# Non-constant ground configurations in the disordered ferromagnet

Michal Bassan  School of Mathematical Sciences, Tel Aviv University, Tel Aviv,
Israel<EMAIL_ADDRESS>,  Shoni Gilboa  Department of
Mathematics, The Open University of Israel, Raanana, Israel<EMAIL_ADDRESS>and  Ron Peled  School of Mathematical Sciences, Tel Aviv University, Tel
Aviv, Israel. School of Mathematics, Institute for Advanced Study and
Department of Mathematics, Princeton University, New Jersey, United States.
<EMAIL_ADDRESS>
###### Abstract.

The disordered ferromagnet is a disordered version of the ferromagnetic Ising
model in which the coupling constants are non-negative quenched random. A
ground configuration is an infinite-volume configuration whose energy cannot
be reduced by finite modifications. It is a long-standing challenge to
ascertain whether the disordered ferromagnet on $\mathbb{Z}^{D}$ admits non-
constant ground configurations. We answer this affirmatively in dimensions
$D\geq 4$, when the coupling constants are sampled independently from a
sufficiently concentrated distribution. The obtained ground configurations are
further shown to be translation-covariant with respect to $\mathbb{Z}^{D-1}$
translations of the disorder.

##  1\. Introduction

Disordered ferromagnet. The Ising model is among the most basic models of
statistical physics. On the hypercubic lattice $\mathbb{Z}^{D}$, it is
described by the formal Hamiltonian

$H^{\text{Ising}}(\sigma):=-\sum_{\\{x,y\\}\in
E(\mathbb{Z}^{D})}\sigma_{x}\sigma_{y}$ (1.1)

on spin _configurations_ $\sigma\colon\mathbb{Z}^{D}\to\\{-1,1\\}$, where we
write $E(\mathbb{Z}^{D})$ for the edge set of $\mathbb{Z}^{D}$.

In this paper we study _the disordered ferromagnet_ (or _ferromagnetic random-
bond Ising model_), a version of the Ising model described by the formal
Hamiltonian

$H^{\eta}(\sigma):=-\sum_{\\{x,y\\}\in
E(\mathbb{Z}^{D})}\eta_{\\{x,y\\}}\sigma_{x}\sigma_{y},$ (1.2)

in which the coupling field $\eta=(\eta_{e})_{e\in E(\mathbb{Z}^{D})}$ is a
_non-negative_ quenched random field. We refer to $\eta$ as the (quenched)
_disorder_ and restrict throughout to the case that it is an _independent_
field. We first consider the _isotropic_ case, in which each $\eta_{e}$ is
sampled independently from the same probability distribution $\nu$, termed
_the disorder distribution_.

Ground configurations. Our primary interest is in the set of _ground
configrations_ , or zero-temperature configurations, of the disordered
ferromagnet111The terminology _ground states_ is commonly used in the
literature, but with two different meanings: while it may refer to the ground
configurations defined here, it may also refer to probability measures over
ground configurations. We use the terminology of ground configurations to
avoid this ambiguity.. Precisely, a configuration $\sigma$ is said to be a
ground configuration for the coupling field $\eta$ if it holds that
$H^{\eta}(\sigma)\leq H^{\eta}(\sigma^{\prime})$ for every configuration
$\sigma^{\prime}$ which differs from $\sigma$ in _finitely_ many places. To
make sense of the definition note that although $H^{\eta}(\sigma)$ is ill-
defined, as a non-convergent infinite sum, the difference
$H^{\eta}(\sigma)-H^{\eta}(\sigma^{\prime})$ is well defined whenever $\sigma$
and $\sigma^{\prime}$ differ in finitely many places.

As the coupling constants are non-negative, it is clear that the constant
configurations, $\sigma\equiv+1$ and $\sigma\equiv-1$, are ground
configurations of the disordered ferromagnet. We study the following basic
challenge:

######  Question 1.1.

Does the disordered ferromagnet admit non-constant ground configurations?

