly
convex, and $C^{3}$ with uniformly bounded derivatives. Previously, similar
limits were considered in [7] in the case where $V$ is quadratic, and
therefore $\mathrm{MM}_{L}$ is Gaussian. Unfortunately, the proof methods
cannot be easily generalized to the case of more general $V$. We introduce a
new approach for the membrane model that allows us to treat all three types of
limits in a non-Gaussian setting and under a unified framework. Our methods
can also be adapted for regimes of general semi-flexible membranes in which
the Laplacian energy dominates. In the case of limits of the rescaled field
$\overline{\phi}$, we prove quantitative Gaussian approximation results and
use the results of [7] to characterize the limit.

We study the membrane model entirely through the distribution of the discrete
Laplacian of the field $\phi$. We consider the equation

$\begin{cases}\Delta\phi(x)=\eta(x),\quad&x\in\Box_{L},\\\
\phi(x)=0,\quad&x\in\partial\Box_{L},\end{cases}$

where $\partial\Box_{L}$ is the outer bounder of $\Box_{L}$ in
$\mathbb{Z}^{d}$, which gives a correspondence between $\phi$ and a field
$\eta\in\mathbb{R}^{\Box_{L}}$. Given the membrane model $\phi$, we can find
$\eta$ by taking the discrete Laplacian, and given $\eta$, we can recover
$\phi$ by solving the above Dirichlet problem. We let $\mathrm{Q}_{L}$ be the
distribution of $\eta$ when $\phi$ is distributed by $\mathrm{MM}_{L}$. The
measure $\mathrm{Q}_{L}$ is itself a Gibbs measure on $\mathbb{R}^{\Box_{L}}$
with energy

$\sum_{x\in\Box_{L}}V(\eta(x))+\sum_{z\in\partial\Box_{L}}V\left(-\sum_{x\in\Box_{L}}P^{z}_{L}(x)\eta(x)\right)$

where $P^{z}_{L}(x)$ is the probability a simple random walk on
$\mathbb{Z}^{d}$ started at $x$ exits $\Box_{L}$ at $z\in\partial\Box_{L}$,
i.e. the Poisson kernel. Note that the latter sum in the energy is present due
to the fact that $\Delta\phi$ is supported on $\Box_{L}\cup\partial\Box_{L}$,
whereas, by definition, $\eta$ consists only of the data of $\Delta\phi$ on
$\Box_{L}$, and so we must recover the values of $\Delta\phi(z)$ for
$z\in\partial\Box_{L}$ from $\eta$ by using the constraint that $\phi$ is
supported on $\Box_{L}$. We use variational methods to understand the Helffer-
Sjöstrand representation associated with $\eta$, which turns out to be more
tractable than that associated with $\phi$. To our knowledge, the approach of
studying $\Delta\phi$, and ultimately $\phi$, through the Gibbs measure
$\mathrm{Q}_{L}$ is new, and we expect this approach may be fruitful for
further investigation of the non-Gaussian membrane model.

###  1.1. Background and Motivation

The membrane model is an instance of a discrete interface model. These are
random fields $\phi\in\mathbb{R}^{\mathbb{Z}^{d}}$ with law proportional to

$\exp(-H(\phi))\prod_{x\in\Box_{L}}d\phi(x)\prod_{x\in\mathbb{Z}^{d}\setminus\Box_{L}}\delta_{0}(d\phi(x)),$

for a Hamiltonian $H:\mathbb{R}^{\mathbb{Z}^{d}}\to\mathbb{R}$. Perhaps the
most studied of such models is the discrete Gaussian Free Field (GFF), which
corresponds to the model with
$H(\phi)\propto\sum_{x\in\mathbb{Z}^{d}}|\nabla\phi(x)|^{2}$ where $\nabla$ is
the discrete gradient. Here, as with the membrane model, it is a natural
question to consider the possible scalings of $\phi$ for which the
$L\to\infty$ limit exists, and for the free field there are many powerful
tools available arising from the Gaussian nature and the gradient form of the
energy. Perhaps most notably, there is a random walk representation of the
covariance of the GFF.

More generally, a major research direction has been to extend results for the
discrete Gaussian free field to the “$\nabla\phi$” model, which corresponds to
$H(\phi)=\sum_{x\in\mathbb{Z}^{d}}V(\nabla\phi(x))$ and $V$ uniformly convex,
symmetric, and smooth. We refer to [9] for a survey of results for this model.

Notably, in [1] methods from elliptic stochastic homogenization were used to
analyze the model. The gradient form of the energy allows for methods from
elliptic regularity, such as the De Giorgi-Nash-Moser theory, to be applied.
We do not know of a replacement for these methods for the membrane model, and
this is a major obstacle in applying a similar approach. Still, we will follow
[1] (which was itself inspired by [16]) in analyzing energies associated to
Helffer-Sjöstrand representations, even though our methods of analysis are
very different.

The physical interest in the membrane model stems partly from its role among
the family of semi-flexible membranes, which are discrete interface models
with

$H(\phi)=\sum_{x\in\mathbb{Z}^{d}}V_{1}(\nabla\phi(x))+V_{2}(\Delta\phi(x))$

where $V_{1}$ and $V_{2}$ are potentials characterizing the membrane’s lateral
tension and bending rigidity (see [8] and references therein). The membrane
model is also interesting due to its scaling properties, especially in the
critical dimension $d=4$, which plays a role analogous to that dimension $2$
plays for the discrete GFF. In this dimension, correlations are logarithmic.

We mention some relevant results about the Gaussian membrane model proved in
[7]. The covariance operator of the Gaussian membrane model is the inverse
discrete bi-Laplacian operator with “pinned” Dirichlet boundary conditions. An
analysis of the scaling limit of this operator in $d=2,3$, proving convergence
to the solution of a continuum problem, was achieved in [15] and is used for
the $d=2,3$ scaling limit results of [7]. Specifically, [7] proves that the
rescaled and interpolated membrane model $\overline{\phi}$ converges weakly as
$L\to\infty$ to a continuum Gaussian process on $[-1,1]^{d}$. In $d\geq 4$,
[7] proves limits of $\overline{\phi}$ using a framework of Gaussian fields
and an extension of a method from [20] on the approximation of discrete
equations by PDE.

Besides for scaling limits, other questions of interest for the membrane model
(and many other interface models) include entropic repulsion, pinning,
wetting, and the behavior of the interface maximum. Entropic repulsion was
addressed in $d\geq 5$ by [11] and [17], in $d=4$ by [13] (and the thesis
[12]), and in $d=2,3$ by [4]. For pinning in $d\geq 4$, some recent results
are given in [18], and pinning in $d=2,3$ is not well understood. The behavior
of the maximum height of the membrane for the critical dimension $d=4$ was
addressed in [19] and for $d\geq 5$ in [6]. These results are all for the
Gaussian model. For the non-Gaussian model, there are very few results.
Notably, [5] proved the scaling limit for the non-Gaussian model in $d=1$
using a random walk representation specific to one-dimension.

###  1.2. Statement of Main Results

We now state the main results of the paper. We assume that the potential $V$
is $C^{3}$ with $\|V\|_{C^{3}(\mathbb{R})}<\infty$, symmetric about $0$, and
$\inf V^{\prime\prime}>0$.

For the infinite volume limit, we characterize the non-Gaussian infinite
volume limit of $\phi$ in $d\geq 5$. In $d\leq 4$, the model experiences a
well-known blow up of variances as $L\to\infty$, and so the infinite volume
limit does not exist.

######  Theorem 1.1 (Infinite Volume Limit).

Let $d\geq 5$, and let
$-\Gamma:\mathbb{Z}^{d}\times\mathbb{Z}^{d}\to\mathbb{R}$ be the Green’s
function for the discrete Laplacian on $\mathbb{Z}^{d}$. Let
$a\in\mathbb{R}^{\mathbb{Z}^{d}}$ have compact support. Let $\nu^{0}$ be a
probability measure on $\mathbb{R}$ with Lebesgue density proportional to
$\exp(-V(\xi))$, and let $\xi$ be a $\nu^{0}$ distributed random variable.

Then we have

(1.1)
$\lim_{L\to\infty}\log\mathbb{E}_{\mathrm{MM}_{L}}\left(\exp\left(\sum_{x\in\mathbb{Z}^{d}}a(x)\phi(x)\right)\right)=\sum_{x\in\mathbb{Z}^{d}}\log\mathbb{E}_{\nu^{0}}\left(\exp\left((\Gamma\ast
a)(x)\cdot\xi\right)\right)$

where $(\Gamma\ast a)(x)=\sum_{y\in\mathbb{Z}^{d}}\Gamma(x,y)a(y)$. In
particular, for any fixed integer $K$, the field $(\phi(x))_{x\in\Box_{K}}$
converges weakly and in moments to a random variable with cumulant generating
function given by the RHS of (1.1).

In the case that $V$ is quadratic, the RHS of (1.1) is
$\sum_{x\in\mathbb{Z}^{d}}\frac{1}{2V^{\prime\prime}}|\Gamma\ast a(x)|^{2}.$
It follows that the limit is a Gaussian field with covariance operator
$\frac{1}{2}(V^{\prime\prime})^{-1}\Gamma\ast\Gamma$. Such a result is well-
known; see e.g. [12]. For general $V$, the limiting distribution is non-
Gaussian, since the RHS of (1.1) is non-quadratic in $a$. Our result implies
that the Laplacian field $\Delta\phi$ converges to an i.i.d. field, and the
limiting distribution of each $\Delta\phi(x)$ is given by
$\nu^{0}\sim\exp(-V)$. The infinite volume law of $\phi$ is in fact the
pushforward of $(\nu^{0})^{\otimes\mathbb{Z}^{d}}$ under convolution by
$\Gamma$, which is well-defined due to the fact that $\Gamma(x,\cdot)\in
L^{2}(\mathbb{Z}^{d})$ in $d\geq 5$.

Our second and third results involves the limit of the rescaled field
$\overline{\phi}(x)=L^{d/2-2}\phi(Lx)$, defined on a lattice of spacing $1/L$.
In $d\geq 4$, the limit as $L\to\infty$ of $\overline{\phi}(x)$ should exist
in a certain negative regularity sense, i.e. if we consider $\overline{\phi}$
summed against smooth test functions. In contrast to the infinite volume limit
of 1.1, the limit will be Gaussian due to the averaging of fluctuations in the
summation. In $d=2,3$, the limit will be a Hölder continuous Gaussian process
after an appropriate interpolation of the interface. The Gaussianity in
$d=2,3$ comes from $\Gamma\not\in L^{2}(\mathbb{Z}^{d})$, and so the field
$\overline{\phi}$ is a large scale average of $\Delta\overline{\phi}$.

In the case that $V$ is quadratic and $d=2,3$, the above scaling limit was
proved on squares and cubes in [7]. In $d\geq 4$, the scaling limit was proved
on smooth domains, i.e. with the family $\Box_{L}$ replaced by
$L\Omega\cap\mathbb{Z}^{d}$ for a smooth domain $\Omega\subset\mathbb{R}^{d}$
and $L\Omega=\\{Lx:x\in\Omega\\}$.

We need a technical assumption, which will allow us to prove in Section 7 that
certain fields do not concentrate on a boundary layer of $\Box_{L}$. For a
given $f:[-1,1]^{d}\to\mathbb{R}$ and $\Delta_{\mathbb{R}^{d}}$ the continuum
Laplacian, consider the solution $u$ to

(1.2)
$\begin{cases}\Delta_{\mathbb{R}^{d}}^{2}u(x)=f(x),\quad&x\in(-1,1)^{d},\\\
u(x)=\partial_{n}u(x)=0,\quad&x\in\partial(-1,1)^{d}.\end{cases}$

Our assumption is that $u\in C^{5}((-1,1)^{d})$.

Our result approximates the cumulant generating function of the non-Gaussian
model by a Gaussian model with an effective covariance and with explicit error
estimates. We are in particular interested in the statistic

$\frac{1}{(2L)^{d}}\sum_{x\in
L^{-1}\mathbb{Z}^{d}\cap[-1,1]^{d}}f(x)\overline{\phi}(x),$

which is a discrete approximation of the continuum $L^{2}([-1,1])$ inner
product. We make $\overline{\phi}$ into an operator on smooth functions in
this way, and we will approximate the law of $\overline{\phi}$, considered in
this way, by a Gaussian law.

######  Theorem 1.2 (Distributional Gaussian Approximation).

Let $d\geq 2$ and $f:[-1,1]^{d}\to\mathbb{R}$ be such that $u\in
C^{5}((-1,1)^{d})$ as defined above. Let
$\overline{\phi}(x)=L^{d/2-2}\phi(Lx)$ be the rescaled membrane, defined on
$[-1,1]^{d}\cap L^{-1}\mathbb{Z}^{d}$, and let $\mathrm{MM}_{L}^{G}$ be the
(Gaussian) membrane model on $\Box_{L}$ with potential
$V^{G}(\xi)=\frac{1}{2}|\xi|^{2}$. Then we have

(1.3) $\displaystyle\log\mathbb{E}_{\mathrm{MM}_{L}}\exp\left(L^{-d}\sum_{x\in
L^{-1}\mathbb{Z}^{d}\cap[-1,1]^{d}}f(x)\overline{\phi}(x)\right)$
$\displaystyle=\log\mathbb{E}_{\mathrm{MM}_{L}^{G}}\exp\left((\mathrm{Var}_{\nu^{0}}\xi)^{1/2}L^{-d}\sum_{x\in
L^{-1}\mathbb{Z}^{d}\cap[-1,1]^{d}}f(x)\overline{\phi}(x)\right)+\mathrm{Error},$

and $|\mathrm{Error}|\leq CL^{-\frac{d-1}{6d-2}}(1+(\log
L)^{3}\mathbf{1}_{d=2})$. The constant $C$ depends on $d$ and continuously on
$\|u\|_{C^{5}}$, $\inf V^{\prime\prime}$, $\sup V^{\prime\prime}$, and
$\sup|V^{(3)}|$. The symbols $\nu_{0}$ and $\xi$ are the same as in 1.1.

While we have stated the above result for the membrane model on $\Box_{L}$ and
test functions on $[-1,1]^{d}$, we are also interested in the membrane model
on smooth domains $L\Omega\cap\mathbb{Z}^{d}$ and test functions on $\Omega$,
particularly due to the importance of this setting in [7]. To that end, we
mention our result also holds, with minimal changes to the proof, in the
setting where $\Omega$ is a bounded, connected open set with $C^{1}$ boundary
(and even weaker conditions on $\Omega$ would suffice).

For the next result, we will need a linear interpolation of the rescaled
membrane $\overline{\phi}$, defined initially on $[-1,1]^{d}\cap
L^{-1}\mathbb{Z}^{d}$, to a continuous function on $[-1,1]^{d}$, for $d=2,3$.
By an abuse of notation, we will also denote the interpolated interface by
$\overline{\phi}$. We refer to [7] for the specific interpolation procedure.
For our main estimate (1.4), essentially any reasonable interpolation
procedure would suffice.

######  Theorem 1.3 (Scaling Limit in $d=2,3$).

Let $d=2,3$ and let $\overline{\phi}(x)=L^{d/2-2}\phi(Lx)$ be the rescaled
membrane on $L^{-1}\mathbb{Z}^{d}$, and interpolate the interface to
$[-1,1]^{d}$ as done in [7]. Let $x_{1},x_{2},\ldots,x_{k}$ be points in
$[-1,1]^{d}$. Then for any $a\in\mathbb{R}^{k}$, we have

(1.4)
$\log\mathbb{E}_{\mathrm{MM}_{L}}\left(\exp(\sum_{i=1}^{k}a_{i}\overline{\phi}(x_{i}))\right)=\log\mathbb{E}_{\mathrm{MM}_{L}^{G}}\left(\exp((\mathrm{Var}_{\nu^{0}}\xi)^{1/2}\sum_{i=1}^{k}a_{i}\overline{\phi}(x_{i}))\right)+\mathrm{Error}$

for $|\mathrm{Error}|\leq CL^{-\frac{d-1}{6d-2}}(1+(\log
L)^{3}\mathbf{1}_{d=2})$. Moreover, there exists a Gaussian process
$\Psi:[-1,1]^{d}\to\mathbb{R}$ such that $\overline{\phi}$ converges in
distribution to $\Psi$ in the space of continuous functions on $[-1,1]^{d}$ as
$L\to\infty$. The process $\Psi$ is centered and a.s. $\alpha$-Hölder
continous for every parameter $\alpha\in(0,1)$ in $d=2$ or $\alpha\in(0,1/2)$
in $d=3$.

The above theorem not only concerns the limit of the finite dimensional
distributions of $\overline{\phi}$, but also the distribution of the maximum
of $\overline{\phi}$. We can only prove the $d=2,3$ scaling limit for the
domains $\Box_{L}$ because we use the results of [7] and the estimates in
[15], which are only known for these domains.

###  1.3. Notation and Proof Outline

Throughout most of the paper, we work on a cube
$\Box_{L}=[-L,L]^{d}\cap\mathbb{Z}^{d}$ assumed to be large. We write $x\sim
y$ if the points $x,y\in\mathbb{Z}^{d}$ are adjacent, i.e. $|x-y|=1$ for the
$\ell^{1}$ norm. The (outer) boundary $\partial\Box_{L}$ of the cube
$\Box_{L}$ is the set of all $z\in\mathbb{Z}^{d}\setminus\Box_{L}$ adjacent to
an element of $\Box_{L}$ in the integer lattice. We define
$\mathrm{cl}_{1}\Box_{L}:=\Box_{L}\cup\partial\Box_{L}$, and we let $\rho_{x}$
denote the $\ell^{1}$ (graph) distance of $x\in\Box_{L}$ to
$\partial\Box_{L}$. The second outer boundary $\partial^{2}\Box_{L}$ consists
of all points in $\mathbb{Z}^{d}\setminus\Box_{L}$ within graph distance $2$
of a point in $\Box_{L}$. For $U$ a finite subset of $\mathbb{Z}^{d}$, we
denote by $L^{2}(U)$ the space $\mathbb{R}^{U}$ with the standard inner
product $\left\langle\cdot,\cdot\right\rangle_{U}$. We often write
$\left\langle\cdot,\cdot\right\rangle$ for
$\left\langle\cdot,\cdot\right\rangle_{\Box_{L}}$. Expectations with respect
to a probability measure $\mu$ are denoted by $\mathbb{E}_{\mu}$. For
$x\in\mathbb{Z}^{d}$, we let $\mathbf{1}_{x}\in\mathbb{R}^{\mathbb{Z}^{d}}$ be
the field with $\mathbf{1}_{x}(y)=1$ if $y=x$ and $\mathbf{1}_{x}(y)=0$ if
$y\neq x$.

When it is clear, we will make no distinction between an element of
$\mathbb{R}^{\mathbb{Z}^{d}}$ supported on $\Box_{L}$ and its restriction to
$\mathbb{R}^{\Box_{L}}$. We will also sometimes restrict an element of
$\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$ to $\mathbb{R}^{\Box_{L}}$ without
writing the restriction explicitly. We often free the notation of parameters
that stay fixed throughout a section; most commonly, the $L$ dependence will
be omitted and the dependence on a parameter $b\in\mathbb{R}^{\Box_{L}}$
(appearing later) will be omitted.

We will approach the study of $\mathrm{MM}_{L}$ through studying the
distribution of

$\Delta\phi(x):=\sum_{y\sim x}(\phi(y)-\phi(x))$

where the sum is over all $y$ adjacent to $x$ in $\mathbb{Z}^{d}$. We note
that our normalization of $\Delta$ differs from some papers, including [7].
More precisely, we let $\Delta_{L}$, as a map
$\mathbb{R}^{\Box_{L}}\to\mathbb{R}^{\Box_{L}}$, be the restriction of
$\Delta$ to functions which are $0$ outside of $\Box_{L}$. The map
$\Delta_{L}$ is a linear bijection: the inverse map $\Delta_{L}^{-1}$ is given
by $\Delta_{L}^{-1}\eta=\phi$ where $\phi$ is the unique solution of

(1.5) $\begin{cases}\Delta\phi(x)=\eta(x),\quad&x\in\Box_{L},\\\
\phi(x)=0,\quad&x\in\partial\Box_{L}.\end{cases}$

We also define for each $z\in\partial\Box_{L}$ the map
$\alpha_{L}^{z}(\eta)=\Delta(\Delta_{L}^{-1}\eta)(z)$. That is, we take
$\phi=\Delta_{L}^{-1}\eta$ for any $\eta\in\mathbb{R}^{\Box_{L}}$, extend
$\phi$ by $0$ outside $\Box_{L}$, and let
$\alpha^{z}_{L}(\eta)=\Delta\phi(z)$. We call $\eta$ the Laplacian field
associated to $\phi$. We sometimes refer to $\eta(x)$ as a “spin”.