Ergodicity implies that existence of non-constant ground configurations is an
event of probability $0$ or $1$ for each dimension and disorder distribution
$\nu$. The case of one dimension ($D=1$) is simple: non-constant ground
configurations exist if and only if $\nu$ has an atom at the bottom of its
support. However, the question has remained open in all higher dimensions.
Much attention has been given to the two-dimensional case ($D=2$), where it is
shown that existence of non-constant ground configurations is equivalent to
the existence of infinite bigeodesics in a dual first-passage percolation
model (see, e.g., [N97]*Page 8). Such bigeodesics are believed not to exist
under mild assumptions on the disorder distribution, whence the answer to
Question 1.1 is expected to be negative in $D=2$. However, so far this is only
proved under assumptions on the model which are still unverified [A20], or for
other, related, models possessing a special integrable structure [BHS22,
BBS20, GJRA21, BS22]. The higher-dimensional case, and the closely-related
question of localization of domain walls of the disordered ferromagnet (see
Question 1.6 below), were studied in the physics and mathematics literature by
several authors, including Huse–Henley [HH85], Bovier–Fröhlich–Glaus [BFG86],
Fisher [F86], Bovier–Picco [BP91], Bovier–Külske [BK94, BK96], Wehr [W97] and
Wehr–Wasielak [WW16]. These studies predict, from non-rigorous methods [HH85,
BFG86, F86] and from rigorous studies of simplified interface models [BP91,
BK94] (see also Section 1.1), an affirmative answer to Question 1.1 in
dimensions $D\geq 4$ for sufficiently concentrated disorder distributions
$\nu$ (see Section 9 for a discussion of other settings). In this work we
confirm this prediction.

Existence result. To state our results, we need to give a precise meaning to
the idea that the disorder distribution $\nu$ is sufficiently concentrated. To
this end, we define a notion of “width” of the distribution, applicable to
either of the following two classes of disorder distributions:

  1. (1)

Compact support: If the distribution $\nu$ has compact support, we set

$\operatorname{diam}(\nu):=\min\\{b-a\,\colon\,b\geq a\text{ and
}\nu([a,b])=1\\}$ (1.3)

and otherwise we set $\operatorname{diam}(\nu):=\infty$.

  2. (2)

Lipschitz image of a Gaussian: If there exists a Lipschitz continuous
$f:\mathbb{R}\to[0,\infty)$ such that $\nu=f(N(0,1))$ (i.e., $\nu$ is the
push-forward through $f$ of the standard normal distribution) then set

$\operatorname{Lip}(\nu):=\inf\\{\operatorname{Lip}(f)\,\colon\,f:\mathbb{R}\to[0,\infty)\text{
is Lipschitz and }\nu=f(N(0,1))\\},$ (1.4)

with $\operatorname{Lip}(f):=\sup_{t\neq s}\frac{|f(t)-f(s)|}{|t-s|}$ being
the Lipschitz constant of $f$. Otherwise, set
$\operatorname{Lip}(\nu):=\infty$.

We then define the “width” of the disorder distribution $\nu$ by

$\operatorname{wid}(\nu):=\min\\{\operatorname{diam}(\nu),\operatorname{Lip}(\nu)\\}.$
(1.5)

We further restrict attention to disorder distributions whose support is
bounded away from zero and to this end we denote by

$\min(\operatorname{supp}(\nu)):=\max\\{\alpha\colon\nu([\alpha,\infty))=1\\}$
(1.6)

the smallest point of the support. Lastly, to avoid issues with uniqueness of
ground configurations in _finite volume_ , we assume that $\nu$ has no atoms.

The following is our main result.

######  Theorem 1.2.

There exists $c>0$ such that the following holds in dimensions $D\geq 4$.
Consider the (isotropic) disordered ferromagnet with disorder distribution
$\nu$. If $\min(\operatorname{supp}(\nu))>0$, $\nu$ has no atoms and

$\frac{\operatorname{wid}(\nu)}{\min(\operatorname{supp}(\nu))}\leq
c\frac{\sqrt{D}}{\log D}$ (1.7)

then the disordered ferromagnet admits non-constant ground configurations.

We make two remarks regarding the theorem:

  1. (1)

Condition (1.7) is our notion of $\nu$ being sufficiently concentrated. It is
invariant to dilations of $\nu$, as it should be since multiplying all
coupling constants $(\eta_{e})$ in the Hamiltonian (1.2) by a positive
constant does not change the set of ground configurations of the disordered
ferromagnet (or its Gibbs states).

The condition becomes easier to satisfy as the dimension $D$ increases. In
particular, the theorem shows that for any non-atomic disorder distribution
$\nu$ satisfying $\min(\operatorname{supp}(\nu))>0$ and
$\operatorname{wid}(\nu)<\infty$ there exists $D_{0}(\nu)\geq 4$ such that the
disordered ferromagnet with disorder distribution $\nu$ admits non-constant
ground configurations in all dimensions $D\geq D_{0}(\nu)$.

Condition (1.7) also becomes easier to satisfy if a positive constant is added
to $\nu$. More precisely, we see that for any non-atomic distribution $\mu$
supported in $[0,\infty)$ and satisfying $\operatorname{wid}(\mu)<\infty$ and
any $D_{0}\geq 4$ there exists $C_{0}(\mu,D_{0})\geq 0$ such that the
following holds: for all $C\geq C_{0}(\mu,D_{0})$, the disordered ferromagnet
with disorder distribution $\nu=\mu+C$ admits non-constant ground
configurations in all dimensions $D\geq D_{0}$ (where $\mu+C$ is the push-
forward of $\mu$ by the translation $x\mapsto x+C$).

As an example, there exists $C>0$ such that the disordered ferromagnet whose
disorder distribution is uniform on the interval $[C,C+1]$ admits non-constant
ground configurations in all dimensions $D\geq 4$.

  2. (2)

For $\operatorname{wid}(\nu)$ to be finite, the disorder distribution needs to
be either of compact support or a Lipschitz image of a Gaussian. The latter
possibility allows some distributions of unbounded support such as the
positive part of a Gaussian (plus a constant, so that
$\min(\operatorname{supp}(\nu))>0$), and can also lead to a smaller value of
$\operatorname{wid}(\nu)$ for some distributions of compact support.

Covariant ground configurations. Once the _existence_ of non-constant ground
configurations has been established, it is natural to turn to more refined
properties. In the non-disordered setup, a key role is played by the notion of
_invariant_ Gibbs states (e.g., translation-invariant Gibbs states). The
corresponding notion in the disordered setup is that of _covariant_ states
(going back at least to the pioneering work of Aizenman–Wehr [AW90], who
further introduced covariant _metastates_). We apply this notion to ground
configurations as follows: Let $G$ be a group of automorphisms of the lattice
$\mathbb{Z}^{D}$ (each $g\in G$ is composed of translations, rotations and
reflections). We naturally define

$g(\eta)_{\\{x,y\\}}:=\eta_{g\\{x,y\\}}=\eta_{\\{gx,gy\\}}\quad\text{and}\quad
g(\sigma)_{x}:=\sigma_{gx}$ (1.8)

for coupling fields $\eta$ and configurations $\sigma$. Let
$\mathcal{C}\subset[0,\infty)^{E(\mathbb{Z}^{D})}$ be a measurable set of
coupling fields which is $G$-invariant in the sense that
$g(\eta)\in\mathcal{C}$ for each automorphism $g\in G$ and coupling field
$\eta\in\mathcal{C}$. A _$G$ -covariant ground configuration_ defined on
$\mathcal{C}$ is a _measurable_ function
$T:\mathcal{C}\to\\{-1,1\\}^{\mathbb{Z}^{D}}$ which satisfies the following
properties for all $\eta\in\mathcal{C}$:

  1. (1)

$T(\eta)$ is a ground configuration for the disordered ferromagnet with
coupling field $\eta$.

  2. (2)

$T(g(\eta))=g(T(\eta))$ for each automorphism $g\in G$.

If, moreover, $T(\eta)$ is non-constant for all $\eta\in\mathcal{C}$ we say
that $T$ is a _non-constant_ $G$-covariant ground configuration defined on
$\mathcal{C}$.

When a disorder distribution $\nu$ has been specified (in the isotropic
setup), we may refer to a $G$-covariant ground configuration without reference
to its domain $\mathcal{C}$. It is then understood that the $(\eta_{e})$ are
sampled independently from $\nu$ and that the $G$-covariant ground
configuration is defined on some $G$-invariant $\mathcal{C}$ satisfying that
$\mathbb{P}(\eta\in\mathcal{C})=1$. The analogous comment applies to the
anistropic setup discussed below, when the two disorder distributions
$\nu^{\parallel}$ and $\nu^{\perp}$ have been specified.