For $z\in\partial\Box_{L}$, we let
$P^{z}_{L}\in\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$ be the Poisson kernel,
which is discrete harmonic in $\Box_{L}$ and equal to $\mathbf{1}_{z}$ on
$\partial\Box_{L}$. We can compute $\alpha^{z}_{L}$ in coordinates as
$\alpha^{z}_{L}(\eta)=-\left\langle P^{z}_{L},\eta\right\rangle_{\Box_{L}}$.
Indeed, we have $P^{z}_{L}(x)=\Gamma_{L}(\tilde{z},x)$ for $x\in\Box_{L}$,
where $-\Gamma_{L}$ is the Dirichlet Green’s function of $\Delta$ on
$\Box_{L}$ and $\tilde{z}\in\Box_{L}$ is the unique interior point adjacent to
$z$. This is because $x\mapsto\Gamma_{L}(\tilde{z},x)+\mathbf{1}_{z}(x)$ is
discrete harmonic in $\Box_{L}$ and has the same boundary values as
$P^{z}_{L}$. It follows that

$\alpha^{z}_{L}(\eta)=-\Delta\left(\sum_{x\in\Box_{L}}\Gamma_{L}(\cdot,x)\eta(x)\right)(z)=-\sum_{x\in\Box_{L}}\Gamma_{L}(\tilde{z},x)\eta(x)=-\left\langle
P^{z}_{L},\eta\right\rangle_{\Box_{L}}.$

With these definitions in place, we pushforward $\mathrm{MM}_{L}$ by
$\Delta_{L}$ to get on $\mathbb{R}^{\Box_{L}}$ the measure

(1.6)
$\mathrm{Q}_{L}(d\eta):=\frac{1}{\mathsf{K}(\Box_{L})}\exp\left(-\sum_{x\in\Box_{L}}V(\eta(x))-\sum_{z\in\partial\Box_{L}}V(\alpha^{z}_{L}(\eta))\right)\prod_{x\in\Box_{L}}d\eta(x)$

for the partition function
$\mathsf{K}(\Box_{L})=|\det\Delta_{L}|\mathrm{Z}(\Box_{L})$. We also introduce
the tilted measure $\mathrm{Q}^{b}_{L}(d\eta)$ for any
$b\in\mathbb{R}^{\Box_{L}}$, defined by

(1.7)
$\mathrm{Q}^{b}_{L}(d\eta):=\frac{\mathsf{K}(\Box_{L})}{\mathsf{K}(\Box_{L},b)}\exp\left(\sum_{x\in\Box_{L}}b(x)\eta(x)\right)\mathrm{Q}_{L}(d\eta),$

where $\mathsf{K}(\Box_{L},b)$ is a constant making $\mathrm{Q}^{b}_{L}$ into
a probability measure. The central object of study is the cumulant generating
function (c.g.f.) of $\mathrm{Q}_{L}$, which is

$a\in\mathbb{R}^{\Box_{L}}\mapsto\log\mathbb{E}_{\mathrm{Q}_{L}}\exp\left(\sum_{x\in\Box_{L}}a(x)\eta(x)\right)=\log\frac{\mathsf{K}(\Box_{L},a)}{\mathsf{K}(\Box_{L})}.$

Note that for any $a^{\prime}\in\mathbb{R}^{\Box_{L}}$ we have

$\sum_{x\in\Box_{L}}a^{\prime}(x)\phi(x)=\sum_{y\in\Box_{L}}\left(\sum_{x\in\Box_{L}}-\Gamma_{L}(y,x)a^{\prime}(x)\right)\eta(y),$

where $\phi,\eta$ satisfy (1.5). If $\phi$ is distributed by
$\mathrm{MM}_{L}$, then $\eta$ will be distributed by $\mathrm{Q}_{L}$, and
vice-versa. It follows that the c.g.f. of the membrane model $\mathrm{MM}_{L}$
evaluated at some $a^{\prime}\in\mathbb{R}^{\Box_{L}}$ is the c.g.f. of
$\mathrm{Q}_{L}$ evaluated at $a=\Delta_{L}^{-1}a^{\prime}$. We will study
$\mathrm{MM}_{L}$ almost entirely through $\mathrm{Q}_{L}$.

######  Remark 1.1.

The $\nabla\phi$ model is commonly studied through the distribution of its
gradient. One benefit is that in the critical dimension $d=2$, the infinite
volume limit of the gradient exists whereas that of the field does not, and
many interesting observables are simple, local functions of the gradient.
However, the gradient is far from an i.i.d. field due to the many linear
dependencies satisfied by gradients, namely that they must sum to $0$ over all
closed loops in $\mathbb{Z}^{d}$. In the membrane model, the situation is
partly reversed: many interesting observables are global functions of the
Laplacian field, and any field on $\Box_{L}$ is a valid Laplacian field. The
former fact will cause difficulties by requiring us to understand the global
structure of the law of $\Delta\phi$, but the latter fact will facilitate this
understanding.

In Section 2, we will derive the Helffer-Sjöstrand representation of the
cumulant generating function.

$\log\frac{\mathsf{K}(\Box_{L},a)}{\mathsf{K}(\Box_{L})}=\int_{0}^{1}(1-r)\mathrm{Var}_{\mathrm{Q}^{ra}}\left(\sum_{x\in\Box_{L}}a(x)\eta(x)\right)dr=\int_{0}^{1}(1-r)\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}dr$

where $\mathrm{HS}^{b}$ is an operator $\mathbb{R}^{\Box_{L}}\to
H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$. We think of $\mathrm{HS}^{b}$ as
an operator dependent on a random environment $\eta$, and averaging over the
randomness $\eta\sim\mathrm{Q}^{b}$ gives
$\mathbb{E}_{\mathrm{Q}^{b}}\mathrm{HS}^{b}$, which is the covariance operator
$\mathbb{R}^{\Box_{L}}\to\mathbb{R}^{\Box_{L}}$ of
$(\eta(x))_{x\in\partial\Box_{L}}$. In the Gaussian case, the random
environment and the $b$ dependence is not present for $\mathrm{HS}^{b}$, which
is itself the covariance matrix.

For a function $v\in H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$, we will
usually consider $v=v(x,\eta)$ as a function of both discrete space
$x\in\Box_{L}$ and the background field $\eta\in\mathbb{R}^{\Box_{L}}$, and we
denote by $\nabla_{\eta(x)}$ the partial derivative in the coordinate
$\eta(x)$. For instance, for each $x,y\in\Box_{L}$, we have that
$\eta\mapsto\nabla_{\eta(x)}v(y,\eta)$ is in $L^{2}(\mathrm{Q}^{b})$.

The operator $\mathrm{HS}^{b}$ has a variational characterization. For
$a,b\in\mathbb{R}^{\Box_{L}}$, define $\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$
on $H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$ by

(1.8) $\displaystyle\mathcal{E}_{a}(v;\mathrm{Q}^{b})$
$\displaystyle=\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x,y\in\Box_{L}}|\nabla_{\eta(x)}v(y,\eta)|^{2}\right]$
$\displaystyle\quad+\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x\in\Box_{L}}V^{\prime\prime}(\eta(x))|v(x,\eta)|^{2}+\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\alpha^{z}(v(\cdot,\eta))|^{2}\right]$
$\displaystyle\quad-\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x\in\Box_{L}}a(x)\eta(x)\right].$

The energy $\mathcal{E}_{a}$ is strictly convex and has a unique minimizer,
which we define as $\mathrm{HS}^{b}a$. Because $\mathcal{E}_{a}$ is quadratic,
the map $\mathrm{HS}^{b}$ is a linear operator.

We will think of each of the four sums within (1.8) as playing a different
role. The first sum, the “smoothing term”, causes the minimizer
$\mathrm{HS}^{b}a$ of $\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$ to depend on the
global shape of $\mathrm{Q}^{b}$; without this term, we could minimize the
terms within the expectation in (1.8) for each realization of $\eta$ alone.
The second sum is the self-interaction term with random positive coefficients
$V^{\prime\prime}(\eta(x))$. The third sum is the boundary term, which
introduces interaction. Without this term, the minimizer’s values at
$(x,\eta)$ would be a function of $\eta(x)$ (and hence we would only need to
understand the one-spin marginals of $\mathrm{Q}^{b}$). The last term is the
perturbation term, without which the minimizer would be $0$.

The main task is to understand the minimizer $\mathrm{HS}^{b}a$ of (1.8). To
do so, we will need some initial understanding of $\mathrm{Q}^{b}$. In Section
4, we will understand the marginal law of a single spin $\eta(x)$ for
$x\in\Box_{L}$ far from the boundary. Using this, we then study the minimizer
of a simpler energy $\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$, which is obtained
from $\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$ by deleting the boundary term.

We find in Section 3 that if $a$ is harmonic on $\Box_{L}$ (after extending
$a$ to $\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$), then there is an alternative
characterization of the minimizer of $\mathcal{E}_{a}$. We use this to show
that $\mathrm{HS}^{b}a$ is small so long as $\|a\|_{L^{2}(\partial\Box_{L})}$
is small.

In Section 5, we consider the boundary term in
$\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$ and estimate it in terms of the
quantities $\left\langle P_{L}^{z},a\right\rangle_{\Box_{L}}$. By subtracting
off an appropriate harmonic function from $a$, we can ensure that
$\left\langle P_{L}^{z},a\right\rangle_{\Box_{L}}$ is small enough for us to
treat the boundary term as negligible as $L\to\infty$. The harmonic function
that we subtract off can be handled by the result in Section 3. Finally, when
the boundary term is small, then $\mathrm{HS}^{b}_{a}$ is approximately equal
to the minimizer of the simpler energy without the boundary term, which was
understood in Section 4.

In Section 6, we synthesize our results to prove the main theorems. In Section
7, we give some basic bounds on the Dirichlet Green’s functions for the
Laplacian, and we give estimates on the aforementioned decomposition of $a$
into a harmonic piece and a remainder.

We now give some further details about the minimization of
$\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$. For simplicity, consider $b=0$.
First, the quadratic functional $\mathcal{E}_{0}(\cdot;\mathrm{Q}^{0})$ is
coercive on $H^{1}(\mathrm{Q}^{0};\mathbb{R}^{\Box_{L}})$, and so if we can
find an ansatz $w$ such that $w$ almost minimizes
$\mathcal{E}_{a}(\cdot;\mathrm{Q}^{0})$, it necessarily follows that $w$
approximates the true minimizer $\mathrm{HS}^{0}a$.

We can construct a natural ansatz $w$ by simply ignoring the boundary term
term in $\mathcal{E}_{a}$ and attempting to minimize the rest. For simplicity,
we also ignore the smoothing term, though in reality we will have to factor it
in. With these conditions, the ansatz is
$w(x,\eta)=(V^{\prime\prime}(\eta(x))^{-1}a(x)$.

What is the typical size of $\left\langle
P^{z}_{L},w\right\rangle_{\Box_{L}}$? Assuming no cancellations, the best we
can estimate is

$|\left\langle
P^{z}_{L},w\right\rangle_{\Box_{L}}|\leq\|P^{z}_{L}\|_{L^{1}(\Box_{L})}\|w\|_{L^{\infty}(\Box_{L})}\approx
L\|w\|_{L^{\infty}(\Box_{L})}\approx L^{-d/2+1}\|a\|_{L^{2}(\Box_{L})}$

where in the last approximation we used $\inf V^{\prime\prime}>0$ and assumed
$a$ is a “macroscopic field”, i.e. $\|a\|_{L^{\infty}}\approx
L^{-d/2}\|a\|_{L^{2}(\Box_{L})}$. The boundary term with this guess is

$\mathbb{E}_{\mathrm{Q}^{0}}\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\left\langle
P^{z}_{L},w\right\rangle_{\Box_{L}}|^{2}\leq
C\|a\|^{2}_{L^{2}(\Box_{L})}\sum_{z\in\partial\Box_{L}}L^{-d+2}\approx
L\|a\|_{L^{2}(\Box_{L})},$

which is very large, meaning that our ansatz is flawed. It turns out that if
$a$ is harmonic and macroscopic (or “smooth”), this is essentially the truth
in the sense that the boundary term dominates, which explains why we are able
to show in Section 3 that the minimizer is small in this case.

To avoid this issue, we instead break down $a$ into two pieces:
$a=K_{L}a+K_{L}^{\perp}a$. Letting $\mathcal{H}(\Box_{L})$ denote fields which
are discrete harmonic on $\Box_{L}$, the first piece is defined by

$K_{L}a=\text{argmin}_{\tilde{a}\in\mathcal{H}(\Box_{L})}\|\tilde{a}-a\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2},$

which is the harmonic Bergman projection, the operator analyzed in Section 7.
Using that $\mathrm{HS}^{0}$ is linear, we can minimize $\mathcal{E}_{K_{L}a}$
and $\mathcal{E}_{K_{L}^{\perp}a}$ separately. We handle the $K_{L}a$ term
with Section 3, and so we set $a=K_{L}^{\perp}a$ in what follows.

Observe that

$\left\langle K_{L}^{\perp}a,P^{z}_{L}\right\rangle_{\Box_{L}}=\left\langle
K_{L}^{\perp}a,P^{z}_{L}\right\rangle_{\mathrm{cl}_{1}\Box_{L}}-K_{L}^{\perp}a(z)=-K_{L}^{\perp}a(z)$

for any $z\in\partial\Box_{L}$, which follows from the fact that
$K_{L}^{\perp}$ is the $L^{2}(\mathrm{cl}_{1}(\Box_{L}))$ projection onto the
orthogonal complement of $\mathcal{H}({\Box_{L}})$, which is the linear span
of $\\{P^{z}_{L}\\}_{z\in\partial\Box_{L}}$. If $a=K^{\perp}_{L}a$ is
macroscopic, we thus expect that typically

$|\left\langle a,P^{z}_{L}\right\rangle_{\Box_{L}}|\approx
L^{-d/2}\|a\|_{L^{2}(\Box_{L})}.$

Returning to our guess $w$, we have

$\left\langle
w,P^{z}_{L}\right\rangle_{\Box_{L}}=\sum_{x\in\Box_{L}}\left(\frac{1}{V^{\prime\prime}(\eta(x))}\right)P^{z}_{L}(x)a(x).$

Since the above sum is over many different sites $x$, we might expect the
random coefficients $(V^{\prime\prime}(\eta(x))^{-1}$ undergo a law of large
numbers effect and so can be treated like a constant. We prove such an effect
and, together with our understanding of the marginals of $\eta(x)$ in Section
4, we can prove that typically

$|\left\langle w,P^{z}_{L}\right\rangle_{\Box_{L}}|\leq C|\left\langle
a,P^{z}_{L}\right\rangle_{\Box_{L}}|+\mathrm{Error}\leq
CL^{-d/2}\|a\|_{L^{2}(\Box_{L})}+\mathrm{Error}$

as $L\to\infty$, for a negligible term $\mathrm{Error}$, so long as we modify
the ansatz $w$ near $\partial\Box_{L}$. Roughly speaking, the boundary term is
then

$\mathbb{E}_{\mathrm{Q}^{0}}\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\left\langle
P^{z}_{L},w\right\rangle_{\Box_{L}}|^{2}\leq
C\|a\|_{L^{2}(\Box_{L})}^{2}\sum_{z\in\partial\Box_{L}}L^{-d}\leq
CL^{-1}\|a\|_{L^{2}(\Box_{L})}^{2},$

which is small. In truth, we get a weaker inequality due to error terms from
other sources, but the above computation illustrates broadly the logic of the
proof.

Acknowledgements. The author was partially supported by NSF grant DMS-2000205.

##  2\. The Helffer-Sjöstrand Equation

In this section, we define the Helffer-Sjöstrand representation for the c.g.f.
and prove well-posedness results. We also define and prove basic properties of
the energies associated to the representation.

Recall the definition of $\mathrm{Q}^{b}$ from (1.6) and (1.7). For any
$b\in\mathbb{R}^{\Box_{L}}$, we define the operator
$\mathcal{L}_{\mathrm{Q}^{b}}$ by

$\mathcal{L}_{\mathrm{Q}^{b}}v=\Delta_{\eta}v+\sum_{x\in\Box_{L}}\nabla_{\eta(x)}\log\mathrm{Q}^{b}\cdot\nabla_{\eta(x)}v,$

where we have also denoted by $\mathrm{Q}^{b}$ the Lebesgue density of
$\mathrm{Q}^{b}$. The operator

$\Delta_{\eta}=\sum_{x\in\Box_{L}}\nabla_{\eta(x)}^{2}$

is the (continuum) Laplacian on $\mathbb{R}^{\Box_{L}}$, not to be mistaken
with the discrete Laplacian. The above definition can easily be extended to
define $\mathcal{L}_{\mu}$ for other measures $\mu$ with a positive, smooth
Lebesgue density.

We compute

(2.1)
$\mathcal{L}_{\mathrm{Q}^{b}}v=\Delta_{\eta}v+\sum_{x\in\Box_{L}}\left(-V^{\prime}(\eta(x))+\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))P^{z}(x)+b(x)\right)\nabla_{\eta(x)}v.$

Note that

(2.2)
$\mathcal{L}_{\mathrm{Q}^{b}}=-\sum_{x\in\Box_{L}}\nabla^{\ast,b}_{\eta(x)}\nabla_{\eta(x)}$

where $\nabla^{\ast,b}_{\eta(x)}$ is the $L^{2}(\mathrm{Q}^{b})$-adjoint of
$\nabla_{\eta(x)}$. Another important operator is the commutator of
$\nabla_{\eta(x)}$ and $\nabla_{\eta(y)}^{\ast,b}$, which is given for any
$x,y\in\Box_{L}$ by

(2.3) $\displaystyle[\nabla_{\eta(x)},\nabla_{\eta(y)}^{\ast,b}]$
$\displaystyle=\nabla_{\eta(x)}\nabla_{\eta(y)}\mathcal{H}^{b}(\eta)$
$\displaystyle=V^{\prime\prime}(\eta(x))\mathbf{1}_{x=y}+\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))P^{z}(x)P^{z}(y).$

Here $\mathcal{H}^{b}$ is the Hamiltonian associated to $\mathrm{Q}^{b}\propto
e^{-\mathcal{H}_{b}}$; see (2.6) below.

We pause to state the Bakry-Emery criterion, which will be used throughout the
paper.

######  Proposition 2.1 (See e.g. [2]).

Let $\mu$ be proportional to $\exp(-W(\eta))\prod_{x\in\Box_{L}}d\eta(x)$, for
some $W\in C^{2}(\mathbb{R}^{\Box_{L}})$. Suppose we can find $\lambda>0$ such
that $\nabla^{2}_{\eta}W\geq\lambda\cdot\mathrm{Id}$, where
$\nabla^{2}_{\eta}W$ denotes the Hessian of $W$. Then $\mu$ satisfies the
Poincaré inequality with constant $\lambda^{-1}$ and the log-Sobolev
inequality with constant $2\lambda^{-1}$. That is,

(2.4) $\mathrm{Var}_{\mu}f\leq\frac{1}{\lambda}\mathbb{E}_{\mu}|\nabla f|^{2}$

for all $f\in H^{1}(\mu)$ and

(2.5) $\mathbb{E}_{\mu}(f^{2}\log
f^{2})\leq\frac{2}{\lambda}\mathbb{E}_{\mu}|\nabla f|^{2}$

for all $f\in H^{1}(\mu)$ with $\mathbb{E}_{\mu}f^{2}=1$.

We can apply the criterion to invert $\mathcal{L}_{\mathrm{Q}^{b}}$.

######  Proposition 2.2.

For any $b\in\mathbb{R}^{\Box_{L}}$, the measure $\mathrm{Q}^{b}$ satisfies
the Poincaré inequality with constant $\inf V^{\prime\prime}$. Furthermore,
the equation $\mathcal{L}_{\mathrm{Q}^{b}}u=f$ admits a unique centered weak
solution $u\in H^{1}(\mathrm{Q}^{b})$ for every $f\in L^{2}(\mathrm{Q}^{b})$
with $\mathbb{E}_{\mathrm{Q}^{b}}f=0$.

###### Proof.

The first part is due to the Bakry-Emery criterion. The Hamiltonian
$\mathcal{H}^{b}$ defining $\mathrm{Q}^{b}$ in (1.6) is

(2.6)
$\mathcal{H}^{b}(\eta)=\sum_{x\in\Box_{L}}V(\eta(x))+\sum_{z\in\partial\Box_{L}}V(\alpha^{z}(\eta))-\sum_{x\in\Box_{L}}b(x)\eta(x)$

which has second derivative

$\nabla_{\eta(x)}\nabla_{\eta(y)}\mathcal{H}^{b}(\eta)=\mathbf{1}_{x=y}V^{\prime\prime}(\eta)+\sum_{z\in\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))P^{z}(x)P^{z}(y).$

Since $\alpha^{z}\otimes\alpha^{z}\geq 0$, we see that
$\nabla^{2}_{\eta}\mathcal{H}^{b}\geq(\inf V^{\prime\prime})\text{Id}$, and so
2.1 applies. We can then invert $\mathcal{L}_{\mathrm{Q}^{b}}$ using the Lax-
Milgram theorem. The Poincaré inequality provides coerciveness of the
quadratic form
$-\mathbb{E}_{\mathrm{Q}^{b}}f\mathcal{L}_{\mathrm{Q}^{b}}f=\mathbb{E}_{\mathrm{Q}^{b}}|\nabla
f|^{2}$ on centered random variables $f$. ∎

We give a formal computation that motivates studying
$\mathcal{L}_{\mathrm{Q}^{b}}$. First, one can show that to compute the c.g.f.
of $\mathrm{Q}$, it is sufficient to understand variances under the tilted
measures $\mathrm{Q}^{b}$. Then, for a centered $F\in H^{1}(\mathrm{Q}^{b})$,
we have

$\mathrm{Var}_{\mathrm{Q}^{b}}F=\mathbb{E}_{\mathrm{Q}^{b}}\left[F(-\mathcal{L}_{\mathrm{Q}^{b}})(-\mathcal{L}_{\mathrm{Q}^{b}})^{-1}F\right]=\sum_{x\in\Box_{L}}\mathbb{E}_{\mathrm{Q}^{b}}\left[\nabla_{\eta(x)}F\cdot\nabla_{\eta(x)}(-\mathcal{L}_{\mathrm{Q}^{b}})^{-1}F\right],$

where in the last equality we used the representation (2.2). Let
$u(x,\eta)=\nabla_{\eta(x)}(-\mathcal{L}_{\mathrm{Q}^{b}})^{-1}F$. If we apply
$\nabla_{\eta(x)}$ to the equation

$(-\mathcal{L}_{\mathrm{Q}^{b}})(-\mathcal{L}_{\mathrm{Q}^{b}})^{-1}F=F$

and commute the gradient and $\mathcal{L}_{\mathrm{Q}^{b}}$, we get

$-\mathcal{L}_{\mathrm{Q}^{b}}u(x,\eta)-[\nabla_{\eta(x)},\mathcal{L}_{\mathrm{Q}^{b}}](-\mathcal{L}_{\mathrm{Q}^{b}})^{-1}F=\nabla_{\eta(x)}F.$

For the commutator above, we compute

$[\nabla_{\eta(x)},-\mathcal{L}_{\mathrm{Q}^{b}}]=\sum_{y\in\Box_{L}}[\nabla_{\eta(x)},\nabla_{\eta(y)}^{\ast,b}\nabla_{\eta(y)}]=\sum_{y\in\Box_{L}}[\nabla_{\eta(x)},\nabla_{\eta(y)}^{\ast,b}]\nabla_{\eta(y)},$

and we use (2.3) on the second term above to see

$-\mathcal{L}_{\mathrm{Q}^{b}}u(x,\eta)+(\nabla^{2}_{\eta}\mathcal{H}^{b}\cdot
u)(x,\eta)=\nabla_{\eta(x)}F.$

This is the Helffer-Sjöstrand equation for $u$. It follows that

$\mathrm{Var}_{\mathrm{Q}^{b}}F=\mathbb{E}_{\mathrm{Q}^{b}}\left[\left\langle
f(\cdot,\eta),u(\cdot,\eta)\right\rangle_{\Box_{L}}\right]$

for $f(x,\eta)=\nabla_{\eta(x)}F(\eta)$, which is the Helffer-Sjöstrand
representation.

The next two lemmas make rigorous the above computation in the special case
$f(x,\eta)=a(x)$.

######  Lemma 2.3.

For any $a,b\in\mathbb{R}^{\Box_{L}}$, the equation

(2.7) $-\mathcal{L}_{\mathrm{Q}^{b}}v+\nabla_{\eta}^{2}\mathcal{H}^{b}\cdot
v=a$

has a unique solution $\mathrm{HS}^{b}a=u^{b}_{a}\in
H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$. Moreover, we have
$u^{b}_{a}(x,\eta)=\nabla_{\eta(x)}v^{b}_{a}(\eta)$, where $v^{b}_{a}$ solves

(2.8)
$-\mathcal{L}_{\mathrm{Q}^{b}}v^{b}_{a}=\mathrm{Const}+\sum_{x\in\Box_{L}}a(x)\eta(x).$

The constant $\mathrm{Const}$ is chosen so that the RHS above has expectation
$0$ under $\mathrm{Q}^{b}$.

###### Proof.

Uniqueness and existence of $u^{b}_{a}$ follows from
$\nabla_{\eta}^{2}\mathcal{H}^{b}\geq(\inf
V^{\prime\prime})\text{Id}_{\Box_{L}\times\Box_{L}}$ a.s. in $\eta$ and from
Lax-Milgram. Given $v^{b}_{a}$ solving (2.8) (such a solution exists by 2.2),
we will show that $\nabla_{\eta(x)}v^{b}_{a}$ satisfies equation (2.7) in a
weak sense, i.e. integrated against
$H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$ functions. From the equation for
$v^{b}_{a}$, we have

$\sum_{x\in\Box_{L}}\mathbb{E}_{\mathrm{Q}^{b}}\left[\nabla_{\eta(x)}v^{b}_{a}\cdot\nabla_{\eta(x)}\varphi\right]=\mathrm{Const}\cdot\mathbb{E}_{\mathrm{Q}^{b}}\varphi+\sum_{x\in\Box_{L}}a(x)\mathbb{E}_{\mathrm{Q}^{b}}\left[\eta(x)\varphi\right]$

for all $\varphi\in H^{2}(\mathrm{Q}^{b})$. Choose
$\varphi=\nabla^{\ast,b}_{\eta(x_{0})}\psi$ for $\psi\in C^{\infty}_{c}$ to
see

(2.9)
$\displaystyle\mathbb{E}_{\mathrm{Q}^{b}}\left[\nabla_{\eta(x)}v^{b}_{a}\cdot\nabla_{\eta(x)}\varphi\right]$
$\displaystyle=\mathbb{E}_{\mathrm{Q}^{b}}\left[\nabla_{\eta(x)}v^{b}_{a}\cdot\left(\nabla^{\ast,b}_{\eta(x_{0})}\nabla_{\eta(x)}+[\nabla_{\eta(x)},\nabla^{\ast,b}_{\eta(x_{0})}]\right)\psi\right]$
$\displaystyle=\mathbb{E}_{\mathrm{Q}^{b}}\left[\nabla_{\eta(x_{0})}v^{b}_{a}\cdot\nabla^{\ast,b}_{\eta(x)}\nabla_{\eta(x)}\psi\right]+\mathbb{E}\left(\nabla_{\eta(x)}v^{b}_{a}\cdot[\nabla_{\eta(x)},\nabla^{\ast,b}_{\eta(x_{0})}]\psi\right).$

In the last line we moved the $\nabla_{\eta(x)}$ derivative onto $\psi$, used
$[\nabla^{\ast,b}_{\eta(x)},\nabla^{\ast,b}_{\eta(x_{0})}]=0$, and then put
the derivative $\nabla_{\eta(x_{0})}$ onto $v^{b}_{a}$. We then use (2.3) and
sum over $x$ to see

$\mathbb{E}_{\mathrm{Q}^{b}}\left[\nabla_{\eta(x_{0})}v^{b}_{a}(-\mathcal{L}_{\mathrm{Q}^{b}}\psi)\right]+\mathbb{E}\left[\sum_{x\in\Box_{L}}\left(\nabla^{2}_{\eta(x),\eta(x_{0})}\mathcal{H}^{b}\right)(\nabla_{\eta(x)}v^{b}_{a})\cdot\psi\right]=a(x_{0})\mathbb{E}_{\mathrm{Q}^{b}}\left[\psi\right].$

Let $\tilde{u}(x_{0},\eta):=\nabla_{\eta(x_{0})}v^{b}_{a}(\eta)$. We now let
$\psi=\psi_{x_{0}}$ depend on $x_{0}$ and sum over $x_{0}$. Note that the
Hessian of $\mathcal{H}^{b}$ is self-adjoint, and so we get

$\mathbb{E}_{\mathrm{Q}^{b}}\left\langle\tilde{u}(\cdot,\eta),(-\mathcal{L}_{\mathrm{Q}^{b}}+\nabla^{2}_{\eta}\mathcal{H}^{b})\psi_{\cdot}(\eta)\right\rangle_{\Box_{L}}=\mathbb{E}_{\mathrm{Q}^{b}}\left\langle
a,\psi_{\cdot}\right\rangle_{\Box_{L}}.$

This is the weak form of (2.7). The matrix $\nabla^{2}\mathcal{H}^{b}$ is
uniformly bounded in $\eta$, and so it follows that $\tilde{u}(x,\cdot)$ is in
$H^{1}(\mathrm{Q}^{b})$ for all $x$. We can then integrate by parts in the
above equation to show that $\tilde{u}$ solves (2.7) weakly in $H^{1}$, and so
by uniqueness we have $u^{b}_{a}=\tilde{u}$. ∎

We will write the solution of (2.7) as $u^{b}_{a}$ or $\mathrm{HS}^{b}a$. When
$b$ is understood, or we are considering solutions of (2.7) where the operator
$\mathcal{L}_{\mathrm{Q}^{b}}$ is replaced by $\mathcal{L}_{\mu}$ for some
reference measure $\mu$, we will often just write $u_{a}$.

######  Lemma 2.4.

We can express the cumulant generating function of $\mathrm{Q}^{0}$ as

(2.10)
$\log\frac{\mathsf{K}(\Box_{L},a)}{\mathsf{K}(\Box_{L},0)}=\int_{0}^{1}(1-r)\mathrm{Var}_{\mathrm{Q}^{ra}}\left[\sum_{x\in\Box_{L}}a(x)\eta(x)\right]dr.$

The variances can be expressed via the Helffer-Sjöstrand representation by

(2.11)
$\mathrm{Var}_{\mathrm{Q}^{ra}}\left[\sum_{x\in\Box_{L}}a(x)\eta(x)\right]=\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}u^{ra}_{a}\right\rangle_{\Box_{L}}=\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}.$

###### Proof.

We start by computing

$\frac{d}{dr}\log\mathsf{K}(\Box_{L},ra)=\mathbb{E}_{\mathrm{Q}^{ra}}\left[\sum_{x\in\Box_{L}}a(x)\eta(x)\right].$

Since $\mathrm{Q}^{0}$ is invariant under $\eta\mapsto-\eta$, we see
$\frac{d}{dr}\log\mathsf{K}(\Box_{L},ra)=0$ at $r=0$. We can take another
derivative in $r$ to see

$\left(\frac{d}{dr}\right)^{2}\log\mathsf{K}(\Box_{L},ra)=\mathrm{Var}_{\mathrm{Q}^{ra}}\left[\sum_{x\in\Box_{L}}a(x)\eta(x)\right].$

We conclude (2.10) by integrating in $r$.

We now prove the variance formula (2.11). Let
$F(\eta)=\sum_{x\in\Box_{L}}a(x)\eta(x)$. We use the formula

$u^{ra}_{a}(x,\cdot)=\nabla_{\eta(x)}(-\mathcal{L}_{\mathrm{Q}^{ra}})^{-1}\left(F-\mathbb{E}_{\mathrm{Q}^{ra}}F\right)$

which was proved in 2.3. Since $\mathcal{L}_{\mathrm{Q}^{ra}}$ is
$L^{2}(\mathrm{Q}^{ra})$ self-adjoint and using the representation (2.2), we
have

$\displaystyle\mathrm{Var}_{\mathrm{Q}^{ra}}F$
$\displaystyle=\mathbb{E}_{\mathrm{Q}^{ra}}\left[F(\eta)(F(\eta)-\mathbb{E}_{\mathrm{Q}^{ra}}F)\right]$
$\displaystyle=-\mathbb{E}_{\mathrm{Q}^{ra}}\left[\mathcal{L}_{\mathrm{Q}^{ra}}F(\eta)(-\mathcal{L}_{\mathrm{Q}^{ra}})^{-1}(F(\eta)-\mathbb{E}_{\mathrm{Q}^{ra}}F)\right]$
$\displaystyle=\mathbb{E}_{\mathrm{Q}^{ra}}\left[\sum_{x\in\Box_{L}}\left(\nabla_{\eta(x)}F(\eta)\right)u^{ra}_{a}(x,\eta)\right]=\mathbb{E}_{\mathrm{Q}^{ra}}\left\langle
a,u^{ra}_{a}(x,\eta)\right\rangle_{\Box_{L}}.$

∎

We will now define three energies and state some basic theorems about their
minimizers. Two of the energies will depend on a background measure
$\mathrm{Q}^{b}$ for some $b\in\mathbb{R}^{\Box_{L}}$, and we will sometimes
omit this dependence. As in (1.8), for any $a\in\mathbb{R}^{\Box_{L}}$, the
energy
$\mathcal{E}_{a}:H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})\to\mathbb{R}$ is
defined by

(2.12) $\displaystyle\mathcal{E}_{a}(v;\mathrm{Q}^{b})$
$\displaystyle=\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x,y\in\Box_{L}}|\nabla_{\eta(y)}v(x,\eta)|^{2}+\sum_{x\in\Box_{L}}V^{\prime\prime}(\eta(x))|v(x,\eta)|^{2}\right]$
$\displaystyle\quad+\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\alpha^{z}_{L}(v(\cdot,\eta))|^{2}\right]-\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x\in\Box_{L}}a(x)v(x,\eta)\right].$

The minimizer of $\mathcal{E}_{a}$ will be denoted by $u_{a}$ (we will see
that it is the same as the solution to the Helffer-Sjöstrand equation
momentarily).

We also introduce a version of $\mathcal{E}_{a}$ without the boundary term:

(2.13)
$\mathcal{F}_{0}(v;\mathrm{Q}^{b})=\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x,y\in\Box_{L}}|\nabla_{\eta(y)}v(x,\eta)|^{2}+\sum_{x\in\Box_{L}}V^{\prime\prime}(\eta(x))|v(x,\eta)|^{2}\right]$

and

(2.14)
$\mathcal{F}_{a}(v;\mathrm{Q}^{b})=\mathcal{F}_{0}(v;\mathrm{Q}^{b})-\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x\in\Box_{L}}a(x)v(x,\eta)\right].$

We denote by $\overline{u}_{a}$ the minimizer of
$\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$.

######  Proposition 2.5.

The functionals $\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$ and
$\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$ are continuous on
$H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$ and strictly convex. We have the
lower bounds

(2.15) $\inf_{v\in H^{1}}\mathcal{E}_{a}(v;\mathrm{Q}^{b})\geq\inf_{v\in
H^{1}}\mathcal{F}_{a}(v;\mathrm{Q}^{b})\geq-\frac{1}{2\inf
V^{\prime\prime}}\|a\|^{2}_{L^{2}(\Box_{L})}.$

The minimizers $u_{a}$ and $\overline{u}_{a}$ are weak solutions of the
following equations:

(2.16)
$\displaystyle-\mathcal{L}_{\mathrm{Q}^{b}}u_{a}(x,\eta)+(\nabla^{2}\mathcal{H}^{b}\cdot
u_{a})(x,\eta)=a(x)$ (2.17)
$\displaystyle-\mathcal{L}_{\mathrm{Q}^{b}}\overline{u}_{a}(x,\eta)+V^{\prime\prime}(\eta(x))\overline{u}_{a}(x,\eta)=a(x).$

For any $v\in H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$, we have the
quadratic response identities

(2.18)
$\mathcal{E}_{a}(v;\mathrm{Q}^{b})-\mathcal{E}_{a}(u_{a};\mathrm{Q}^{b})=\mathcal{E}_{0}(v-u_{a};\mathrm{Q}^{b}),\quad\mathcal{F}_{a}(v;\mathrm{Q}^{b})-\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})=\mathcal{F}_{0}(v-\overline{u}_{a};\mathrm{Q}^{b}).$

Consequently (letting $v=0$), we have

(2.19)
$\mathcal{E}_{0}(u_{a};\mathrm{Q}^{b})\leq\mathcal{F}_{0}(\overline{u}_{a};\mathrm{Q}^{b})\leq\frac{1}{2\inf
V^{\prime\prime}}\|a\|^{2}_{L^{2}}.$

Furthermore, $\overline{u}_{a}(x,\eta)$ is between $(\inf
V^{\prime\prime})^{-1}a(x)$ and $(\sup V^{\prime\prime})^{-1}a(x)$ almost
surely (in $\eta$) for all $x\in\Box_{L}$.

###### Proof.

Continuity and strict convexity of the functionals follow easily from $0<\inf
V^{\prime\prime}\leq\sup V^{\prime\prime}<\infty$.

The inequalities in (2.15) follow from ignoring all terms in the energy except
for

$\mathbb{E}_{\mathrm{Q}^{b}}\left[\frac{1}{2}\sum_{x\in\Box_{L}}V^{\prime\prime}(\eta(x))|v(x,\eta)|^{2}-\sum_{x\in\Box_{L}}a(x)v(x,\eta)\right].$

The equations for $u_{a}$ and $\overline{u}_{a}$ follow from standard first
variation arguments. The quadratic response identities follow immediately from
expanding the square within the energies and using the equations. The bound
(2.19) follows from plugging $v=0$ into the quadratic response identities and
using (2.15).

Finally, we prove the bound on $\overline{u}_{a}$. Let $[c_{x},d_{x}]$ be the
closed interval between $(\inf V^{\prime\prime})^{-1}a(x)$ and $(\sup
V^{\prime\prime})^{-1}a(x)$. Define $\chi_{x}:\mathbb{R}\to\mathbb{R}$ by
$\chi_{x}(y)=\max(c_{x},\min(y,d_{x}))$. Then
$(x,\eta)\mapsto\chi_{x}(\overline{u}_{a}(x,\eta))$ is a minimizer of
$\mathcal{F}_{a}$. This is because
$\nabla_{\eta(x)}\chi_{x}(\overline{u}_{a}(x,\eta)=0$ almost surely when
$\chi(\overline{u}_{a})$ is $c$ or $d$, and

$\displaystyle\frac{1}{2}\sum_{x\in\Box_{L}}V^{\prime\prime}(\eta(x))|\chi_{x}(\overline{u}_{a}(x,\eta))|^{2}-\sum_{x\in\Box_{L}}a(x)\chi_{x}(\overline{u}_{a}(x,\eta))$
$\displaystyle\leq\frac{1}{2}\sum_{x\in\Box_{L}}V^{\prime\prime}(\eta(x))|\overline{u}_{a}(x,\eta)|^{2}$
$\displaystyle\quad-\sum_{x\in\Box_{L}}a(x)\overline{u}_{a}(x,\eta).$

By uniqueness of the minimizer, we then have
$\chi_{x}(\overline{u}_{a}(x,\eta))=\overline{u}_{a}(x,\eta)$ a.s. ∎

The main importance of $\overline{u}_{a}$ is that it is significantly easier
to understand than $u_{a}$, and so it serves as a building block for
constructing near-minimizers of $\mathcal{E}_{a}$. In particular, we will
approximate $\overline{u}_{a}$ it in terms of the following 1D object. For any
$\beta\in\mathbb{R}$, we let $\nu^{\beta}$ be the probability measure on
$\mathbb{R}$ proportional to $\exp(-V(\xi)+\beta\xi)d\xi$. Then, for $v\in
H^{1}(\nu^{\beta};\mathbb{R})$ we let

(2.20)
$\mathcal{G}_{0}(v;\nu^{\beta})=\mathbb{E}_{\nu^{\beta}}\left[\frac{1}{2}|\nabla_{\xi}v|^{2}+\frac{1}{2}V^{\prime\prime}(\xi)|v(\xi)|^{2}\right]$

and, for any $\gamma\in\mathbb{R}$, we let
$\mathcal{G}_{\gamma}(v;\nu^{\beta})=\mathcal{G}_{0}(v;\nu^{\beta})-\gamma\mathbb{E}_{\nu^{\beta}}v$.
We let $U^{\beta}$ be the minimizer of $\mathcal{G}_{1}(\cdot;\nu^{\beta})$.

######  Proposition 2.6.

The energy $\mathcal{G}_{1}(\cdot;\nu^{\beta})$ is continuous on
$H^{1}(\nu^{\beta};\mathbb{R})$ and strictly convex. The associated minimizer
has $(\sup V^{\prime\prime})^{-1}\leq U^{\beta}\leq(\inf
V^{\prime\prime})^{-1}$ a.s. and $\|\nabla_{\xi}U^{\beta}\|_{L^{\infty}}\leq
C$.

###### Proof.

The proof of $(\sup V^{\prime\prime})^{-1}\leq U^{\beta}\leq(\inf
V^{\prime\prime})^{-1}$ is exactly the same as the corresponding bound on
$\overline{u}_{a}$ in 2.5. To prove the bound on the gradient of $U^{\beta}$,
consider the dynamics

$d\xi_{t}=(-V^{\prime}(\xi_{t})+\beta)dt+\sqrt{2}dB_{t}$

for a standard Brownian motion $B_{t}$. The operator
$-\mathcal{L}_{\nu^{\beta}}=-\Delta+(V^{\prime}(\xi)-\beta)\nabla$ is the
generator of the dynamics. Then we have the Feynman-Kac representation

$U^{\beta}(\xi)=\mathbb{E}^{\xi_{0}=\xi}\left[\int_{0}^{\infty}\exp\left(-\int_{0}^{t}V^{\prime\prime}(\xi_{s})ds\right)dt\right].$

And so

$\nabla_{\xi}U^{\beta}(\xi)=-\mathbb{E}^{\xi_{0}=\xi}\left[\int_{0}^{\infty}\left(\exp\left(-\int_{0}^{t}V^{\prime\prime}(\xi_{s})ds\right)\int_{0}^{t}V^{(3)}(\xi_{s})J_{0,s}(\xi)ds\right)dt\right]$

where $J_{0,s}(\xi)$ is the “derivative” flow of $\xi_{t}$, which solves

$\begin{cases}\partial_{t}J_{0,t}(\xi)=-V^{\prime\prime}(\xi_{t})J_{0,t}(\xi),\\\
J_{0,0}(\xi)=1.\end{cases}$

It is clear that $J_{0,t}(\xi)$ is bounded uniformly in $(\xi_{t})_{t\geq 0}$
and $t$ since its equation is contractive, and $V^{(3)}$ is bounded by
assumption. Thus

$|\nabla_{\xi}U^{\beta}(\xi)|\leq
C\mathbb{E}^{\xi_{0}=\xi}\left[\int_{0}^{\infty}\exp\left(-\int_{0}^{t}V^{\prime\prime}(\xi_{s})ds\right)tdt\right]\leq
C\int_{0}^{\infty}te^{-\inf V^{\prime\prime}t}dt\leq C.$

∎

##  3\. Minimization for Smooth Harmonic Perturbations

In this section, we let $a=\sum_{z\in\partial\Box_{L}}\gamma_{z}P^{z}$ for
some coefficients $(\gamma_{z})_{z\in\partial\Box_{L}}$ and the Poisson kernel
$P^{z}=P^{z}_{L}$ on $\Box_{L}$, and we want to prove that the minimizer
$u_{a}$ of $\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$ is small. This will prove
$\mathrm{HS}^{b}a$ is small and ultimately allow us to discard harmonic
components of more general $a$ in our calculations. We omit the dependence on
$\mathrm{Q}^{b}$ below, and all constants $C$ are independent of $b$.

We first reduce the problem to the same problem for a Gaussian membrane mode.
To that end, let $\mathcal{E}^{\prime}_{a}$ be the energy

(3.1) $\displaystyle\mathcal{E}^{\prime}_{a}(v)$
$\displaystyle=\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\left(\sum_{x\in\Box_{L}}(\inf
V^{\prime\prime})|v(x,\eta)|^{2}+\sum_{z\in\partial\Box_{L}}(\inf
V^{\prime\prime})|\alpha^{z}(v(\cdot,\eta))|^{2}\right)$
$\displaystyle\quad-\mathbb{E}_{\mathrm{Q}^{b}}\left(\sum_{x\in\Box_{L}}a(x)v(x,\eta)\right).$

Note that $\mathcal{E}^{\prime}_{0}(v)$, and its perturbed versions
$\mathcal{E}^{\prime}_{a}$, correspond to the energies $\mathcal{E}_{a}$ with
the quadratic potential $x\mapsto(\inf V^{\prime\prime})|x|^{2}$ in place of
$V$. We do not include the smoothing term $|\nabla_{\eta}v|^{2}$ because all
minimizers are independent of $\eta$ (it would also be of no consequence to
include it). We have

(3.2) $\mathcal{E}_{a}(v)\geq\mathcal{E}^{\prime}_{a}(v).$

Let $u_{a}$ be the minimizer of $\mathcal{E}_{a}$ and let $u^{\prime}_{a}$ be
the minimizer of $\mathcal{E}^{\prime}_{a}$.