Wehr–Wasielak [WW16] prove, in all dimensions $D\geq 1$, that when the
disorder distribution $\nu$ is non-atomic and has finite mean (or, more
generally, sufficiently light tail) then there are no non-constant
$\mathbb{Z}^{D}$-translation-covariant ground configurations
($\mathbb{Z}^{D}$-translation-covariant means that the group $G$ is the
translation group of $\mathbb{Z}^{D}$). More generally, they prove that
$\mathbb{Z}^{D}$-translation-covariant ground metastates (i.e., $G$-covariant
_probability distributions_ on ground configurations) must be supported on the
constant configurations.

Our next result shows that non-constant $G$-covariant ground configurations
may exist already when $G$ is one rank lower than the full translation group
of $\mathbb{Z}^{D}$.

######  Theorem 1.3.

Let $G^{D-1}$ be the group of automorphisms of $\mathbb{Z}^{D}$ which preserve
the last coordinate, i.e.,

$G^{D-1}:=\\{g\text{ automorphism of
$\mathbb{Z}^{D}$}\colon(gx)_{D}=x_{D}\text{ for all
}x=(x_{1},\ldots,x_{D})\\}.$ (1.9)

Under the assumptions of Theorem 1.2 (for a sufficiently small $c>0$), there
exists a non-constant $G^{D-1}$-covariant ground configuration.

The anisotropic disordered ferromagnet. Our proof naturally extends to an
_anisotropic_ setup, in which there is a distinguished lattice axis and the
coupling constants of edges in the direction of that axis are sampled from a
different disorder distribution. We next describe this setup (distinguishing
the $D$th axis).

It is sometimes convenient to identify edges of $\mathbb{Z}^{D}$ with their
dual plaquettes, i.e., to identify $\\{x,y\\}\in E(\mathbb{Z}^{D})$ with the
$(D-1)$-dimensional plaquette separating the unit cubes centered at $x$ and
$y$. With this identification in mind we partition the plaquettes into those
which are parallel to the hyperplane spanned by the first $D-1$ coordinate
axes and those which are perpendicular to it. Thus we define

$\begin{split}E^{\parallel}(\mathbb{Z}^{D}):=\\{\\{x,y\\}\in
E(\mathbb{Z}^{D})\colon x-y\in\\{-e_{D},e_{D}\\}\\},\\\
E^{\perp}(\mathbb{Z}^{D}):=\\{\\{x,y\\}\in E(\mathbb{Z}^{D})\colon
x-y\notin\\{-e_{D},e_{D}\\}\\},\end{split}$ (1.10)

where $e_{1},e_{2},\ldots,e_{D}$ are the standard unit vectors in
$\mathbb{Z}^{D}$.

By the _anisotropic disordered ferromagnet_ we mean that the disorder $\eta$
is sampled independently from two disorder distributions, $\nu^{\parallel}$
and $\nu^{\perp}$. Precisely, $(\eta_{e})_{e\in E(\mathbb{Z}^{D})}$ are
independent, with $\eta_{e}$ sampled from $\nu^{\parallel}$ when $e\in
E^{\parallel}(\mathbb{Z}^{D})$, and sampled from $\nu^{\perp}$ when $e\in
E^{\perp}(\mathbb{Z}^{D})$. The isotropic setup is recovered when
$\nu^{\parallel}=\nu^{\perp}$.

Our standard assumptions on the disorder distributions are

$\displaystyle\min(\operatorname{supp}(\nu^{\parallel}))>0,\quad$
$\displaystyle\operatorname{wid}(\nu^{\parallel})<\infty\quad\text{and
$\nu^{\parallel}$ has no atoms},$ (1.11)
$\displaystyle\min(\operatorname{supp}(\nu^{\perp}))>0,$
$\displaystyle\operatorname{wid}(\nu^{\perp})<\infty.$

We do not assume that $\nu^{\perp}$ has no atoms (and, in fact, the case that
$\nu^{\perp}$ is supported on a single point is of interest as it leads to the
disordered Solid-On-Solid model; see Remark 1.11).