There is a small technical issue in that the representation
$a=\sum_{z\in\partial\Box_{L}}\gamma_{z}P^{z}$ is not unique given just the
data $a\in\mathbb{R}^{\Box_{L}}$. It can happen that $P^{z}=P^{w}$ in
$\Box_{L}$ for $z\neq w$, for example if $z$ and $w$ are adjacent to the same
corner of the square $\Box_{L}$. The below proposition will be valid for any
choice of extension of $a$ to $\mathrm{cl}_{1}\Box_{L}$.

######  Proposition 3.1.

Let $a=\sum_{z\in\partial\Box_{L}}\gamma_{z}P^{z}$ and
$u_{a}=\mathrm{HS}^{b}a$. Then there is a constant $C$ such that

(3.3) $\mathcal{E}_{0}(u_{a};\mathrm{Q}^{b})\leq
C\|a\|_{L^{2}(\Box_{L})}\|\gamma\|_{L^{2}(\partial\Box_{L})}.$

Note that the estimate (3.3) has the boundary term
$\|\gamma\|_{L^{2}(\partial\Box_{L})}$, which should be much smaller than
$\|a\|_{L^{2}(\Box_{L})}$ provided $a$ is macroscopic or smooth in a sense.
This makes the estimate much stronger than the trivial estimate (2.19) with
RHS $C\|a\|_{L^{2}(\Box_{L})}^{2}$.

###### Proof.

It is enough to provide a lower bound for $\mathcal{E}_{a}$, since we have
$\mathcal{E}_{0}(u_{a})=-\mathcal{E}_{a}(u_{a})$ by comparison with $v=0$ and
quadratic response. By (3.2), it is enough to bound
$\mathcal{E}^{\prime}_{a}(u^{\prime}_{a})$. The proof proceeds by rewriting
the minimization problem in terms of the boundary values $\gamma_{z}$ of $a$.

Define a strictly convex, continuous functional
$\mathcal{B}_{\gamma}:\mathbb{R}^{\partial\Box_{L}}\to\mathbb{R}$ by

$\mathcal{B}_{\gamma}(\varepsilon)=\frac{1}{2}(\inf
V^{\prime\prime})\|\sum_{z\in\partial\Box_{L}}\varepsilon_{z}P^{z}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}-\sum_{z\in\partial\Box_{L}}\varepsilon_{z}\gamma_{z}.$

Let $\lambda\in\mathbb{R}^{\partial\Box_{L}}$ be the minimizer of
$\mathcal{B}_{\gamma}$. We claim that
$\overline{\lambda}=\sum_{z\in\partial\Box_{L}}\lambda_{z}P^{z}\in\mathbb{R}^{\Box_{L}}$
is equal to the minimizer $u^{\prime}_{a}$ of $\mathcal{E}^{\prime}_{a}$
within $\Box_{L}$. Indeed, the critical point $\lambda$ solves

$\sum_{w\in\partial\Box_{L}}\lambda_{w}\left\langle
P^{z},P^{w}\right\rangle_{\mathrm{cl}_{1}\Box_{L}}=\frac{\gamma_{z}}{\inf
V^{\prime\prime}}\quad\forall z\in\partial\Box_{L},$

which we rewrite to

(3.4) $\lambda_{z}+\sum_{w\in\partial\Box_{L}}\lambda_{w}\left\langle
P^{z},P^{w}\right\rangle_{\Box_{L}}=\frac{\gamma_{z}}{\inf
V^{\prime\prime}}\quad\forall z\in\partial\Box_{L}.$

After multiplying (3.4) by $P^{z}(x)$ and summing over $z$, we get that

$\overline{\lambda}(x)+\sum_{z\in\partial\Box_{L}}\left\langle
P^{z},\overline{\lambda}\right\rangle_{\Box_{L}}P^{z}(x)=\frac{a(x)}{\inf
V^{\prime\prime}}\quad\forall x\in\Box_{L}.$

This equation also characterizes minimizers of $\mathcal{E}^{\prime}_{a}$, so
we conclude $\overline{\lambda}=u^{\prime}_{a}$.

Note that

$\mathcal{B}_{\gamma}(\varepsilon)\geq\frac{1}{2}(\inf
V^{\prime\prime})\|\varepsilon\|_{L^{2}(\partial\Box_{L})}^{2}-\|\varepsilon\|_{L^{2}(\partial\Box_{L})}\|\gamma\|_{L^{2}(\partial\Box_{L})}\geq-\frac{1}{2(\inf
V^{\prime\prime})}\|\gamma\|^{2}_{L^{2}(\partial\Box_{L})}.$

By quadratic response we have

$\mathcal{B}_{0}(\lambda)\leq\mathcal{B}_{\gamma}(0)-\mathcal{B}_{\gamma}(\lambda)\leq\frac{1}{2(\inf
V^{\prime\prime})}\|\gamma\|^{2}_{L^{2}(\partial\Box_{L})},$

and thus

(3.5) $\|u^{\prime}_{a}\|_{L^{2}(\Box_{L})}^{2}\leq\frac{2}{\inf
V^{\prime\prime}}\mathcal{B}_{0}(\lambda)\leq\frac{1}{(\inf
V^{\prime\prime})^{2}}\|\gamma\|^{2}_{L^{2}(\partial\Box_{L})}.$

Therefore, we have

$\mathcal{E}^{\prime}_{a}(u^{\prime}_{a})\geq-\left\langle
u^{\prime}_{a},a\right\rangle_{\Box_{L}}\geq-C\|a\|_{L^{2}(\Box_{L})}\|\gamma\|_{L^{2}(\partial\Box_{L})},$

and the claim (3.3) is established. ∎

##  4\. Marginals and the Simplified Energy

The goal of this section is to understand the energy
$\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$ defined in (2.14). To do this, we will
need to understand the marginal law $\mu^{b}_{x_{0}}$ of $\eta(x_{0})$ under
$\mathrm{Q}^{b}$ for $x_{0}\in\Box_{L}$. Specifically, we will prove an
approximation

(4.1)
$\mu^{b}_{x_{0}}(d\eta(x_{0}))\approx\nu^{b(x_{0})}(d\eta(x_{0})):=\frac{1}{\mathrm{J}(b(x_{0}))}\exp(b(x_{0})\eta(x_{0})-V(\eta(x_{0})))d\eta(x_{0})$

where $\mathrm{J}(b(x_{0}))$ is a normalizing constant and $x_{0}$ is not too
close to $\partial\Box_{L}$. This approximation would be exact if the boundary
term were not present in the energy $\mathcal{H}^{b}$ defining
$\mathrm{Q}^{b}$.

We will also show that the “off-diagonal” derivatives
$\nabla_{\eta(y)}v(x,\eta)$ for $x\neq y$ within the smoothing term of
$\mathcal{F}_{a}(v;\mathrm{Q}^{b})$ are not consequential for the minimization
problem. After these off-diagonal terms are eliminated and
$\mu^{b}_{x_{0}}\approx\nu^{b(x_{0})}$ is applied, the energy
$\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$ just becomes a weighted sum of simple
energies like $\mathcal{G}_{1}(\cdot;\nu^{b(x)})$ which can be minimized
independently.

Our main tool for both of these results is a special profile
$e_{x_{0}}\in\mathbb{R}^{\Box_{L}}$, which we now describe.

###  4.1. The Special Profile

To motivate the construction of the special profile, consider the following
computation of the derivative of the marginal of $\eta(x_{0})$. We have

$\mu^{b}_{x_{0}}(\eta(x_{0}))=\frac{1}{\mathsf{K}(\Box_{L},b)}\int_{\mathbb{R}^{\Box_{L}\setminus\\{x_{0}\\}}}e^{-\mathcal{H}^{b}(\eta)}\prod_{x\neq
x_{0}}d\eta(x).$

where we denote by $\mu^{b}_{x_{0}}(\eta(x_{0}))$ the Lebesgue density of the
marginal, by abuse of notation. Then for every $\varepsilon\in\mathbb{R}$, we
have

$\mu^{b}_{x_{0}}(\eta(x_{0})+\varepsilon)-\mu^{b}_{x_{0}}(\eta(x_{0}))=\mathbb{E}_{\mathrm{Q}^{b}_{c}}\left[e^{\mathcal{H}^{b}(\eta)-\mathcal{H}^{b}(\eta+\varepsilon\mathbf{1}_{x_{0}})}-1\right]$

where $\mathrm{Q}^{b}_{c}$ means $\mathrm{Q}^{b}$ conditioned on
$\eta(x_{0})$. And so

$\displaystyle\nabla_{\eta(x_{0})}\mu^{b}_{x_{0}}(\eta(x_{0}))$
$\displaystyle=-\mathbb{E}_{\mathrm{Q}^{b}_{c}}\left[\nabla_{\eta(x_{0})}\mathcal{H}^{b}(\eta)\right]$
$\displaystyle=-V^{\prime}(\eta(x_{0}))+b(x_{0})+\mathbb{E}_{\mathrm{Q}^{b}_{c}}\left[\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))P^{z}(x_{0})\right].$

The first two terms on the RHS are consistent with our approximation
$\mu^{b}_{x_{0}}(\eta(x_{0}))\approx\nu^{b(x_{0})}$, but the last boundary
term needs to be small. We do not know a way to bound this quantity directly
to satisfactory accuracy, particularly in $d=2,3$ where the summability
properties of $P^{z}$ are less desirable.

If instead, we apply a change of variables $\eta\mapsto\eta-\varepsilon
e_{x_{0}}$ to the integral defining
$\mu^{b}_{x_{0}}(\eta(x_{0})+\varepsilon)$, we can “move” the derivative off
of the boundary term. We get

$\mu^{b}_{x_{0}}(\eta(x_{0})+\varepsilon)-\mu^{b}_{x_{0}}(\eta(x_{0}))=\mathbb{E}_{\mathrm{Q}^{b}_{c}}\left[e^{\mathcal{H}^{b}(\eta)-\mathcal{H}^{b}(\eta+\varepsilon\mathbf{1}_{x_{0}}+\varepsilon
e_{x_{0}})}-1\right],$

and so

$\displaystyle\nabla_{\eta(x_{0})}\mu^{b}_{x_{0}}(\eta(x_{0}))$
$\displaystyle=-V^{\prime}(\eta(x_{0}))+b(x_{0})+\mathbb{E}_{\mathrm{Q}^{b}_{c}}\left[\sum_{x\neq
x_{0}}(-V^{\prime}(\eta(x))+b(x))e_{x_{0}}(x)\right]$
$\displaystyle\quad+\mathbb{E}_{\mathrm{Q}^{b}_{c}}\left[\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))\left\langle
P^{z},\mathbf{1}_{x_{0}}+e_{x_{0}}\right\rangle_{\Box_{L}}\right].$

If we construct $e_{x_{0}}$ such that
$\left\langle\mathbf{1}_{x_{0}}+e_{x_{0}},P^{z}\right\rangle=0$, the boundary
term now vanishes. The extra term will be much easier to bound, particularly
since $e_{x_{0}}$ will be small. The above computation shows that an a priori
strong interaction between $\eta(x_{0})$ and the $O(L^{d-1})$ boundary terms
can be mitigated through a very small variation by $O(L^{d})$ other spins
$\eta(x)$, $x\neq x_{0}$.

######  Proposition 4.1.

Recall that $\rho_{x_{0}}$ denotes the distance between $x_{0}$ and
$\partial\Box_{L}$. For each ${x_{0}\in\Box_{L}}$, there exists
$e_{x_{0}}\in\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$ such that
$\|e_{x_{0}}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}\leq C\rho_{x_{0}}^{-d/2}$ and

$\left\langle P^{z}_{L},e_{x_{0}}\right\rangle_{\Box_{L}}=\left\langle
P^{z}_{L},e_{x_{0}}\right\rangle_{\mathrm{cl}_{1}\Box_{L}}=-P^{z}_{L}(x_{0})$

for all $z\in\partial\Box_{L}$.

###### Proof.

For $x\in\mathrm{cl}_{1}\Box_{L}$, let $K_{x}$ be an integer such that
$x\in\partial\Box_{K_{x}}$, if it exists. Let $P^{x}_{K_{x}}$ then denote the
Poisson kernel of $\Box_{K_{x}}$ at boundary point $x\in\partial\Box_{K_{x}}$.
Define $e=e_{x_{0}}\in\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$ by

$e(x)=-\varepsilon(K_{x})P^{x}_{K_{x}}(x_{0})$

for some choice of values $\varepsilon(k)$ such that
$\sum_{k=0}^{L}\varepsilon(k)=1$ and $\varepsilon(k)=0$ if $k\leq K_{x_{0}}$.
We will also choose $\varepsilon(L)=0$, and WLOG assume $x_{0}$ is not
adjacent to $\partial\Box_{L}$ (in which case, we choose
$e_{x_{0}}=-\mathbf{1}_{x_{0}}$). If $x$ is such that no $K_{x}$ exists,
define $e(x)=0$.

We claim that $\left\langle P^{z}_{L},e\right\rangle_{\Box_{L}}=\left\langle
P^{z}_{L},e\right\rangle_{\mathrm{cl}_{1}\Box_{L}}=-P^{z}_{L}(x_{0})$. To see
this, note that

$\left\langle
e,P^{z}_{L}\right\rangle_{\mathrm{cl}_{1}\Box_{L}}=-\sum_{k=K_{x_{0}}+1}^{L}\varepsilon(k)\sum_{x\in\mathrm{cl}_{1}\Box_{L}:K_{x}=k}P^{x}_{k}(x_{0})P^{z}_{L}(x)=-\sum_{k=0}^{L}\varepsilon(k)P^{z}_{L}(x_{0})=-P^{z}_{L}(x_{0}),$

where we have used the interpretation of the Poisson kernel as the exit
location distribution of a simple random walk and the strong Markov property.

We now choose $\varepsilon$ to try to nearly minimize $\|e\|_{L^{2}}$. Using
basic bounds on the Poisson kernel from 7.1, we have

(4.2) $\displaystyle\|e\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}$
$\displaystyle=\sum_{k=K_{x_{0}}+1}^{L}|\varepsilon(k)|^{2}\sum_{x\in\mathrm{cl}_{1}\Box_{L}:K_{x}=K}|P^{x}_{k}(x_{0})|^{2}$
$\displaystyle\leq
C\sum_{k=K_{x_{0}}+1}^{L}|\varepsilon(k)|^{2}\mathrm{dist}(x_{0},\partial\Box_{k})^{-d+1}\leq
C\sum_{k=1}^{\rho_{x_{0}}}|\varepsilon(k+K_{x_{0}})|^{2}k^{-d+1}.$

Note that

$\lambda:=\sum_{k=1}^{\rho_{x_{0}}}k^{d/2-1/2}\geq
C^{-1}\rho_{x_{0}}^{d/2+1/2},$

and so if we choose
$\varepsilon(k)\sim\lambda^{-1}\mathrm{dist}(x_{0},\partial\Box_{k})^{d/2-1/2}$
for $K_{x_{0}}+1\leq k<L$, we can ensure $\sum\varepsilon(k)=1$ and

$\|e\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}\leq
C\lambda^{-2}\sum_{k=1}^{\rho_{x_{0}}}1\leq C\rho_{x_{0}}^{-d}.$

This finishes the proof. ∎

######  Remark 4.2.

In the case that the domains $\Box_{L}$ are replaced by smooth domains
$\Omega_{L}$, we can carry out a similar construction and achieve the same
estimates. Instead of defining $e_{x_{0}}$ with reference to discrete
hypercubes $\Box_{k}$ centered at $0$, we can center the hypercubes at
$x_{0}$.

###  4.2. Single-spin Law

The goal of this subsection is to prove the approximation
$\mu^{b}_{x_{0}}\approx\nu^{b(x_{0})}$ for $x_{0}$ not close to
$\partial\Box_{L}$. We do this through a change of variables and entropy
argument.

Recall the definition of $\mathrm{J}(b(x_{0}))$ from (4.1). For
$f:\mathbb{R}\to\mathbb{R}$ measurable and bounded, we have

$\mathbb{E}_{\mu^{b}_{x_{0}}}f(\eta(x_{0}))=\frac{\mathrm{J}(b(x_{0}))}{\mathsf{K}(b)}\mathbb{E}_{\nu^{b(x_{0})}}\left[\mathsf{K}(\eta(x_{0}),x_{0};b)f(\eta(x_{0}))\right]$

where $\mathsf{K}(b)$ is the partition function of $\mathrm{Q}^{b}$ and

$\displaystyle\mathsf{K}(\xi,x_{0};b)$
$\displaystyle=\int_{\mathbb{R}^{\Box_{L}}}\exp\left(-\sum_{x\neq
x_{0}}\left(V(\eta(x))-b(x)\eta(x)\right)-\sum_{z\in\partial\Box_{L}}V(\alpha^{z}(\eta))\right)d\delta_{\xi}(\eta(x_{0}))\prod_{x\neq
x_{0}}d\eta(x).$

The measure $\delta_{\xi}(\eta(x_{0}))$ is the Dirac delta function
$\eta(x_{0})=\xi$.

The main task is to prove that $\mathsf{K}(\xi,x_{0};b)$ depends only weakly
on $\xi$. By the change of variables
$\eta\mapsto\eta-\varepsilon(\mathbf{1}_{x_{0}}+e_{x_{0}})$,

$\displaystyle\mathsf{K}(\xi+\varepsilon,x_{0};b)$
$\displaystyle=\int_{\mathbb{R}^{\Box_{L}}}e^{-\sum_{x\neq
x_{0}}\left(V((\eta+\varepsilon e_{x_{0}})(x))-b(x)(\eta+\varepsilon
e_{x_{0}})(x)\right)-\sum_{z\in\partial\Box_{L}}V(\alpha^{z}(\eta))}d\delta_{\xi}(\eta(x_{0}))\prod_{x\neq
x_{0}}d\eta(x).$

Thus

(4.3)
$\frac{1}{\mathsf{K}(\xi,x_{0};b)}\frac{d}{d\xi}\mathsf{K}(\xi,x_{0};b)=-\mathbb{E}_{\mathrm{Q}^{b}}\left[\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)-b(x)e_{x_{0}}(x)\ \bigg{|}\
\eta(x_{0})=\xi\right].$

We use this computation in the next theorem, which shows that, far from the
boundary, the one-spin marginal law does not feel strongly the effect of the
boundary interaction.

######  Theorem 4.1.

The distance between $\mu^{b}_{x_{0}}$ and $\nu^{b(x_{0})}$ in total variation
norm is bounded by $C(1+\|b\|_{L^{2}})\rho_{x_{0}}^{-d/2}$. In particular, for
any bounded and measurable $f$, we have

(4.4) $|\mathbb{E}_{\mu^{b}_{x_{0}}}f-\mathbb{E}_{\nu^{b(x_{0})}}f|\leq
C\|f\|_{L^{\infty}}(1+\|b\|_{L^{2}(\Box_{L})})\rho_{x_{0}}^{-d/2}.$

###### Proof.

Define

$g(\xi):=\frac{\mu^{b}_{x_{0}}(d\xi)}{\nu^{b(x_{0})}(d\xi)}=\frac{\mathrm{J}(b(x_{0}))\mathsf{K}(\xi,x_{0};b)}{\mathsf{K}(b)}.$

Our strategy is to bound the relative entropy
$H(\mu^{b}_{x_{0}}|\nu^{b(x_{0})})=\mathbb{E}_{\nu^{b(x_{0})}}g(\xi)\log
g(\xi)$. Pinsker’s inequality will then allow us to bound the total variation
distance.

We apply the log-Sobolev inequality, as stated in 2.1, to see

$H(\mu^{b}_{x_{0}}|\nu^{b(x_{0})}):=\mathbb{E}_{\nu^{b(x_{0})}}g(\xi)\log
g(\xi)\leq
C\mathbb{E}_{\nu^{b(x_{0})}}\left(\frac{\nabla_{\xi}g(\xi)}{\sqrt{g(\xi)}}\right)^{2}.$

Let
$\mathcal{I}(\xi)=(\mathsf{K}(\xi,x_{0};b))^{-1}\frac{d}{d\xi}K(\xi,x_{0};b)$.
Then the RHS above can be written as

(4.5)
$\displaystyle\mathbb{E}_{\nu^{b(x_{0})}}\left(\frac{\nabla_{\xi}g(\xi)}{\sqrt{g(\xi)}}\right)^{2}$
$\displaystyle=\mathbb{E}_{\mu^{b}_{x_{0}}}\left(\frac{\nabla_{\xi}g(\xi)}{{g(\xi)}}\right)^{2}=\mathbb{E}_{\mu^{b}_{x_{0}}}|\mathcal{I}(\xi)|^{2}$
$\displaystyle\leq\mathbb{E}_{\mathrm{Q}^{b}}\bigg{|}\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)-b(x)e_{x_{0}}(x)\bigg{|}^{2}.$

In the last inequality above, we used the expression (4.3) and Jensen’s
inequality. Continuing, we split the RHS above and apply Cauchy-Schwarz to get

(4.6) $H(\mu^{b}_{x_{0}}|\nu^{b(x_{0})})\leq
2\mathbb{E}_{\mathrm{Q}^{b}}\bigg{|}\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\bigg{|}^{2}+2\|b\|_{L^{2}}^{2}\|e_{x_{0}}\|_{L^{2}}^{2}.$

To bound the first term on the RHS, we first bound the mean

$m(b)=\mathbb{E}_{\mathrm{Q}^{b}}\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x).$

For any $r\in(0,1)$, we compute

(4.7) $\displaystyle\frac{d}{dr}m(rb)$
$\displaystyle=\mathrm{Cov}_{\mathrm{Q}^{rb}}\left(\sum_{x\in\Box_{L}}b(x)\eta(x),\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\right)$
$\displaystyle\leq\left(\mathrm{Var}_{\mathrm{Q}^{rb}}\left(\sum_{x\in\Box_{L}}b(x)\eta(x)\right)\right)^{1/2}\left(\mathrm{Var}_{\mathrm{Q}^{rb}}\left(\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\right)\right)^{1/2}.$

Now the Poincaré inequality implies
$\mathrm{Var}_{\mathrm{Q}^{rb}}\left(\sum_{x\in\Box_{L}}b(x)\eta(x)\right)\leq
C\|b\|_{L^{2}}^{2}$ and

(4.8) $\mathrm{Var}\left(\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\right)\leq C\sum_{x\neq
x_{0}}\mathbb{E}_{\mathrm{Q}^{rb}}|V^{\prime\prime}(\eta(x))e_{x_{0}}(x)|^{2}\leq
C\|e_{x_{0}}\|_{L^{2}}^{2}.$

Together with $m(0)=0$ by symmetry of $\mathrm{Q}^{0}$ and $V$, we conclude
$m(b)\leq C\|b\|_{L^{2}}\|e_{x_{0}}\|_{L^{2}}$. The estimate (4.8) shows

(4.9) $\displaystyle\mathbb{E}_{\mathrm{Q}^{b}}\bigg{|}\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\bigg{|}^{2}$
$\displaystyle=\mathrm{Var}\left(\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\right)+|m(b)|^{2}$ $\displaystyle\leq
C(1+\|b\|_{L^{2}}^{2})\|e_{x_{0}}\|_{L^{2}}^{2}.$

We now apply this estimate to (4.6) and use $\|e_{x_{0}}\|_{L^{2}}^{2}\leq
C\rho_{x_{0}}^{-d}$ to conclude

$H(\mu^{b}_{x_{0}}|\nu^{b(x_{0})})\leq
C(1+\|b\|_{L^{2}}^{2})\rho_{x_{0}}^{-d},$

and the total variation distance between $\mu^{b}_{x_{0}}$ and
$\nu^{b(x_{0})}$ is less than a constant times the square root of the relative
entropy by Pinsker’s inequality. ∎

###  4.3. Minimization of $\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$

Recall that $\overline{u}_{a}\in H^{1}(\mathrm{Q}^{b};\mathbb{R}^{\Box_{L}})$
is the minimizer of the functional $\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$
defined in (2.14). By linearity of the map $a\mapsto\overline{u}_{a}$, it will
be sufficient to understand $\overline{u}_{\mathbf{1}_{x}}$. Moreover, we have
$\overline{u}_{\mathbf{1}_{x}}(y,\eta)=0$ almost surely if $y\neq x$ by 2.5,
so we will define $\overline{u}_{x}\in H^{1}(\mathrm{Q}^{b};\mathbb{R})$ by
$\overline{u}_{x}(\eta)=\overline{u}_{\mathbf{1}_{x}}(x,\eta)$. By an abuse of
notation, we will say $\overline{u}_{x}$ is the minimizer of
$\mathcal{F}_{\mathbf{1}_{x}}(\cdot;\mathrm{Q}^{b})$. Let $U_{x}:=U^{b(x)}\in
H^{1}(\nu^{b(x)};\mathbb{R})$ be the minimizer of
$\mathcal{G}_{1}(\cdot;\nu^{b(x)})$, which was defined in (2.20). We note that
$\overline{u}_{x}$ and $U_{x}$ depend on $b$, but we will mostly omit the
dependence. However, all constants $C$ are independent of $b$.

The goal of this subsection is to prove $\overline{u}_{x}\approx U_{x}$ in a
certain sense, when $x$ is not close to $\partial\Box_{L}$. Recalling 2.5, we
see that $\overline{u}_{x}$ solves

$-\Delta_{\eta}\overline{u}_{x}+\sum_{y\in\Box_{L}}\left(V^{\prime}(\eta(y))-\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))P^{z}(y)-b(y)\right)\nabla_{\eta(y)}\overline{u}_{x}+V^{\prime\prime}(\eta(x))\overline{u}_{x}=1.$

The equation for $U_{x}$ (as a function of $\eta(x)$) is

$-\Delta_{\eta(x)}U_{x}+\left(V^{\prime}(\eta(x))-b(x)\right)\nabla_{\eta(x)}U_{x}+V^{\prime\prime}(\eta(x))U_{x}=1.$

We define $w_{x}(\eta):=\overline{u}_{x}(\eta)-U_{x}(\eta(x))$ and write the
equation that $w_{x}$ solves:

$-\mathcal{L}_{\mathrm{Q}^{b}}w_{x}+V^{\prime\prime}(\eta(x))w_{x}=-\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))P^{z}(x)\nabla_{\eta(x)}U_{x}.$

Integrating against $w_{x}\mathrm{Q}^{b}$ shows

(4.10)
$2\mathcal{F}_{0}(w_{x};\mathrm{Q}^{b})=-\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x}\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))P^{z}(x)\nabla_{\eta(x)}U_{x}\right].$

Our goal is to show that the RHS is smaller than
$\mathcal{F}_{0}(w_{x};\mathrm{Q}^{b})$ plus some small constant. We cannot
use Hölder’s inequality directly on (4.10), because in $d=2,3$ the summability
properties of $P^{z}(x)$ are too weak (we can however get a strong enough
result in $d\geq 4$). Instead, we must use the special profile
$e_{x}\in\mathbb{R}^{\Box_{L}}$ constructed in 4.1.

######  Lemma 4.3.

We have

(4.11) $\mathcal{F}_{0}(\overline{u}_{x}-U_{x};\mathrm{Q}^{b})\leq
C(1+\|b||_{L^{2}}^{2})\rho_{x_{0}}^{-d}$

where $U_{x}$ is a considered as a function of $\eta(x)$. Consequently

(4.12)
$|\mathbb{E}_{\mathrm{Q}^{b}}\overline{u}_{x_{0}}-\mathbb{E}_{\nu^{b(x_{0})}}U_{x_{0}}|\leq
C(1+\|b||_{L^{2}})\rho_{x_{0}}^{-d/2}.$

###### Proof.

We will write $\mathrm{Q}^{b}(\eta)$ for the density of $\mathrm{Q}^{b}$ at
$\eta$. Note that

(4.13)
$\left(\sum_{z\in\partial\Box_{L}}V^{\prime}(\alpha^{z}(\eta))P^{z}(x_{0})\right)\mathrm{Q}^{b}(\eta)=-\nabla_{\eta(x_{0})}\mathrm{Q}^{b}(\eta)-V^{\prime}(\eta(x_{0}))\mathrm{Q}^{b}(\eta)+b(x_{0})\mathrm{Q}^{b}(\eta).$

We break (4.10) into pieces

(4.14)
$\displaystyle\mathcal{I}_{1}=\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x_{0}}\nabla_{\eta(x_{0})}U_{x_{0}}\frac{\nabla_{\eta(x_{0})}\mathrm{Q}^{b}(\eta)}{\mathrm{Q}^{b}(\eta)}\right]$
$\displaystyle\mathcal{I}_{2}=\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x_{0}}\nabla_{\eta(x_{0})}U_{x_{0}}\left(V^{\prime}(\eta(x_{0}))-b(x_{0})\right)\right]$

so that
$\mathcal{F}_{0}(w_{x_{0}};\mathrm{Q}^{b})=\mathcal{I}_{1}+\mathcal{I}_{2}$.
After performing a change of variables $\eta^{\prime}=\eta+\varepsilon
e_{x_{0}}$, we have

(4.15)
$\displaystyle\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x_{0}}\nabla_{\eta(x_{0})}U_{x_{0}}\frac{\mathrm{Q}^{b}(\eta+\varepsilon\mathbf{1}_{x_{0}})}{\mathrm{Q}^{b}(\eta)}\right]$
$\displaystyle=\int_{\mathbb{R}^{\Box_{L}}}w_{x_{0}}(\eta-\varepsilon
e_{x_{0}})\nabla_{\eta(x_{0})}U_{x_{0}}(\eta(x_{0}))Q^{b}(\eta+\varepsilon\mathbf{1}_{x_{0}}-\varepsilon
e_{x_{0}})\prod d\eta(x)$

for any $\varepsilon\in\mathbb{R}$, and forming a difference quotient as
$\varepsilon\to 0$ shows that

(4.16) $\displaystyle\mathcal{I}_{1}$
$\displaystyle=-\mathbb{E}_{\mathrm{Q}^{b}}\left[\left(\sum_{x\neq
x_{0}}e_{x_{0}}(x)\nabla_{\eta(x)}w_{x_{0}}\right)\nabla_{\eta(x_{0})}U_{x_{0}}\right]$
$\displaystyle\quad+\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x_{0}}\nabla_{\eta(x_{0})}U_{x_{0}}\left(\sum_{x\neq
x_{0}}(V^{\prime}(\eta(x))-b(x))e_{x_{0}}(x)\right)\right]$
$\displaystyle\quad-\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x_{0}}\nabla_{\eta(x_{0})}U_{x_{0}}(V^{\prime}(\eta(x_{0}))-b(x_{0}))\right].$

The last line of (4.16) cancels with $\mathcal{I}_{2}$, so we just need to
bound the first two lines. On the first term use the $L^{\infty}$ bound in 2.6
on $\nabla_{\eta(x_{0})}U_{x_{0}}$ to see

(4.17) $\displaystyle\left|\mathbb{E}_{\mathrm{Q}^{b}}\left[\left(\sum_{x\neq
x_{0}}e_{x_{0}}(x)\nabla_{\eta(x)}w_{x_{0}}\right)\nabla_{\eta(x_{0})}U_{x_{0}}\right]\right|$
$\displaystyle\leq C\sum_{x\neq
x_{0}}e_{x_{0}}(x)\mathbb{E}_{Q^{b}}{|\nabla_{\eta(x)}w_{x_{0}}|}$
$\displaystyle\leq\frac{1}{8}\sum_{x\neq
x_{0}}\mathbb{E}_{\mathrm{Q}^{b}}|\nabla_{\eta(x)}w_{x_{0}}|^{2}+2C\|e_{x_{0}}\|_{L^{2}}^{2}$

Similarly on the second term we have

(4.18)
$\left|\mathbb{E}_{\mathrm{Q}^{b}}\left[w_{x_{0}}\nabla_{\eta(x_{0})}U_{x_{0}}\left(\sum_{x\neq
x_{0}}(V^{\prime}(\eta(x))-b(x))e_{x_{0}}(x)\right)\right]\right|\\\
\leq\frac{1}{8\inf
V^{\prime\prime}}\mathbb{E}_{\mathrm{Q}^{b}}w_{x_{0}}^{2}+C\mathbb{E}_{\mathrm{Q}^{b}}\left(\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\right)^{2}.$

Putting it all together into (4.10), we get

$2\mathcal{F}_{0}(w_{x_{0}};\mathrm{Q}^{b})\leq\frac{1}{2}\mathcal{F}_{0}(w_{x_{0}};\mathrm{Q}^{b})+C\|e_{x_{0}}\|_{L^{2}}^{2}+C\mathbb{E}_{\mathrm{Q}^{b}}\left(\sum_{x\neq
x_{0}}V^{\prime}(\eta(x))e_{x_{0}}(x)\right)^{2}.$

The last term on the RHS was bounded by
$C(1+\|b\|_{L^{2}}^{2})\|e_{x_{0}}\|_{L^{2}}^{2}$ in (4.9), and so (4.11) is
proved. For (4.12), note that

$|\mathbb{E}_{\mathrm{Q}^{b}}\overline{u}_{x_{0}}-\mathbb{E}_{\mu^{b}_{x_{0}}}U_{x_{0}}|=|\mathbb{E}_{\mathrm{Q}^{b}}w_{x_{0}}|\leq
C\left(\mathcal{F}_{0}(w_{x_{0}};\mathrm{Q}^{b})\right)^{1/2},$

and, by (4.4) and boundedness of $U_{x_{0}}$ we have

$|\mathbb{E}_{\mu^{b}_{x_{0}}}U_{x_{0}}-\mathbb{E}_{\nu^{b(x_{0})}}U_{x_{0}}|\leq
C(1+\|b\|_{L^{2}})\rho_{x_{0}}^{-d/2}.$

The triangle inequality finishes the proof. ∎

Since the map $a\mapsto\overline{u}_{a}$, where $\overline{u}_{a}$ minimizes
$\mathcal{F}_{a}(\cdot;\mathrm{Q}^{b})$, is linear, the lemma can be used to
show $\overline{u}_{a}(x,\eta)$ and $U_{x}(\eta)a(x)$ are approximately equal
so long as $a$ does not have its mass concentrate near $\partial\Box_{L}$.

To conclude the section, we consider the dependence of $U_{x_{0}}$ on $b$.

######  Lemma 4.4.

Let $U^{\beta}$ denote the minimizer of $\mathcal{G}_{1}(\cdot;\nu^{\beta})$,
and let $U^{0}$ denote the minimizer of $\mathcal{G}_{1}(\cdot;\nu^{0})$. We
have

(4.19) $|\mathbb{E}_{\nu^{\beta}}[U^{\beta}]-\mathbb{E}_{\nu^{0}}[U^{0}]|\leq
C\beta$

for some constant $C$.

###### Proof.

First, the difference $w:=U^{\beta}-U^{0}$ satisfies the equation

$-\Delta w+V^{\prime}(\xi)\nabla w+V^{\prime\prime}(\xi)w=\beta\nabla
U^{\beta}$

over $\mathbb{R}$. Integrating against $w\nu^{0}$, and using the bound on
$\nabla U^{\beta}$ from 2.6, we obtain

$\mathcal{G}_{0}(w;\nu^{0})\leq C|\beta|\mathbb{E}_{\nu^{0}}(|w|)\leq
C|\beta|\sqrt{\mathcal{G}_{0}(w;\nu^{0})}\leq C\beta^{2},$

and so

$\left|\mathbb{E}_{\nu^{0}}U^{\beta}-\mathbb{E}_{\nu^{0}}U^{0}\right|^{2}\leq\mathbb{E}_{\nu^{0}}|U^{\beta}-U^{0}|^{2}\leq
C\mathcal{G}_{0}(w;\nu^{0})\leq C\beta^{2}.$

By interpolation and the Poincaré inequality, we have

$|\mathbb{E}_{\nu^{\beta}}U^{\beta}-\mathbb{E}_{\nu^{0}}U^{\beta}|\leq\int_{0}^{1}\left|\mathrm{Cov}_{\nu^{r\beta}}\left[\beta\xi,U^{\beta}(\xi)\right]\right|dr\leq|\beta|\left(\mathrm{Var}_{\nu^{r\beta}}U^{\beta}\right)^{1/2}\leq
C|\beta|.$

We conclude from the triangle inequality. ∎

##  5\. Minimization of the Helffer-Sjöstrand Energy

In this section, we will take general $a\in\mathbb{R}^{\Box_{L}}$, but we have
particular interest in the case where $a$ is almost orthogonal to the family
$\\{P^{z}\\}_{z\in\partial\Box_{L}}$ in the $L^{2}(\Box_{L})$ inner product.
We consider $(\eta(x))_{x\in\Box_{L}}$ distributed according to the measure
$\mathrm{Q}^{b}$. For any $x\in\Box_{L}$, we let $\mu^{b}_{x}$ be the marginal
distribution of $\eta(x)$ under $\mathrm{Q}^{b}$, and let $\nu^{b(x)}$ be the
probability measure on $\mathbb{R}$ proportional to
$\exp(-V(\xi)+b(x)\xi)d\xi$.

We seek a precise understanding of $\mathcal{E}_{a}(\cdot;\mathrm{Q}^{b})$ and
its minimizer. In the case of the Gaussian membrane model and $\left\langle
a,P^{z}\right\rangle_{\Box_{L}}=0$, the minimizer is simply
$u_{a}(x,\eta)=ca(x)$ for some constant $c$. Critically, the fact that $a$ is
“boundary orthogonal”, i.e. orthogonal to the $\\{P^{z}\\}$, means that the
boundary terms within $\mathcal{E}_{a}$ are irrelevant for the Gaussian
energy.

In the non-Gaussian case, even if $\left\langle
a,P^{z}\right\rangle_{\Box_{L}}=0$, the boundary terms are still active due to
the random background measure $\mathrm{Q}^{b}$ and random coefficients
involving $V^{\prime\prime}$ within $\mathcal{E}_{a}$. We pursue the idea,
mentioned at the end of Section 1, that there should be a homogenization/law
of large numbers effect due to the random environment $\eta$ in the boundary
terms within $\mathcal{E}_{a}$. It will allow us to show the minimizer of
$\mathcal{E}_{a}$ is close to that of $\mathcal{F}_{a}$, which was understood
in Section 4.

The question of dependence of $u_{a}=\mathrm{HS}^{b}a$ on $b$ must also be
addressed. We find that if $|a|^{2}|b|$ is small in $L^{1}(\Box_{L})$, the
dependence is also small. This condition will hold for the limits involving
the rescaled field $\overline{\phi}$, but not for the infinite volume limit in
$d\geq 5$. For the infinite volume limit 1.1, we will consider $a$ and $b$
with mass concentrated on a single point in 5.4.

Fix $\varepsilon\in(0,1)$ and $\ell\in\mathbb{Z}$ such
$L^{1-\varepsilon}\leq\ell\leq 2L^{1-\varepsilon}$, and let
$\Lambda=\Box_{L}\setminus\Box_{L-\ell}$ be a boundary layer of thickness
$\ell$. Define $w\in H^{1}(Q^{b};\mathbb{R}^{\Box_{L}})$ by

(5.1) $w(x,\eta)=\begin{cases}U_{x}(\eta(x))a(x),&\quad x\in\Box_{L-\ell},\\\
(\mathbb{E}_{\nu_{0}}U^{0})a(x),&\quad x\in\Lambda.\end{cases}$

Here $U_{x}=U^{b(x)}$ is the minimizer of
$\mathcal{G}_{1}(\cdot;\mathrm{Q}^{b(x)})$. We claim that $w$ is an
approximate minimizer of $\mathcal{E}_{a}$, and therefore a good approximation
to the true minimizer $u_{a}$ by coercivity of $\mathcal{E}_{a}$. We address
the boundary energy of $w$ first.

######  Lemma 5.1.

With $w$ as in (5.1), there is a constant $C>0$ such that

(5.2)
$\displaystyle\mathbb{E}_{\mathrm{Q}^{b}}\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\left\langle
P^{z},w\right\rangle_{\Box_{L}}|^{2}$ $\displaystyle\leq
C(1+\|b\|_{L^{2}(\Box_{L})}^{2})\|a\|_{L^{2}(\Box_{L})}^{2}L^{\varepsilon(3d-2)}L^{-(d-1)}+C\cdot
B_{L}(a).$

where

(5.3) $B_{L}(a):=\sum_{z\in\partial\Box_{L}}|\left\langle
P^{z},a\right\rangle_{\Box_{L}}|^{2}.$

is the “boundary energy” of $a$.

###### Proof.

For $z\in\partial\Box_{L}$, we decompose
$\mathbb{E}_{\mathrm{Q}^{b}}\left\langle P^{z},w\right\rangle_{\Box_{L}}$ into
two pieces:

$\mathbb{E}_{\mathrm{Q}^{b}}\left\langle
P^{z},w\right\rangle_{\Box_{L}}=\mathbb{E}_{\nu^{0}}U^{0}\left\langle
P^{z},a\right\rangle_{\Box_{L}}+\sum_{x\in\Box_{L-\ell}}P^{z}(x)(\mathbb{E}_{\mu^{b}_{x}}U_{x}-\mathbb{E}_{\nu^{0}}U^{0})a(x).$

Applying (4.4) for $x\in\Box_{L-\ell}$, we see

$|\mathbb{E}_{\mu^{b}_{x}}U_{x}-\mathbb{E}_{\nu^{b(x)}}U_{x}|\leq
C(1+\|b\|_{L^{2}(\Box_{L})})\ell^{-d/2}$

and from (4.19) we have

$|\mathbb{E}_{\nu^{b}_{x}}U_{x}-\mathbb{E}_{\nu^{0}}U^{0}|\leq C|b(x)|.$

Thus, bounding $|P^{z}(x)|\leq C\ell^{-(d-1)}$ and using Cauchy-Schwarz, we
have

(5.4) $\displaystyle|\mathbb{E}_{\mathrm{Q}^{b}}\left\langle
P^{z},w\right\rangle_{\Box_{L}}-\mathbb{E}_{\nu^{0}}U^{0}\left\langle
P^{z},a\right\rangle_{\Box_{L}}|$ $\displaystyle\leq
C\sum_{x\in\Box_{L-\ell}}\ell^{-d+1}(|b(x)|+(1+\|b\|_{L^{2}})\ell^{-d/2})|a(x)|$
$\displaystyle\leq
C\ell^{-(d-1)}\|b\|_{L^{2}(\Box_{L})}\|a\|_{L^{2}(\Box_{L})}+C(1+\|b\|_{L^{2}(\Box_{L})})\ell^{-\frac{3}{2}d+1}\|a\|_{L^{1}(\Box_{L})}$
$\displaystyle\leq
C\ell^{-(d-1)}\|b\|_{L^{2}(\Box_{L})}\|a\|_{L^{2}(\Box_{L})}+C(1+\|b\|_{L^{2}(\Box_{L})})\ell^{-(d-1)}L^{\varepsilon
d/2}\|a\|_{L^{2}(\Box_{L})}$ $\displaystyle\leq
CL^{\varepsilon(3d/2-1)}L^{-(d-1)}(1+\|b\|_{L^{2}(\Box_{L})})\|a\|_{L^{2}(\Box_{L})}:=\mathrm{Error}_{1}.$

We conclude $|\mathbb{E}_{\mathrm{Q}^{b}}\left\langle
P^{z},w\right\rangle|^{2}\leq 2(\mathrm{Error}_{1})^{2}+C|\left\langle
P^{z},a\right\rangle_{\Box_{L}}|^{2}$.