In the anisotropic setup, condition (1.7) of Theorem 1.2 is replaced by
condition (1.14) below, which is based on the following quantity:

$\kappa(\nu^{\parallel},\nu^{\perp},d):=\left(\frac{1}{\underline{\alpha}^{\parallel}\underline{\alpha}^{\perp}}+\frac{1}{d(\underline{\alpha}^{\perp})^{2}}\right)\operatorname{wid}(\nu^{\parallel})^{2}+\frac{1}{(\underline{\alpha}^{\perp})^{2}}\operatorname{wid}(\nu^{\perp})^{2}$
(1.12)

where, for brevity, we denote

$\underline{\alpha}^{\parallel}:=\min(\operatorname{supp}(\nu^{\parallel}))\quad\text{and}\quad\underline{\alpha}^{\perp}:=\min(\operatorname{supp}(\nu^{\perp})).$
(1.13)

######  Theorem 1.4.

There exists $c_{0}>0$ such that the following holds in dimensions $D\geq 4$.
In the anisotropic disordered ferromagnet, suppose the disorder distributions
$\nu^{\parallel}$ and $\nu^{\perp}$ satisfy (1.11) and

$\kappa(\nu^{\parallel},\nu^{\perp},D-1)\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)\leq
c_{0}\frac{D}{(\log D)^{2}}.$ (1.14)

Then the anisotropic disordered ferromagnet admits non-constant ground
configurations.

Moreover, there exists a non-constant $G^{D-1}$-covariant ground
configuration, where $G^{D-1}$ is given by (1.9).

Theorem 1.2 and Theorem 1.3 arise as the special case of Theorem 1.4 in which
$\nu^{\parallel}=\nu^{\perp}$. We thus focus in the sequel on the anisotropic
setup.

Similarly to condition (1.7), we make the following remark regarding condition
(1.14). For any pair of disorder distributions $\nu^{\parallel}$ and
$\nu^{\perp}$ satisfying $\min(\operatorname{supp}(\nu^{\parallel}))>0$,
$\min(\operatorname{supp}(\nu^{\perp}))>0$,
$\operatorname{wid}(\nu^{\parallel})<\infty$ and
$\operatorname{wid}(\nu^{\perp})<\infty$, condition (1.14) will be satisfied
in either sufficiently high dimensions $D$, or in any fixed dimension $D\geq
4$ provided $\nu^{\perp}$ and $\nu^{\parallel}$ are replaced by
$\nu^{\perp}+C$ and $\nu^{\parallel}+C$, respectively, for a sufficiently
large $C$.

Dobrushin boundary conditions. Our proof of Theorem 1.4 proceeds through an
explicit construction of a non-constant ground configuration. Specifically, we
will show that the infinite-volume limit of the model with _Dobrushin boundary
conditions_ leads to such a ground configuration. In this section we explain
the relevant result, following required notation.

We first generalize the notion of ground configuration to allow boundary
conditions. Let $\Delta\subset\mathbb{Z}^{D}$ and
$\rho\colon\mathbb{Z}^{D}\to\\{-1,1\\}$. The configuration space in the domain
$\Delta$ with boundary conditions $\rho$ is

$\Omega^{\Delta,\rho}:=\\{\sigma\colon\mathbb{Z}^{D}\to\\{-1,1\\}\colon\text{$\sigma_{x}=\rho_{x}$
for $x\notin\Delta$}\\}.$ (1.15)

A _ground configuration in $\Omega^{\Delta,\rho}$_ (for the coupling field
$\eta$) is any $\sigma\in\Omega^{\Delta,\rho}$ with the property that
$H^{\eta}(\sigma)\leq H^{\eta}(\sigma^{\prime})$ for every configuration
$\sigma^{\prime}\in\Omega^{\Delta,\rho}$ which differs from $\sigma$ in
finitely many places. (noting, again, that the difference
$H^{\eta}(\sigma)-H^{\eta}(\sigma^{\prime})$ is then well defined). It is
possible for multiple ground configurations to exist even when $\Delta$ is
finite, though this will not be the case when $\eta$ is generic in a suitable
sense (see Section 4.1.1).