Next, we look at the variances of $\left\langle
P^{z},w\right\rangle_{\Box_{L}}$ under $\mathrm{Q}^{b}$. Applying the Poincaré
inequality for $\mathrm{Q}^{b}$ gives

(5.5)
$\displaystyle\sum_{z\in\partial\Box_{L}}\mathrm{Var}_{\mathrm{Q}^{b}}\left\langle
P^{z},w\right\rangle_{\Box_{L}}$ $\displaystyle\leq
C\mathbb{E}_{\mathrm{Q}^{b}}\sum_{x\in\Box_{L-\ell}}\left(\sum_{z}|P^{z}(x)|^{2}\right)|a(x)|^{2}|\nabla_{\eta(x)}U_{x}(\eta(x))|^{2}$
$\displaystyle\leq
C\ell^{-d+1}\|a\|_{L^{2}(\Box_{L})}^{2}\sup_{x\in\Box_{L-\ell}}\mathbb{E}_{\mathrm{Q}^{b}}|\nabla
U_{x}(\eta(x))|^{2}$ $\displaystyle\leq
CL^{\varepsilon(d-1)}L^{-(d-1)}\|a\|_{L^{2}(\Box_{L})}^{2}.$

Here we used the estimate

$\sum_{z\in\partial\Box_{L}}|P^{z}(x)|^{2}\leq\sup_{z}P^{z}(x)\sum_{z\in\partial\Box_{L}}P^{z}(x)=\sup_{z}P^{z}(x)\leq
C\ell^{-d+1}.$

Thus the boundary term in the energy is

(5.6)
$\displaystyle\frac{1}{2}\mathbb{E}_{\mathrm{Q}^{b}}\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\left\langle
P^{z},w\right\rangle_{\Box_{L}}|^{2}$ $\displaystyle\leq
C\sum_{z}\left(\mathrm{Var}_{\mathrm{Q}^{b}}\left\langle
P^{z},w\right\rangle_{\Box_{L}}+|\mathbb{E}_{\mathrm{Q}^{b}}\left\langle
P^{z},w\right\rangle_{\Box_{L}}|^{2}\right)$ $\displaystyle\leq
CB_{L}(a)+CL^{d-1}(\text{Error}_{1})^{2}+CL^{\varepsilon(d-1)}L^{-(d-1)}\|a\|_{L^{2}(\Box_{L})}^{2}$
$\displaystyle\leq
CB_{L}(a)+C(1+\|b\|_{L^{2}(\Box_{L})}^{2})\|a\|_{L^{2}(\Box_{L})}^{2}L^{\varepsilon(3d-2)}L^{-(d-1)}.$

This completes the proof. ∎

Next, we show that $w$ nearly minimizes $\mathcal{F}_{a}$.

######  Lemma 5.2.

We have

(5.7)
$\mathcal{F}_{a}(w;\mathrm{Q}^{b})\leq\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})+\mathrm{Error}$

where

$|\mathrm{Error}|\leq
C\|a\|_{L^{2}(\Lambda)}^{2}+C(1+\|b\|_{L^{2}(\Box_{L})}^{2})\|a\|_{L^{2}(\Box_{L})}^{2}L^{\varepsilon
d}L^{-d}$

for some constant $c_{d}$. Assuming $\|a\|_{L^{2}(\Lambda)}^{2}\leq
CL^{-\varepsilon}$ and $\|a\|_{L^{2}(\Box_{L})}\leq C$, we obtain

$|\mathrm{Error}|\leq
CL^{-\varepsilon}+C(1+\|b\|_{L^{2}(\Box_{L})}^{2})L^{\varepsilon d}L^{-d}.$

###### Proof.

Note that $\overline{u}_{a}(x,\eta)=a(x)\overline{u}_{x}(\eta)$, where
$\overline{u}_{x}$ is the minimizer of
$\mathcal{F}_{\mathbf{1}_{x}}(\cdot;\mathrm{Q}^{b})$. We have

(5.8)
$\displaystyle\mathcal{F}_{a}(w;\mathrm{Q}^{b})-\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})$
$\displaystyle=\mathcal{F}_{0}(w-\overline{u}_{a};\mathrm{Q}^{b})$
$\displaystyle=\sum_{x\in\Lambda}|a(x)|^{2}\mathcal{F}_{0}((\mathbb{E}_{\nu^{0}}U^{0})\mathbf{1}_{x}-\overline{u}_{x};\mathrm{Q}^{b})$
$\displaystyle\quad+\sum_{x\in\Box_{L-\ell}}|a(x)|^{2}\mathcal{F}_{0}(U_{x}-\overline{u}_{x};\mathrm{Q}^{b}).$

We now control the two sums on the RHS above, which we label $S_{1}$ and
$S_{2}$, respectively. For the first, we have

$\mathcal{F}_{0}((\mathbb{E}_{\nu^{0}}U^{0})\mathbf{1}_{x}-\overline{u}_{x};\mathrm{Q}^{b})\leq
2\mathcal{F}_{0}(\overline{u}_{x};\mathrm{Q}^{b})+2\mathcal{F}_{0}((\mathbb{E}_{\nu^{0}}U^{0})\mathbf{1}_{x};\mathrm{Q}^{b})\leq
C,$

and so $S_{1}\leq C\|a\|_{L^{2}(\Lambda)}^{2}$. The term $S_{2}$ is controlled
by (4.11). Specifically, we have

$\mathcal{F}_{0}(U_{x}-\overline{u}_{x};\mathrm{Q}^{b})\leq
C(1+\|b\|_{L^{2}(\Box_{L})}^{2})\ell^{-d}$

and inserting this bound into $S_{2}$ finishes the proof. ∎

The main result of the section follows.

######  Proposition 5.3.

Let $w$ be defined as in (5.1), and recall the definition of $B_{L}(a)$ in
(5.3). For $u_{a}=\mathrm{HS}^{b}a$, we have

(5.9) $\displaystyle\mathcal{E}_{0}(w-u_{a};\mathrm{Q}^{b})$
$\displaystyle\leq C\|a\|_{L^{2}(\Lambda)}^{2}+CB_{L}(a)$
$\displaystyle\quad+C(1+\|b\|_{L^{2}(\Box_{L})}^{2})\|a\|_{L^{2}(\Box_{L})}^{2}L^{(3d-2)\varepsilon}L^{-(d-1)},$

and

(5.10)
$\displaystyle\|\mathbb{E}_{\mathrm{Q}^{b}}u_{a}-(\mathbb{E}_{\nu^{0}}U^{0})a\|_{L^{2}(\Box_{L})}$
$\displaystyle\leq
C(1+\|b\|_{L^{2}(\Box_{L})})\|a\|_{L^{2}(\Box_{L})}L^{\varepsilon(3d/2-1)}L^{-(d-1)/2}$
$\displaystyle\quad+C\|a\|_{L^{2}(\Lambda)}+C({B_{L}(a)})^{1/2}+C\left(\sum_{x\in\Box_{L}}|a(x)|^{2}|b(x)|\right)^{1/2}.$

In the case that $\|a\|_{L^{2}(\Lambda)}^{2}\leq CL^{-\varepsilon}$,
$\|a\|_{L^{2}(\Box_{L})}\leq C$, $\sup|b(x)|\leq CL^{-d/2}$ and $B_{L}(a)\leq
CL^{-1}$, we get

$\|\mathbb{E}_{\mathrm{Q}^{b}}u_{a}-(\mathbb{E}_{\nu^{0}}U^{0})a\|_{L^{2}(\Box_{L})}\leq
CL^{\varepsilon(3d/2-1)}L^{-(d-1)/2}+CL^{-\varepsilon/2}+CL^{-d/4}+CL^{-1/2}.$

We can then optimize by choosing $\varepsilon=\frac{d-1}{3d-1}$ to get a rate
of $CL^{-\frac{d-1}{6d-2}}$ on the RHS.

###### Proof.

Using quadratic response and $\inf\mathcal{F}_{a}\leq\inf\mathcal{E}_{a}$, we
start from

$\mathcal{E}_{0}(w-u_{a};\mathrm{Q}^{b})=\mathcal{E}_{a}(w;\mathrm{Q}^{b})-\mathcal{E}_{a}(u_{a};\mathrm{Q}^{b})\leq\mathcal{E}_{a}(w;\mathrm{Q}^{b})-\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b}).$

We bound the boundary term inside $\mathcal{E}_{a}(w;\mathrm{Q}^{b})$ using
5.1, and 5.2 bounds the remainder:

(5.11) $\displaystyle\mathcal{E}_{a}(w;\mathrm{Q}^{b})$
$\displaystyle\leq\mathcal{F}_{a}(w;\mathrm{Q}^{b})+C(1+\|b\|^{2}_{L^{2}})\|a\|_{L^{2}}^{2}L^{\varepsilon(3d-2)}L^{-(d-1)}+CB_{L}(a)$
$\displaystyle\leq\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})+C\|a\|_{L^{2}(\Lambda)}^{2}+C(1+\|b\|^{2}_{L^{2}})\|a\|_{L^{2}}^{2}L^{\varepsilon(3d-2)}L^{-(d-1)}+CB_{L}(a),$

and so

$\mathcal{E}_{0}(w-u_{a};\mathrm{Q}^{b})\leq
C\|a\|_{L^{2}(\Lambda)}^{2}+C(1+\|b\|_{L^{2}}^{2})\|a\|_{L^{2}}^{2}L^{\varepsilon(3d-2)}L^{-(d-1)}+CB_{L}(a).$

Clearly,

$\sum_{x\in\Box_{L}}|\mathbb{E}_{\mathrm{Q}^{b}}u_{a}(x)-\mathbb{E}_{\mathrm{Q}^{b}}w(x)|^{2}\leq
C\mathcal{E}_{0}(w-u_{a};\mathrm{Q}^{b}),$

and by (4.4), which shows $\nu^{b(x)}$ approximates the marginal of
$\mathrm{Q}^{b}$ at $\eta(x)$, we have

$\sum_{x\in\Box_{L}}|\mathbb{E}_{\mathrm{Q}^{b}}w(x)-\mathbb{E}_{\nu^{b(x)}}w|^{2}\leq
C(1+\|b\|_{L^{2}}^{2})\|a\|_{L^{2}}^{2}\ell^{-d}.$

Finally, by (4.19)

$\sum_{x\in\Box_{L}}|\mathbb{E}_{\nu^{b(x)}}w(x)-a(x)\mathbb{E}_{\nu^{0}}U^{0}|^{2}\leq
C\sum_{x\in\Box_{L}}|a(x)|^{2}|b(x)|.$

Combining the last three displays and (5.9) with the triangle inequality
proves (5.10). ∎

The bound (5.10) is satisfactory when the perturbations $a,b$ have the
following properties: (1) they are bounded in $L^{2}$, (2) $a$ does not
concentrate on a thin boundary layer, and (3) $|a|^{2}|b|$ is small in
$L^{1}$. Condition (3) does not hold in the infinite volume limit 1.1, so we
now provide a different construction that works in this case.

It will turn out to be sufficient to assume that $a$ concentrates almost all
of its mass on a single point $x_{0}$. In this case, the minimizer $u_{a}$ of
$\mathcal{E}_{a}$ is similarly concentrated on $x_{0}$, and there is a simple
approximation for $\mathbb{E}_{\mathrm{Q}^{b}}u_{a}(x_{0})$ given by the
energy $\mathcal{G}_{1}$.

######  Proposition 5.4.

Let $a\in\mathbb{R}^{\Box_{L}}$ and let $u_{a}=\mathrm{HS}^{b}a$, and recall
the definition of $B_{L}(a)$ from (5.3). We have

(5.12)
$\mathbb{E}_{\mathrm{Q}^{b}}\|u_{a}\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}\leq
C\|a\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}+CB_{L}(a),$

and

(5.13)
$|\mathbb{E}_{\mathrm{Q}^{b}}u_{a}(x_{0})-a(x_{0})\mathbb{E}_{\nu^{b(x_{0})}}U_{x_{0}}|^{2}\leq
C(1+\|b\|_{L^{2}}^{2})|a(x_{0})|^{2}\rho_{x_{0}}^{-d}+C\|a\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}+CB_{L}(a).$

###### Proof.

Let $w(x,\eta)=\overline{u}_{\mathbf{1}_{x_{0}}}(x_{0},\eta)a(x)$ for all
$x\in\Box_{L}$, where $\overline{u}_{\mathbf{1}_{x_{0}}}$ minimizes
$\mathcal{F}_{\mathbf{1}_{x_{0}}}(\cdot;\mathrm{Q}^{b})$. Observe that

$\mathbb{E}_{\mathrm{Q}^{b}}\sum_{z\in\partial\Box_{L}}V^{\prime\prime}(\alpha^{z}(\eta))|\left\langle
P^{z},w\right\rangle_{\Box_{L}}|^{2}=\sum_{z\in\partial\Box_{L}}\mathbb{E}_{\mathrm{Q}^{b}}\left(V^{\prime\prime}(\alpha^{z}(\eta))|\overline{u}_{\mathbf{1}_{x_{0}}}(x_{0},\eta)|^{2}\right)|\left\langle
P^{z},a\right\rangle_{\Box_{L}}|^{2},$

and this is bounded by $CB_{L}(a)$ by (2.19). We can apply the above to see

$\mathcal{E}_{a}(w;\mathrm{Q}^{b})-\mathcal{E}_{a}(u_{a};\mathrm{Q}^{b})\leq\mathcal{E}_{a}(w;\mathrm{Q}^{b})-\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})=\mathcal{F}_{a}(w;\mathrm{Q}^{b})-\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})+CB_{L}(a),$

Next, we have

$\mathcal{F}_{a}(w;\mathrm{Q}^{b})-\mathcal{F}_{a}(\overline{u}_{a};\mathrm{Q}^{b})=\mathcal{F}_{0}(w-\overline{u}_{a};\mathrm{Q}^{b})=\sum_{x\in\Box_{L}}|a(x)|^{2}\mathcal{F}_{0}(\overline{u}_{\mathbf{1}_{x_{0}}}(x_{0},\cdot)\mathbf{1}_{x}(\cdot)-\overline{u}_{\mathbf{1}_{x}};\mathrm{Q}^{b}).$

For $x\neq x_{0}$, we bound

$\mathcal{F}_{0}(\overline{u}_{\mathbf{1}_{x_{0}}}(x_{0},\cdot)\mathbf{1}_{x}(\cdot)-\overline{u}_{\mathbf{1}_{x}};\mathrm{Q}^{b})\leq
C$

and for $x=x_{0}$, we have

$\mathcal{F}_{0}(\overline{u}_{\mathbf{1}_{x_{0}}}(x_{0},\cdot)\mathbf{1}_{x}(\cdot)-\overline{u}_{\mathbf{1}_{x}};\mathrm{Q}^{b})=0.$

due to the fact $\overline{u}_{\mathbf{1}_{x}}(y,\eta)=0$ a.s. for $y\neq x$.
Thus

$\mathcal{F}_{0}(w-\overline{u}_{a};\mathrm{Q}^{b})\leq
C\|a\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}.$

It follows that

(5.14)
$\mathbb{E}_{\mathrm{Q}^{b}}\|u_{a}\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}\leq
C\mathbb{E}_{\mathrm{Q}^{b}}\|w\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}+C\mathcal{E}_{0}(w-u_{a};\mathrm{Q}^{b})\leq
C\|a\|_{L^{2}(\Box_{L}\setminus\\{x_{0}\\})}^{2}+CB_{L}(a),$

as desired.

We now prove (5.13). We compute, by the triangle inequality and Jensen’s
inequality, that

$|\mathbb{E}_{\mathrm{Q}^{b}}u_{a}(x_{0})-a(x_{0})\mathbb{E}_{\nu^{b(x_{0})}}U_{x_{0}}|^{2}\leq
C\mathbb{E}_{\mathrm{Q}^{b}}\|u_{a}-w\|_{L^{2}(\Box_{L})}^{2}+C|a(x_{0})|^{2}|\mathbb{E}_{\mathrm{Q}^{b}}\overline{u}_{x_{0}}-\mathbb{E}_{\nu^{b(x_{0})}}U_{x_{0}}|^{2}.$

The first term on the RHS above is dominated by
$C\mathcal{E}_{0}(w-u_{a};\mathrm{Q}^{b})$, which we have already estimated.
By (4.12), we can bound the second term by

$|\mathbb{E}_{\mathrm{Q}^{b}}\overline{u}_{x_{0}}-\mathbb{E}_{\nu^{b(x_{0})}}U^{b}_{x_{0}}|^{2}\leq
C(1+\|b\|^{2}_{L^{2}})\rho_{x_{0}}^{-d}.$

The proof is finished. ∎

##  6\. Proofs of the Main Theorems

In this section, we prove the main theorems listed in Section 1. The general
process for all the results is as follows.

  1. (1)

Figure out the relevant $a\in\mathbb{R}^{\Box_{L}}$ at which we need to
compute the cumulant generating function (c.g.f.) of $\mathrm{Q}$.

  2. (2)

Break $a$ down into a harmonic part $K_{L}a$ and an “almost boundary-
orthogonal” part $K_{L}^{\perp}a$ with the Bergman projection $K_{L}$, which
is the $L^{2}(\mathrm{cl}_{1}\Box_{L})$ projection onto the space of functions
which are discrete harmonic on $\Box_{L}$.

  3. (3)

Estimate the sizes of $a$, $K_{L}a$, $K_{L}^{\perp}a$, especially in a
boundary layer of $\Box_{L}\cup\partial\Box_{L}$. The estimates for $K_{L}a$
and $K_{L}^{\perp}a$ are major tasks in Section 7.

  4. (4)

Apply the estimates from Section 3 and Section 5 to characterize the solutions
of Helffer-Sjöstrand equations.

  5. (5)

Apply the Helffer-Sjöstrand representation of the c.g.f. given by (2.10) and
(2.11).

  6. (6)

For the limits of $\overline{\phi}$, this process applies equally well to the
Gaussian membrane model, so we can compare the non-Gaussian to the Gaussian
case and achieve our results.

###  6.1. Infinite Volume Limit

We first prove the infinite volume limit 1.1 in $d\geq 5$. Fix a
$K\in\mathbb{N}$ to be held constant as $L\to\infty$, and let
$a^{\prime}\in\mathbb{R}^{\mathbb{Z}^{d}}$ be supported within $\Box_{K}$. Let
$a\in\mathbb{R}^{\Box_{L}}$ be defined by
$a(x)=-\sum_{y\in\Box_{K}}\Gamma_{L}(x,y)a^{\prime}(y)$, where $-\Gamma_{L}$
is the Dirichlet Green’s function for $\Delta$ on $\Box_{L}$. We allow
constants denoted by $C$ to depend on $K$ and $\|a^{\prime}\|_{L^{2}}$, but
they remain uniform as $L\to\infty$.

We let $K_{L}$ be the $L^{2}(\Box_{L}\cup\partial\Box_{L})$ projection onto
the linear span of $\\{P^{z}\\}_{z\in\partial\Box_{L}}$ and
$K_{L}^{\perp}=\text{Id}-K_{L}$. These operators are analyzed in Section 7.

######  Theorem 6.1.

For any $\varepsilon>0$ fixed, $d\geq 5$, and $a$ as above, we have

(6.1) $\mathbb{E}_{\mathrm{Q}^{ra}}\left\langle
a,\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}=\sum_{x\in\Box_{L}}|a(x)|^{2}\mathbb{E}_{\nu^{ra(x)}}U^{ra(x)}+O(L^{-d/2+2+\varepsilon})$

as $L\to\infty$. Consequently, 1.1 holds, characterizing the infinite volume
limit of the membrane model.

###### Proof.

We will first prove (6.1) and then show how it implies 1.1.

We give some routine bounds on $a$. By 7.1, we have

(6.2) $\displaystyle\|a\|_{L^{2}(\Box_{L})}^{2}$ $\displaystyle\leq
C\sum_{x\in\Box_{L}}\sum_{y\in\Box_{K}}|a^{\prime}(y)|^{2}|\Gamma_{L}(x,y)|^{2}$
$\displaystyle\leq
C\sum_{x\in\mathbb{Z}^{d}}\sup_{y\in\Box_{K}}|\Gamma(x,y)|^{2}\leq
C\sum_{x\in\mathbb{Z}^{d}}\frac{1}{(1+|x|)^{2d-4}}\leq C.$

Next we estimate $a$ on a boundary layer. Fix $\varepsilon\in(0,1/4)$, and let
$\ell\in\mathbb{Z}$ be between $L^{1-\varepsilon}$ and $2L^{1-\varepsilon}$.
Let $\Lambda=\Box_{L}\setminus\Box_{L-\ell}$ be a boundary layer of thickness
$\ell$. Then the same computation as before shows

(6.3) $\|a\|_{L^{2}(\Lambda)}^{2}\leq
C\sum_{x\in\Lambda}\sum_{y\in\Box_{K}}|a^{\prime}(y)|^{2}|\Gamma_{L}(x,y)|^{2}\leq
C\sum_{x\in\Lambda}\frac{1}{|x|^{2d-4}}\leq CL^{-d+4-\varepsilon}.$

Similarly we bound the $L^{1}(\Box_{L})$ norm of $a$ by

$\|a\|_{L^{1}(\Box_{L})}\leq
C\sup_{y\in\Box_{K}}\sum_{x\in\Box_{L}}\frac{1}{(1+|x-y|)^{d-2}}\leq CL^{2}.$

We now compute $\mathbb{E}_{\mathrm{Q}^{ra}}\left\langle
a,\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}$. Recall the boundary energy
$B_{L}(a)$ defined in (5.3) and the special profile $e_{x}$ constructed in 4.1
for any $x\in\Box_{L}$. Recall also that $\rho_{x}$ denotes the distance from
$x$ to $\partial\Box_{L}$.

We use the decomposition

(6.4)
$\mathrm{HS}^{ra}a=\mathrm{HS}^{ra}(a\mathbf{1}_{\Lambda})+\sum_{x\in\Box_{L-\ell}}a(x)\mathrm{HS}^{ra}({K}_{L}\mathbf{1}_{x})+\sum_{x\in\Box_{L-\ell}}a(x)\mathrm{HS}^{ra}({K}_{L}^{\perp}\mathbf{1}_{x}).$

Only the last term should make a non-negligible contribution toward (6.1).
Indeed, the first term is a boundary layer term, and we will use the profile
$e_{x}$ to show that $K_{L}\mathbf{1}_{x}$ is very small in $L^{2}$ when $x$
is not close to $\partial\Box_{L}$.