We proceed to discuss ground configurations with Dobrushin boundary
conditions. Assume henceforth that the dimension $D\geq 2$ and introduce the
convenient notation

$d:=D-1.$ (1.16)

We often tacitly use the convention to denote vertices of $\mathbb{Z}^{D}$ by
a pair $(v,k)$ with $v\in\mathbb{Z}^{d}$ and $k\in\mathbb{Z}$. By Dobrushin
boundary conditions we mean
$\rho^{\operatorname{Dob}}\colon\mathbb{Z}^{D}\to\\{-1,1\\}$ given by

$\rho^{\operatorname{Dob}}_{(v,k)}:=\operatorname{sign}(k-1/2)$ (1.17)

where $\operatorname{sign}$ denotes the sign function.

The following simple lemma shows that there is a unique ground configuration
with Dobrushin boundary conditions on infinite cylinders of the form
$\Lambda\times\mathbb{Z}$ for $\Lambda\subset\mathbb{Z}^{d}$ finite, under
suitable conditions on the disorder distributions.

######  Lemma 1.5.

In the anisotropic disordered ferromagnet, suppose the disorder distributions
$\nu^{\parallel}$ and $\nu^{\perp}$ satisfy (1.11). For each finite
$\Lambda\subset\mathbb{Z}^{d}$ there exists almost surely a _unique_ ground
configuration in
$\Omega^{\Lambda\times{\mathbb{Z}},\rho^{\operatorname{Dob}}}$, that we denote
$\sigma^{\eta,\Lambda,\operatorname{Dob}}$. Moreover,
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{x}=\rho^{\operatorname{Dob}}_{x}$
for all but finitely many $x\in\mathbb{Z}^{D}$.

The ground configuration $\sigma^{\eta,\Lambda,\operatorname{Dob}}$
necessarily contains an interface separating the $+1$ boundary values imposed
at the “top” of the cylinder $\Lambda\times\mathbb{Z}$ from the $-1$ boundary
values imposed at the “bottom” of the cylinder. To derive the existence of
non-constant ground configurations in the whole of $\mathbb{Z}^{D}$ from these
semi-infinite-volume ground configurations, the fundamental issue to be
understood is whether this interface remains localized in the sense of the
following question.

######  Question 1.6.

Is the interface under Dobrushin boundary conditions localized, uniformly in
the finite volume $\Lambda$? More precisely, is it the case that for each
$\varepsilon>0$ there exists a finite $\Delta\subset\mathbb{Z}^{D}$ such that

$\mathbb{P}(\sigma^{\eta,\Lambda,\operatorname{Dob}}\text{ is constant on
}\Delta)<\varepsilon\quad\text{for all finite
$\Lambda\subset\mathbb{Z}^{d}$}.$ (1.18)

For the set $\Delta$ in the question, one may have in mind, e.g., the set
$\\{(0,\ldots,0,j)\colon-k\leq j\leq k\\}$ with $k$ large.

A positive answer to Question 1.6 implies a positive answer to Question 1.1.
Indeed, by compactness, the distribution of the pair
$(\eta,\sigma^{\eta,\Lambda,\operatorname{Dob}})$ admits sub-sequential limits
along any sequence of finite volumes $(\Lambda_{n})$ increasing to
$\mathbb{Z}^{d}$. Any such limiting distribution is supported on pairs
$(\eta^{\prime},\sigma)$ with $\eta^{\prime}$ having the distribution of
$\eta$ and $\sigma$ an infinite-volume ground configuration for
$\eta^{\prime}$. A positive answer to the question ensures that $\sigma$ is
almost surely non-constant.

The answer to Question 1.6 is known to be negative for $D=2$ in the isotropic
setup under mild assumptions on the disorder distribution $\nu$; this is
essentially the Benjamini–Kalai–Schramm midpoint problem [BKS03], resolved
conditionally by Damron–Hanson [DH17], unconditionally by Ahlberg–Hoffman
[AH16] and quantitatively by Dembin, Elboim and the third author [DEP22]. The
following theorem, our main result on Dobrushin interfaces, proves that the
answer is positive in dimensions $D\geq 4$ for sufficiently concentrated
disorder distributions. Further discussion on Question 1.6, including the
possibility of a roughening transition in the disorder concentration, is in
Section 9.