We handle the first term by

$\displaystyle|\mathbb{E}_{\mathrm{Q}^{ra}}\left\langle
a,\mathrm{HS}^{ra}(a\mathbf{1}_{\Lambda})\right\rangle_{\Box_{L}}|$
$\displaystyle\leq\|a\|_{L^{2}(\Box_{L})}\|\mathrm{HS}^{ra}(a\mathbf{1}_{\Lambda})\|_{L^{2}(\Box_{L})}$
$\displaystyle\leq
C\|a\|_{L^{2}(\Box_{L})}\|a\mathbf{1}_{\Lambda}\|_{L^{2}(\Box_{L})}\leq
CL^{-d/2+2-\varepsilon/2},$

where we used that $\mathrm{HS}^{ra}$ is bounded as an operator
$L^{2}(\Box_{L})$ uniformly in $L$ (by (2.19) and dominating the $L^{2}$ norm
by the energy $C\mathcal{E}_{0}$). For the second term, observe that
${K}_{L}(\mathbf{1}_{x}+e_{x})=0$ since $\left\langle
P^{z},\mathbf{1}_{x}+e_{x}\right\rangle_{\mathrm{cl}_{1}(\Box_{L})}=0$ by
construction. We have

(6.5)
$\|{K}_{L}\mathbf{1}_{x}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}=\|{K}_{L}e_{x}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}\leq
C\rho_{x}^{-d/2}$

where we used that ${K}_{L}$ is an $L^{2}$ projection and
$\|e_{x}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}\leq C\rho_{x}^{-d/2}$. Thus

(6.6) $\displaystyle\sum_{x\in\Box_{L-\ell}}\left\langle
a,\mathrm{HS}^{ra}{K}_{L}(a\mathbf{1}_{x})\right\rangle_{\Box_{L}}$
$\displaystyle\leq
C\sum_{x\in\Box_{L-\ell}}\|a\|_{L^{2}(\Box_{L})}|a(x)|\rho_{x}^{-d/2}$
$\displaystyle\leq
C\|a\|_{L^{2}(\Box_{L})}\|a\|_{L^{1}(\Box_{L})}\ell^{-d/2}\leq CL^{\varepsilon
d/2}L^{-d/2+2}.$

The contribution of the second term in (6.4) is thus negligible as
$L\to\infty$ provided $\varepsilon$ is chosen small.

For the third term, we first apply (5.12) to see

(6.7) $\sum_{x\in\Box_{L-\ell}}a(x)\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}{K}_{L}^{\perp}\mathbf{1}_{x}\right\rangle_{\Box_{L}}=\sum_{x\in\Box_{L-\ell}}|a(x)|^{2}\left(\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}{K}_{L}^{\perp}\mathbf{1}_{x}\right)(x)+\mathrm{Error}_{1}$

where

$\displaystyle\mathrm{Error}_{1}$ $\displaystyle\leq
C\sum_{x\in\Box_{L-\ell}}|a(x)|\|a\|_{L^{2}(\Box_{L})}\|\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}K_{L}^{\perp}\mathbf{1}_{x}\|_{L^{2}(\Box_{L}\setminus\\{x\\})}$
$\displaystyle\leq
C\sum_{x\in\Box_{L-\ell}}\left(|a(x)|\|a\|_{L^{2}(\Box_{L})}[\|{K}_{L}^{\perp}\mathbf{1}_{x}\|_{L^{2}(\Box_{L}\setminus\\{x\\})}+\sqrt{B_{L}(K_{L}^{\perp}\mathbf{1}_{x})}]\right).$

Since ${K}_{L}^{\perp}\mathbf{1}_{x}(y)=-{K}_{L}\mathbf{1}_{x}(y)$ for $y\neq
x$, (6.5) proves the bound

$\bigg{|}\sum_{x\in\Box_{L-\ell}}|a(x)|\|a\|_{L^{2}(\Box_{L})}\|{K}_{L}^{\perp}\mathbf{1}_{x}\|_{L^{2}(\Box_{L}\setminus\\{x\\})}\bigg{|}\leq
C\|a\|_{L^{2}(\Box_{L})}\|a\|_{L^{1}(\Box_{L})}\ell^{-d/2}\leq
CL^{-d/2+2+\varepsilon d/2}$

which controls part of $\mathrm{Error}_{1}$. We now control
$B_{L}(K_{L}^{\perp}\mathbf{1}_{x})$. By definition, we have

$\left\langle
K_{L}^{\perp}\mathbf{1}_{x},P^{z}\right\rangle_{\mathrm{cl}_{1}\Box_{L}}=0$

for all $z\in\partial\Box_{L}$. Thus

$\left\langle
K_{L}^{\perp}\mathbf{1}_{x},P^{z}\right\rangle_{\Box_{L}}=\left\langle
K_{L}^{\perp}\mathbf{1}_{x},P^{z}\right\rangle_{\mathrm{cl}_{1}\Box_{L}}-K_{L}^{\perp}\mathbf{1}_{x}(z)=-K_{L}^{\perp}\mathbf{1}_{x}(z)=K_{L}\mathbf{1}_{x}(z).$

Thus we can again apply (6.5) to get

(6.8)
$B_{L}(K_{L}^{\perp}\mathbf{1}_{x}):=\sum_{z\in\partial\Box_{L}}\bigg{|}\left\langle
K_{L}^{\perp}\mathbf{1}_{x},P^{z}\right\rangle_{\Box_{L}}\bigg{|}^{2}\leq\|K_{L}\mathbf{1}_{x}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}\leq
C\rho_{x}^{-d}.$

We use this to get a final bound

$\mathrm{Error}_{1}\leq CL^{-d/2+2+\varepsilon d/2}.$

Next, by (5.13), we have

(6.9)
$\sum_{x\in\Box_{L-\ell}}|a(x)|^{2}\left(\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}{K}_{L}^{\perp}\mathbf{1}_{x}\right)(x)=\sum_{x\in\Box_{L-\ell}}|a(x)|^{2}\mathbb{E}_{\nu^{ra(x_{0})}}U^{ra(x_{0})}+\mathrm{Error}_{2}$

where

$\mathrm{Error}_{2}\leq
C\sum_{x\in\Box_{L-\ell}}|a(x)|^{2}(|{K}_{L}^{\perp}\mathbf{1}_{x}(x)|\ell^{-d/2}+\|{K}_{L}^{\perp}\mathbf{1}_{x}\|_{L^{2}(\Box_{L}\setminus\\{x\\})}+\sqrt{B_{L}(K^{\perp}_{L}\mathbf{1}_{x})})$

The term $\mathrm{Error}_{2}$ is bounded by $C\ell^{-d/2}$ by the same method
as the bound for $\mathrm{Error}_{1}$, i.e. by (6.5) and (6.8).

Combining (6.4), (6.7), and (6.9), we have almost proved (6.1), except we are
summing over $\Box_{L-\ell}$ instead of $\Box_{L}$. This is fixed by noting

$\bigg{|}\sum_{x\in\Lambda}|a(x)|^{2}\mathbb{E}_{\nu^{ra(x)}}U^{ra(x)}\bigg{|}\leq\|a\|_{L^{2}(\Lambda)}^{2}\leq
L^{-d+4-\varepsilon},$

and (6.1) is proved.

Next, we explain how to conclude 1.1. First, we translate (1.1) to the level
of $\mathrm{Q}^{0}$ and apply Helffer-Sjöstrand representation in (2.10) and
(2.11):

$\log\mathbb{E}_{\mathrm{MM}_{L}}\exp\left(\sum_{x\in\Box_{K}}a^{\prime}(x)\phi(x)\right)=\log\mathbb{E}_{\mathrm{Q}^{0}}\exp\left\langle
a,\eta\right\rangle_{\Box_{L}}=\int_{0}^{1}(1-r)\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}dr.$

We apply (6.1) to get

$\log\mathbb{E}_{\mathrm{MM}_{L}}\exp\left(\sum_{x\in\Box_{K}}a^{\prime}(x)\phi(x)\right)=\sum_{x\in\Box_{L}}|a(x)|^{2}\int_{0}^{1}(1-r)\mathbb{E}_{\nu^{ra(x)}}U^{ra(x)}dr+\text{Error}.$

The Helffer-Sjöstrand representation associated to the measure $\nu^{ra(x)}$,
analogous to (2.10) and (2.11), is

$\mathrm{Var}_{\nu^{ra(x)}}(\xi)=\mathbb{E}_{\nu^{ra(x)}}U^{ra(x)},\quad\log\mathbb{E}_{\nu^{0}}\exp(a(x)\xi)=\int_{0}^{1}(1-r)\mathrm{Var}_{\nu^{ra(x)}}(a(x)\xi)dr,$

and so

(6.10)
$\log\mathbb{E}_{\mathrm{MM}_{L}}\exp\left(\sum_{x\in\Box_{K}}a^{\prime}(x)\phi(x)\right)=\sum_{x\in\Box_{L}}\log\mathbb{E}_{\nu^{0}}\exp((\Gamma_{L}\ast
a^{\prime})(x)\xi)+\text{Error}.$

All that remains is to pass to $L\to\infty$ in the RHS. Let
$\psi(\lambda)=\log\mathbb{E}_{\nu^{0}}e^{\lambda\xi}$ be the c.g.f. of
$\nu^{0}$, which has locally bounded derivatives. We have

$|\psi(\Gamma_{L}\ast a^{\prime}(x))-\psi(\Gamma\ast a^{\prime}(x))|\leq
C|\Gamma_{L}\ast a^{\prime}(x)-\Gamma\ast a^{\prime}(x)|$

where $C$ depends only on $a^{\prime}$. Using the estimates in 7.1 and (7.4),
we can bound the RHS by

$\displaystyle|\Gamma_{L}\ast a^{\prime}(x)-\Gamma\ast a^{\prime}(x)|$
$\displaystyle\leq\sum_{z\in\partial\Box_{L}}\sum_{y\in\Box_{K}}P^{z}(x)\Gamma(z,y)|a^{\prime}(y)|$
$\displaystyle\leq
C\|a^{\prime}\|_{L^{1}(\Box_{K})}\sup_{z\in\partial\Box_{L},y\in\Box_{K}}\frac{1}{|z-y|^{d-2}}\leq
CL^{-d+2}.$

Observe that $\Gamma_{L}\ast a^{\prime}(x)$ decays like $|x|^{-d+2}$ as
$|x|\to\infty$, uniformly in $L$, and $\mathbb{E}_{\nu^{0}}\xi=0$. It follow
that the c.g.f. of $\nu^{0}$ at $\Gamma_{L}\ast a^{\prime}(x)$ decays like
$|x|^{-2d+4}$, and so, for any fixed $\beta>0$, we have

$\sum_{x\in\mathbb{Z}^{d}\setminus\Box_{L^{\beta}}}|\psi(\Gamma_{L}\ast
a^{\prime}(x))|\leq
C\sum_{x\in\mathbb{Z}^{d}\setminus\Box_{L^{\beta}}}\frac{1}{|x|^{2d-4}}=O(L^{-\beta(d-4)})$

as $L\to\infty$. The same estimate holds for $\Gamma$ in place of
$\Gamma_{L}$. Combining the above, we see

$\displaystyle\sum_{x\in\Box_{L}}\psi(\Gamma_{L}\ast a^{\prime}(x))$
$\displaystyle=\sum_{x\in\Box_{L^{\beta}}}\psi(\Gamma_{L}\ast
a^{\prime}(x))+O(L^{-\beta(d-4)})$
$\displaystyle=\sum_{x\in\Box_{L^{\beta}}}\psi(\Gamma\ast
a^{\prime}(x))+O(L^{-d+2}L^{\beta d})+O(L^{-\beta(d-4)})$
$\displaystyle=\sum_{x\in\mathbb{Z}^{d}}\psi(\Gamma\ast
a^{\prime}(x))+O(L^{-d+2}L^{\beta d})+O(L^{-\beta(d-4)}).$

Choosing $\beta>0$ sufficiently small finishes the proof of 1.1. ∎

###  6.2. Limits of the rescaled model

Next, we prove the scaling limit in $d\geq 2$. Let $f:[-1,1]^{d}\to\mathbb{R}$
and let $u$ be the solution to continuum problem

(6.11)
$\begin{cases}{\Delta_{\mathbb{R}^{d}}^{2}}u(x)=f(x),\quad&x\in(-1,1)^{d},\\\
u(x)=\partial_{n}u(x)=0,\quad&x\in\partial(-1,1)^{d}.\end{cases}$

Depending on the dimension $d$, we will make one of the two following
assumptions:

(6.12) $u\in C^{5}((-1,1)^{d})\text{ with }\|u\|_{C^{5}((-1,1)^{d})}\leq
C<\infty$

or

(6.13) $f\in C^{0}((-1,1)^{d})\text{ with }\|f\|_{L^{\infty}((-1,1)^{d})}\leq
C<\infty.$

Condition (6.13) is clearly weaker than (6.12), and it suffices in the case of
square/cubic domains in $d=2,3$ using estimates in [15]. We conjecture that it
is also sufficient in all other cases. We set
$a(x)=-L^{-d/2-2}\sum_{y\in\Box_{L}}f(y/L)\Gamma_{L}(x,y)$.

We now cite some basic properties of $a$ and ${K}_{L}$ proved in Section 7.
The field ${K}_{L}a$ is harmonic in $\Box_{L}$. Let
$\Lambda^{\prime}=\mathrm{cl}_{1}\Box_{L}\setminus\Box_{L-\ell^{\prime}}$ for
some integer $\ell^{\prime}\in[1,L]$, i.e. $\Lambda^{\prime}$ is a boundary
layer of $\mathrm{cl}_{1}\Box_{L}$. If we assume (6.12) holds or if we assume
$d=2,3$ and (6.13) holds, then by either (7.16) or 7.1, we have

(6.14)
$\|{K}_{L}a\|_{L^{2}(\Lambda^{\prime})}^{2}+\|(\text{Id}-{K}_{L})a\|_{L^{2}(\Lambda^{\prime})}^{2}\leq\frac{C\cdot(\ell^{\prime}+1)}{L}(1+\mathbf{1}_{d=2}(\log
L)^{2}).$

The same inequality holds for $\Lambda^{\prime}=\mathrm{cl}_{1}\Box_{L}$ if we
set $\ell^{\prime}=L$. It is important that $\Lambda^{\prime}$ contains
$\partial\Box_{L}$, unlike the layer $\Lambda$ considered in the previous
subsection, and the constant $C$ depends only on the dimension and the size of
the constants in (6.12) or (6.13). We also have

(6.15) $\|a\|_{L^{\infty}(\mathrm{cl}_{1}\Box_{L})}\leq
CL^{-d/2}\|f\|_{L^{\infty}}(1+\mathbf{1}_{d=2}\log L)\leq
CL^{-d/2}(1+\mathbf{1}_{d=2}\log L),$

which follows easily from a standard estimate on $\Gamma_{L}$ stated in (7.1).

######  Theorem 6.2.

Let $a$ be as above and assume either (6.13) holds and $d=2,3$ or (6.12) holds
and $d\geq 2$. Then we have

(6.16) $\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}=\mathbb{E}_{\nu^{0}}U^{0}\|(\mathrm{Id}-{K}_{L})a\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}+\mathrm{Error}$

as $L\to\infty$, where $\mathrm{Error}$ satisfies

$|\mathrm{Error}|\leq CL^{-\frac{d-1}{6d-2}}(1+\mathbf{1}_{d=2}(\log L)^{3}).$

As a consequence, 1.2 holds.

###### Proof.

We first prove (6.16) and then explain how it implies 1.2.

We let $\varepsilon>0$ be a parameter to be fixed later. Let
$\Lambda=\Box_{L}\setminus\Box_{L-\ell}$ be a boundary layer of width $\ell$
between $L^{1-\varepsilon}$ and $2L^{1-\varepsilon}$. Let
$\Lambda^{\prime}=\Lambda\cup\partial\Box_{L}$.

Our strategy for approximating $\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}a$
is to decompose $a$ as ${K}_{L}a+{K}_{L}^{\perp}a$ and compute the linear
operator $\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}$ on each piece.

The first piece ${K}_{L}a$ is a harmonic function which does not concentrate
on $\partial\Box_{L}$ by (6.14). The estimate (3.3) from 3.1 therefore
applies, and we have

$\displaystyle\|\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}{K}_{L}a\|_{L^{2}(\Box_{L})}^{2}$
$\displaystyle\leq
C\mathcal{E}_{0}(\mathrm{HS}^{ra}{K}_{L}a)=C\mathcal{E}_{0}(u_{{K}_{L}a})$
$\displaystyle\leq
C\|{K}_{L}a\|_{L^{2}(\Box_{L})}\|{K}_{L}a\|_{L^{2}(\partial\Box_{L})}\leq
CL^{-1/2}(1+\mathbf{1}_{d=2}(\log L)^{2}).$

In the last line, we used (6.14) on the layer
$\Lambda^{\prime}=\partial\Box_{L}$ of width $1$ to estimate the boundary
term.

Next, we consider the term coming from ${K}_{L}^{\perp}a$. Applying the result
in equation (5.10) gives

(6.17)
$\displaystyle\|\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}{K}_{L}^{\perp}a-\mathbb{E}_{\nu^{0}}U^{0}\cdot{K}_{L}^{\perp}a\|_{L^{2}(\Box_{L})}$
$\displaystyle\leq CL^{-d/2+1/2+c_{d}\varepsilon}(1+(\log
L)^{2}\mathbf{1}_{d=2})+C\|{K}_{L}^{\perp}a\|_{L^{2}(\Lambda)}$
$\displaystyle\quad+C\sqrt{B_{L}(K_{L}^{\perp}a)}+C\left(\sum_{x\in\Box_{L}}|a(x)||{K}_{L}^{\perp}a(x)|^{2}\right)^{1/2},$

where $c_{d}=\frac{3}{2}d-1$ and $B_{L}$ is defined in (5.3). We rewrite
$B_{L}(K_{L}^{\perp}a)$ like in (6.8) to get

$B_{L}(K_{L}^{\perp}a)=\sum_{z\in\partial\Box_{L}}\bigg{|}\left\langle
K_{L}^{\perp}a,P^{z}\right\rangle_{\Box_{L}}\bigg{|}^{2}=\|K_{L}^{\perp}a\|_{L^{2}(\partial\Box_{L})}^{2}.$

Considering the estimates (6.14) and (6.15) on $a$, it is straightforward to
bound the RHS of (6.17) by

$C(L^{-d/2+1/2+c_{d}\varepsilon}+L^{-\frac{1}{2}\varepsilon}+L^{-d/4})(1+(\log
L)^{2}\mathbf{1}_{d=2}).$

We put together the preceding estimates along with Cauchy-Schwarz to conclude

$\left\langle
a,\mathbb{E}_{\mathrm{Q}^{ra}}\mathrm{HS}^{ra}a\right\rangle_{\Box_{L}}=\mathbb{E}_{\nu^{0}}U^{0}\left\langle
a,{K}_{L}^{\perp}a\right\rangle_{\Box_{L}}+\text{Error}$

where Error satisfies

$|\text{Error}|\leq
C\left(L^{-d/2+1/2+(3d/2-1)\varepsilon}+L^{-\varepsilon/2}+L^{-d/4}+L^{-1/4}\right)(1+\mathbf{1}_{d=2}(\log
L)^{3}).$

We optimize by choosing $\varepsilon=\frac{d-1}{3d-1}$, which gives

$|\text{Error}|\leq CL^{-\frac{d-1}{6d-2}}(1+\mathbf{1}_{d=2}(\log L)^{3}).$

Finally, since ${K}_{L}$ is an $L^{2}(\mathrm{cl}_{1}\Box_{L})$ projection and
$a(z)=0$ for $z\in\partial\Box_{L}$, we have

$\left\langle a,{K}_{L}^{\perp}a\right\rangle_{\Box_{L}}=\left\langle
a,{K}_{L}^{\perp}a\right\rangle_{\mathrm{cl}_{1}\Box_{L}}=\|{K}_{L}^{\perp}a\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2},$

which concludes the proof of (6.16).

To conclude 1.2, notice that $\mathbb{E}_{\nu^{0}}U^{0}$ is equal to the
variance of a variable $\xi$ with law $\nu^{0}$, and the approximation (6.16)
can also be done for a Gaussian membrane model. Then the representations
(2.10) and (2.11) relate (6.16) to the cumulant generating function, and allow
us to conclude the theorem by comparing the approximations for the Gaussian
and non-Gaussian model. ∎

Our theory makes no serious distinction between the first parts 1.2 and 1.3,
and we now prove the latter.

###### Proof of 1.3.

The proof is the same as that of 6.2, except with a different form for $a$,
which is $a(x)=L^{d/2-2}\sum_{i=1}^{k}\Gamma_{L}(x,L\cdot y_{i})c_{i}$ for
some numbers $c_{i}$ and points $y_{i}\in[-1,1]^{d}$. The values $c_{i}$ and
$y_{i}$ change with $L$ because of the linear interpolation of the interface
$\overline{\phi}$, but we have effectively the same estimates on $a$ as
(6.14). In particular, we have

$\|a\|_{L^{\infty}}\leq CL^{d/2-2}(1+\mathbf{1}_{d=2}\log
L),\quad\|a\|_{L^{2}}\leq C(1+\mathbf{1}_{d=2}\log L)$

uniformly over $L$. And for a boundary layer $\Lambda^{\prime}$ of width
$\ell$, using 7.1, we have

(6.18) $\displaystyle\|{K}_{L}^{\perp}a(x)\|_{L^{2}(\Lambda^{\prime})}^{2}$
$\displaystyle\leq C(1+\mathbf{1}_{d=2}(\log
L)^{2})L^{d-4}\sup_{i}\sum_{x\in\Lambda^{\prime}}\frac{1}{(1+|x-Ly_{i}|)^{2d-4}}$
$\displaystyle\leq C(1+\mathbf{1}_{d=2}(\log
L)^{2})L^{d-4}\ell\sum_{x\in\mathbb{Z}^{d-1},\|x\|\leq
L}\frac{1}{(1+|x|)^{2d-4}}$ $\displaystyle\leq C(1+\mathbf{1}_{d=2}(\log
L)^{2})\frac{\ell}{L}.$

On the middle line above, we dominated the sum over the boundary layer by
$\ell$ times a sum over a $d-1$ dimensional plane of length $L$. By the same
idea, we have

$\|a\|_{L^{2}(\Lambda^{\prime})}^{2}\leq CL^{-1}\ell(1+\mathbf{1}_{d=2}(\log
L)^{2}).$

The bounds above are exactly the same in $d=2$ as that used in the proof of
6.2. For $d=3$, we have a weaker $L^{\infty}$ control on $a$ than before, but
it is inconsequential for the final error estimate. We omit the rest of the
proof the main estimate.

In order to prove the existence of a Gaussian distributional limit in the
space of continuous functions on $[-1,1]^{d}$, we must address tightness of
the distribution of $\overline{\phi}$ on $C([-1,1]^{d})$.

Let $x,y\in[-1,1]^{d}$. We wish to bound from above the typical size of
$|\overline{\phi}(x)-\overline{\phi}(y)|$ in terms of $|x-y|$. The result we
desire is proved in [7] for the Gaussian membrane model, so we seek to use
Gaussian domination coming from $\inf V^{\prime\prime}>0$. To do this, we need
a version of the Brascamp-Lieb inequality from [3], Theorem 5.1, which is a
generalization of the Poincaré inequality 2.1. The Hessian of the general
membrane model energy is larger than the Hessian of the Gaussian membrane
model associated to the single-spin potential $\xi\mapsto C^{-1}\xi^{2}$,
where $C$ depends on $\inf V^{\prime\prime}$. We have

$\mathbb{E}_{\mathrm{MM}_{L}}\left[|\overline{\phi}(x)-\overline{\phi}(y)]|^{2\beta}\right]\leq
C\mathbb{E}_{\mathrm{MM}^{G}_{L}}\left[|\overline{\phi}(x)-\overline{\phi}(y)]|^{2\beta}\right]$

where $\mathrm{MM}^{G}_{L}$ is a Gaussian membrane model and $2\beta\geq 1$.
It follows from Lemma 2.6 in [7] that

$\mathrm{Var}_{\mathrm{MM}_{L}^{G}}(\overline{\phi}(x)-\overline{\phi}(y))\leq
C|x-y|^{1+\lambda}$

for any fixed $\lambda\in(0,1)$ in $d=2$ or $\lambda=0$ in $d=3$. Since the
model is Gaussian, it follows

$\mathbb{E}_{\mathrm{MM}^{G}_{L}}\left[|\overline{\phi}(x)-\overline{\phi}(y)]|^{2\beta}\right]\leq
C|x-y|^{\beta(1+\lambda)},$

and so we conclude the same inequality for $\mathrm{MM}_{L}$, except with a
potentially larger constant $C$.

Tightness and the Hölder continuity properties of the limit follow from the
Kolmogorov-Chentsov criterion (see e.g. [10], Corollary 16.9). Since [7]
proves the limit of $\overline{\phi}$ under $\mathrm{MM}^{G}_{L}$, and we know
these limits must agree up to a scaling factor, the proof is concluded. ∎

##  7\. Appendix: Bergman projections and Green’s function estimates

In this section, we collect already known estimates for the Laplacian Green’s
functions $-\Gamma_{L}:\Box_{L}\times\Box_{L}\to\mathbb{R}$. We also prove a
non-concentration result for the discrete harmonic Bergman projection. Many of
the estimates are not sharp, but we include them to be self-contained and to
show that, for the most part, only crude bounds are needed for our results. We
also could not find a reference for the discrete harmonic Bergman projection
results.

###  7.1. Laplacian Green’s function Estimates

The first result gives some non-optimal bounds on the Poisson kernel and
Green’s function of a simple random walk. We have chosen to give crude bounds
because the proof is easy to adapt to other domains than $\Box_{L}$.

######  Proposition 7.1.

Let $d\geq 2$, and let $-\Gamma_{L}$ be the Dirichlet Green’s function for the
discrete Laplacian on $\Box_{L}$. Then there is a constant $C$, depending only
on $d$, such that

(7.1) $|\Gamma_{L}(x,y)|\leq C(1+|x-y|)^{2-d}(1+\mathbf{1}_{d=2}\log L)$

for all $x,y\in\Box_{L}$. We also have (recall that
$\rho_{y}=\mathrm{dist}(y,\partial\Box_{L})$)

(7.2) $|D_{y}\Gamma_{L}(x,y)|)\leq C(1+|x-y|)^{1-d}+C\rho_{y}^{1-d}$

where $D_{y}$ denotes the discrete derivative in $y$. We can use this to bound
the Poisson kernel $P^{z}\in\mathbb{R}^{\Box_{L}}$:

(7.3) $|P^{z}(y)|\leq C\rho_{y}^{-d+1}.$

###### Proof.

We consider dimensions $d\geq 3$ first. Let $\Gamma$ be the Green’s function
for $-\Delta$ on $\mathbb{Z}^{d}$ and $\Gamma_{L}$. We have the representation

(7.4)
$\Gamma_{L}(x,y)=\Gamma(x,y)-\sum_{z\in\partial\Box_{L}}P^{z}(x)\Gamma(z,y),$

which can be proved by noticing the RHS solves the same discrete Poisson
equation as $\Gamma_{L}$. Estimates $\eqref{gam_sup}$ and
$\eqref{greens_deriv}$ are true with $\Gamma$ in place of $\Gamma_{L}$, as
proved in [14]. Estimate $\eqref{gam_sup}$ follows then directly from the
positivity of $\Gamma_{L}$ and $\Gamma$. Taking the derivative of the
representation and applying estimates for $D_{y}\Gamma$, we can write

(7.5) $\displaystyle|D_{y}\Gamma_{L}(x,y)|$
$\displaystyle\leq|D_{y}\Gamma(x,y)|+\sum_{z\in\partial\Box_{L}}P^{z}(x)|D_{y}\Gamma(z,y)|$
$\displaystyle\leq
C(1+|x-y|)^{1-d}+C\rho_{y}^{-d+1}\sum_{z\in\partial\Box_{L}}P^{z}(x)=C(1+|x-y|)^{1-d}+C\rho_{y}^{-d+1}.$

In the last inequality, we used the fact that $\sum_{z}P^{z}(x)=1$ is the
probability that a simple random walk starting at $x$ exits $\Box_{L}$ in
finite time.

For $d=2$, the infinite volume Green’s function does not exist, but the
potential kernel $a(y-x)=\lim_{L\to\infty}(\Gamma_{L}(x,y)-\Gamma_{L}(x,x))$
exists and depends only on $y-x$. The representation formula (7.4) holds in
the sense

$\Gamma_{L}(x,y)=\bigg{(}\sum_{z\in\partial\Box_{L}}P^{z}(x)a(z-y)\bigg{)}-a(y-x).$

The estimates (7.1) and (7.2) follow in the same manner as before, now
substituting the estimates $|a(x)|\leq C\log|x|$ and $|D_{x}a(x)|\leq
C|x|^{-1}$.

Note that $P^{z}(y)=\frac{1}{2d}\Gamma_{L}(y,\tilde{z})$, where
$\tilde{z}\in\Box_{L}$ is the unique point adjacent to $z\in\partial\Box_{L}$.
Indeed, the function $y\mapsto(2d)^{-1}\Gamma_{L}(y,\tilde{z})$ is harmonic in
$\Box_{L}$ with boundary condition $\mathbf{1}_{z}$, and so it must coincide
with $P^{z}$. The estimate (7.3) follows from (7.2) by noting
$\Gamma_{L}(y,z)=0$. ∎

###  7.2. The harmonic Bergman projection

In this subsection, we give estimates for the Bergman projection $K_{L}$
applied to $a$ of the form that we encounter in the course of the proofs of
our main theorems.

Let $\mathcal{H}(\Box_{L})$ denote the subspace of functions on
$\Box_{L}\cup\partial\Box_{L}$ which are discrete harmonic in $\Box_{L}$. The
Bergman projection
$K_{L}:\mathbb{R}^{\Box_{L}\cup\partial\Box_{L}}\to\mathcal{H}(\Box_{L})$ is
defined by

(7.6)
$K_{L}a=\text{argmin}_{\tilde{a}\in\mathcal{H}(\Box_{L})}\|\tilde{a}-a\|_{L^{2}(\Box_{L}\cup\partial\Box_{L})}^{2}.$

We can relate $K_{L}$ to the bi-Laplacian Green’s function $G_{L}$, which is
defined as the solution to

$\begin{cases}\Delta^{2}_{x}G_{L}(x,y)=\mathbf{1}_{x=y},\quad&x\in\Box_{L},\\\
G_{L}(x,y)=0,\quad&x\in\mathbb{Z}^{d}\setminus\Box_{L}.\end{cases}$

By a slight abuse of notation, we can consider $G_{L}$ as a convolutional
operator
$\mathbb{R}^{\Box_{L}}\to\mathbb{R}^{\Box_{L}\cup\partial^{2}\Box_{L}}$.
Recall that $\partial^{2}\Box_{L}$ consists of all points in
$\mathbb{Z}^{d}\setminus\Box_{L}$ within $\ell^{1}$ distance $2$ of
$\Box_{L}$. We can also consider the Laplacian $\Delta$ as an operator
$\mathbb{R}^{\Box_{L}\cup\partial\Box_{L}}\to\mathbb{R}^{\Box_{L}}$ or
$\mathbb{R}^{\Box_{L}\cup\partial^{2}\Box_{L}}\to\mathbb{R}^{\Box_{L}\cup\partial\Box_{L}}$.
Under these identifications, we now verify that $K^{\perp}_{L}=\Delta
G_{L}\Delta$, where $K^{\perp}_{L}=\text{Id}-K_{L}$.

Indeed, for any $v$ in the image of $K^{\perp}_{L}$, we have that $v$ is
$L^{2}(\Box_{L}\cup\partial\Box_{L})$-orthogonal to the kernel of
$\Delta:\mathbb{R}^{\Box_{L}\cup\partial\Box_{L}}\to\mathbb{R}^{\Box_{L}}$,
and so is in the image of the adjoint
$\Delta:\mathbb{R}^{\Box_{L}}\to\mathbb{R}^{\Box_{L}\cup\partial\Box_{L}}$
given by extension by $0$ outside $\Box_{L}$. That is, $v=\Delta u$ for some
$u\in\mathbb{R}^{\mathbb{Z}^{d}}$ with $\mathrm{supp}(u)\subset\Box_{L}$. And
so $\Delta G_{L}\Delta v=\Delta G_{L}\Delta^{2}u=\Delta u=v$. We also clearly
have $(\Delta G_{L}\Delta)(\mathcal{H}(\Box_{L}))=\\{0\\}$, and so the claim
$K^{\perp}_{L}=\Delta G_{L}\Delta$ is established.

For our main theorems, we need to estimate $K_{L}a$ for $a=\Gamma_{L}\ast
f_{L}$ where $f_{L}$ is a “nice” element of
$\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$. Specifically, we need to prove that
the $L^{2}$ mass of $K_{L}a$ does not concentrate in a boundary layer of
$\Box_{L}$.

One convenient condition on $f_{L}$ is that it is a rescaled version of a
continuum function $f$ for which the continuum bi-Laplacian problem has
regular solution. This is assumption (6.12) in Section 6.

For $f:[-1,1]^{d}\to\Box_{L}$, define $f_{L}(x)=L^{-d/2-2}f(x/L)$ as a mapping
$\Box_{L}\to\mathbb{R}$. Let $u:[-1,1]^{d}\to\Box_{L}$ solve

(7.7)
$\begin{cases}{\Delta}_{\mathbb{R}^{d}}^{2}u(x)=f(x),\quad&x\in(-1,1)^{d},\\\
u(x)=\partial_{n}u(x)=0,\quad&x\in\partial(-1,1)^{d},\end{cases}$

where $\partial_{n}$ denotes the normal derivative and
${\Delta}_{\mathbb{R}^{d}}$ denotes the continuum Laplacian. Define
$u_{L}:\Box_{L}\cup\partial^{2}\Box_{L}\to\mathbb{R}$ by
$u_{L}(x)=L^{-d/2+2}u(x/(L+2))$. Finally, let
$v:\mathrm{cl}_{2}\Box_{L}\to\mathbb{R}$ be defined by

(7.8) $\begin{cases}\Delta^{2}v(x)=f_{L}(x),\quad&x\in\Box_{L},\\\
v(x)=0,\quad&x\in\partial^{2}\Box_{L},\end{cases}$

where $\Delta^{2}$ is the discrete bi-Laplacian operator.

The following proposition shows that $\Delta v=\Delta G_{L}f_{L}$ and $\Delta
u_{L}$ are close in $L^{2}$ assuming $u$ is regular. Note that
$K_{L}=\text{Id}-\Delta G_{L}\Delta$, and, using the below proposition, we may
estimate $K_{L}\Gamma_{L}\ast f_{L}=\Gamma_{L}\ast f_{L}-\Delta v$ by
comparing to a continuum object.

######  Proposition 7.2.

Let $u,u_{L},v\in\mathbb{R}^{\mathrm{cl}_{2}\Box_{L}}$ be defined as above,
and assume $u$ is five times differentiable in $(-1,1)^{d}$. Then we have

(7.9) $\|\Delta u_{L}-\Delta v\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}\leq
CL^{-1/2}\|u\|_{C^{5}((-1,1)^{d})}$

for a constant $C$ dependent only on $d$.

###### Proof.

Let $w=u_{L}-v$. The proof will follow two steps: (1) we can bound $\|\Delta
w\|_{L^{2}}$ by $\|\Delta^{2}w\|_{L^{2}}$ and the boundary values of $w$ on
$\partial^{2}\Box_{L}$, and (2) $w$ is almost a biharmonic function with
Dirichlet boundary conditions.

By discrete integration by parts, we have

$\|\Delta w\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}=\left\langle
w,\Delta^{2}w\right\rangle_{\Box_{L}}+\sum_{x\in\partial^{2}\Box_{L},y\in\mathrm{cl}_{1}\Box_{L}}\Delta(x,y)w(x)\Delta
w(y)$

where $\Delta(x,y)$ is $1$ if $x\sim y$ and $-2d$ if $x=y$ and $0$ otherwise.
Thus

(7.10) $\displaystyle\|\Delta w\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}$
$\displaystyle\leq\|w\|_{L^{2}(\Box_{L})}\|\Delta^{2}w\|_{L^{2}(\Box_{L})}+C\|w\|_{L^{\infty}(\partial^{2}\Box_{L})}\sum_{y\in\mathrm{cl}_{1}\Box_{L}\setminus\Box_{L-1}}|\Delta
w(y)|$
$\displaystyle\leq\|w\|_{L^{2}(\Box_{L})}\|\Delta^{2}w\|_{L^{2}(\Box_{L})}+CL^{(d-1)/2}\|w\|_{L^{\infty}(\partial^{2}\Box_{L})}\|\Delta
w\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}.$

We extend $w$ by $0$ to $\Box_{L+10}$ (say) and apply the discrete Poincaré
inequality twice to get

(7.11) $\displaystyle\|w\|_{L^{2}(\Box_{L})}\leq
CL^{2}\|\nabla^{2}w\|_{L^{2}(\Box_{L+10})}$ $\displaystyle=CL^{2}\|\Delta
w\|_{L^{2}(\Box_{L+10})}$ $\displaystyle\leq CL^{2}\|\Delta
w\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}+CL^{2+(d-1)/2}\|w\|_{L^{\infty}(\partial^{2}\Box_{L})}.$

The last inequality follows from the fact that $|\Delta w(x)|\leq
4d\max_{y,\mathrm{dist}(y,x)\leq 1}|w(y)|$. Applying Young’s inequality to
(7.10) gives

(7.12) $\|\Delta
w\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}\leq\frac{\varepsilon^{2}}{L^{4}}\|w\|_{L^{2}(\Box_{L})}^{2}+C_{\varepsilon}L^{4}\|\Delta^{2}w\|_{L^{2}(\Box_{L})}+\varepsilon^{2}\|\Delta
w\|_{L^{2}(\mathrm{cl}_{1}(\Box_{L}))}^{2}+C_{\varepsilon}L^{d-1}\|w\|_{L^{\infty}(\partial^{2}\Box_{L})}^{2}.$

for any $\varepsilon>0$. Choosing $\varepsilon^{2}<1/C$ and using (7.11) shows

(7.13) $\|\Delta w\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}^{2}\leq
CL^{4}\|\Delta^{2}w\|_{L^{2}(\Box_{L})}^{2}+CL^{d-1}\|w\|_{L^{\infty}(\partial^{2}\Box_{L})}^{2},$

which completes step (1) of the proof.

We now bound $\Delta^{2}w$ in $\Box_{L}$. Let $L^{\prime}=L+2$ to lighten
notation. Applying repeatedly the fundamental theorem of calculus, we compute

(7.14) $\displaystyle\Delta u_{L}(x)$
$\displaystyle=L^{-d/2+2}(L^{\prime})^{-2}{\Delta}_{\mathbb{R}^{d}}u(x/L^{\prime})$
$\displaystyle\quad+L^{-d/2+2}(L^{\prime})^{-2}\sum_{i=1}^{d}\sum_{\sigma=\pm
1}\int_{0}^{1}(1-s)s\left(\nabla^{2}_{i,i}u\left(\frac{x+\sigma
se_{i}}{L^{\prime}}\right)-\nabla^{2}_{i,i}u\left(\frac{x}{L^{\prime}}\right)\right)ds.$

where $\nabla^{2}_{i,i}$ denotes the continuum second derivative in the
standard basis direction $e_{i}$. The term on the last line can be bounded by
$CL^{-d/2-1}\|\nabla_{\mathbb{R}^{d}}^{3}u\|_{L^{\infty}}$. Moreover, a
similar argument applied to the integrand shows the last term has second
discrete derivative bounded by
$L^{-d/2-3}\|\nabla_{\mathbb{R}^{d}}^{5}u\|_{L^{\infty}}$. So we can iterate
the above identity to see

$\Delta^{2}u_{L}(x)=L^{-d/2-2}(\Delta_{\mathbb{R}^{d}}^{2}u)(x/L)+\text{Error}=f_{L}(x)+\text{Error}_{x}$

where

$|\text{Error}_{x}|\leq
CL^{-d/2-3}\|\nabla_{\mathbb{R}^{d}}^{5}u\|_{L^{\infty}}+CL^{-d/2-3}\|\Delta_{\mathbb{R}^{d}}^{2}u\|_{L^{\infty}}+CL^{-d/2-3}\|\nabla_{\mathbb{R}^{d}}f\|_{L^{\infty}}$

for a constant $C$ independent of $L$ and $u$. Above, we have also changed
$L^{\prime}$ into $L$ and generated the corresponding error terms. Using
$\Delta^{2}v=f_{L}$, we conclude that

$\|\Delta^{2}w\|_{L^{2}(\Box_{L})}\leq\|\text{Error}\|_{L^{2}(\Box_{L})}\leq
CL^{-3}\left(\|\nabla_{\mathbb{R}^{d}}^{5}u\|_{L^{\infty}}+\|\Delta_{\mathbb{R}^{d}}^{2}u\|_{L^{\infty}}+\|\nabla_{\mathbb{R}^{d}}f\|_{L^{\infty}}\right).$

Next, since $w(x)=L^{-d/2+2}u(x/(L+2))$ for $x\in\partial^{2}\Box_{L}$ and by
the boundary conditions of $u$, we have

$\|w\|_{L^{\infty}(\partial^{2}\Box_{L})}\leq
CL^{-d/2}\|\nabla_{\mathbb{R}^{d}}^{2}u\|_{L^{\infty}}.$

This completes step (2) of the proof. We conclude from the estimates on $w$
and $\Delta^{2}w$ and (7.13) that

$\|\Delta u_{L}-\Delta v\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}\leq
CL^{-1/2}\|u\|_{C^{5}((-1,1)^{d})},$

and the proof is complete. ∎

We now bound the Bergman projection $K_{L}$ on a boundary layer.

######  Proposition 7.3.

Let $a=\Gamma_{L}\ast f_{L}\in\mathbb{R}^{\mathrm{cl}_{1}\Box_{L}}$ and
$f_{L}=L^{-d/2-2}f(x/L)$ for a function $f:(-1,1)^{d}\to\mathbb{R}$ such that
the solution $u$ to

(7.15)
$\begin{cases}\Delta_{\mathbb{R}^{d}}^{2}u(x)=f(x),\quad&x\in(-1,1)^{d},\\\
u(x)=\partial_{n}u(x)=0,\quad&x\in\partial(-1,1)^{d}.\end{cases}$

has $\|u\|_{C^{5}(-1,1)^{d}}\leq C$. Let
$\Lambda=\mathrm{cl}_{1}\Box_{L}\setminus\Box_{L-\ell}$ be a boundary layer of
width $\ell+1$. Then we have

(7.16)
$\|{K}_{L}a\|^{2}_{L^{2}(\Lambda)}+\|a\|_{L^{2}(\Lambda)}^{2}\leq\frac{C(\ell+1)}{L}(1+\mathbf{1}_{d=2}\log
L)$

for a constant $C$ dependent only on $d$ and $\|u\|_{C^{5}(-1,1)^{d}}$.

###### Proof.

We use $K_{L}a=a-\Delta G_{L}\ast f_{L}$ and treat these two terms separately.
By the triangle inequality, we have

$\|\Delta G_{L}\ast f_{L}\|_{L^{2}(\Lambda^{\prime})}\leq\|\Delta G_{L}\ast
f_{L}-\Delta u_{L}\|_{L^{2}(\mathrm{cl}_{1}\Box_{L})}+\|\Delta
u_{L}\|_{L^{2}(\Lambda^{\prime})}.$

for $u_{L}=L^{-d/2+2}u(x/(L+2))$. We can bound the first term using (7.9) by
$CL^{-1/2}$. Moreover, we have

$\Delta
u_{L}=L^{-d/2}(\Delta_{\mathbb{R}^{d}}u)(x/(L+2))+O(L^{-d/2-1})\|u\|_{C^{3}}\leq
CL^{-d/2}$

and so $\|\Delta u_{L}\|^{2}_{L^{2}(\Lambda^{\prime})}\leq C(\ell+1)L^{-1}$.

We can bound $\|a\|_{L^{2}(\Lambda)}$ from the following estimate on
$\|a\|_{L^{\infty}}$:

$|a(x)|\leq\sum_{y\in\Box_{L}}\Gamma_{L}(x,y)f_{L}(y)\leq\|f_{L}\|_{L^{\infty}}\sum_{y\in\Box_{L}}|\Gamma_{L}(x,y)|\leq
CL^{-d/2}(1+\mathbf{1}_{d=2}\log L).$

The last inequality follows from summing the inequality (7.1) in 7.1. Putting
all the above estimates together yields (7.16). ∎

We pause now to note that our condition on $f_{L}$ is certainly not sharp, and
can be improved in dimensions $2$ and $3$ on square/cubic domains using the
results of [15], one of which we now state. The following estimate on $G_{L}$
is readily seen to imply the result we need in the context of assumption