######  Theorem 1.7 (Localization of Dobrushin interface).

There exist $c_{0},c>0$ such that the following holds in dimensions $D\geq 4$.
In the anisotropic disordered ferromagnet, suppose the disorder distributions
$\nu^{\parallel}$ and $\nu^{\perp}$ satisfy (1.11) and condition (1.14) holds
(with the constant $c_{0}$). Then for all finite
$\Lambda\subset\mathbb{Z}^{d}$ and all $(v,k)\in\mathbb{Z}^{D}$,

$\mathbb{P}\left(\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(v,k)}\neq\rho^{\operatorname{Dob}}_{(v,k)}\right)\leq\exp\left(-\frac{c}{d^{2}\kappa}|k|^{\frac{d-2}{d-1}}\right)$
(1.19)

with $\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d)$ defined in (1.12).
Moreover, for small $k$ we have the improved dependence on dimension: if
$|k|\leq\frac{4^{d-1}}{d}$ then

$\mathbb{P}\left(\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(v,k)}\neq\rho^{\operatorname{Dob}}_{(v,k)}\right)\leq\exp\left(-\frac{c}{\kappa}|k|\right).$
(1.20)

We add that the theorem resolves a version of [DG23, Open question 1] (see
Section 1.1.5). We also remark that the power of $|k|$ in (1.19) is probably
not optimal and may be increased with further optimization of the proof.

As explained above, Theorem 1.7 already suffices to deduce the existence of
non-constant ground configurations. To go further and prove the existence of a
non-constant _covariant_ ground configuration we employ the next result, which
proves the almost sure convergence of the semi-infinite-volume ground
configurations with Dobrushin boundary conditions to an infinite-volume limit.

######  Theorem 1.8 (Convergence).

Under the assumptions of Theorem 1.7 (for a sufficiently small $c_{0}>0$)
there exists a configuration $\sigma^{\eta,\operatorname{Dob}}$ such that for
every fixed sequence $(\Lambda_{n})$ of finite subsets of $\mathbb{Z}^{d}$
satisfying that $\Lambda_{n}\supset\\{-n,-n+1,\ldots,n\\}^{d}$ for each $n$,
almost surely,

$\text{for each $v\in\mathbb{Z}^{d}$ there exists $n_{0}$ such that
$\sigma^{\eta,\Lambda_{n},\operatorname{Dob}}|_{\\{v\\}\times\mathbb{Z}}\equiv\sigma^{\eta,\operatorname{Dob}}|_{\\{v\\}\times\mathbb{Z}}$
for all $n\geq n_{0}$}.$ (1.21)

The following is an immediate consequence of Theorem 1.7 and Theorem 1.8.

######  Corollary 1.9.

There exists $c>0$ such that the following holds under the assumptions of
Theorem 1.7 (for a sufficiently small $c_{0}>0$). The configuration
$\sigma^{\eta,\operatorname{Dob}}$ of Theorem 1.8 (possibly modified on a set
of zero probability) is a non-constant $G^{D-1}$-covariant ground
configuration, where $G^{D-1}$ is given by (1.9). In addition, for all
$(v,k)\in\mathbb{Z}^{D}$,

$\mathbb{P}\left(\sigma^{\eta,\operatorname{Dob}}_{(v,k)}\neq\rho^{\operatorname{Dob}}_{(v,k)}\right)\leq\exp\left(-\frac{c}{d^{2}\kappa}|k|^{\frac{d-2}{d-1}}\right)$
(1.22)

with $\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d)$ defined in (1.12).
Moreover, for small $k$ we have the improved dependence on dimension: if
$|k|\leq\frac{4^{d-1}}{d}$ then

$\mathbb{P}\left(\sigma^{\eta,\operatorname{Dob}}_{(v,k)}\neq\rho^{\operatorname{Dob}}_{(v,k)}\right)\leq\exp\left(-\frac{c}{\kappa}|k|\right).$
(1.23)

Theorem 1.4 is an immediate consequence of the last corollary.

Our techniques further allow to quantify the rate of convergence in Theorem
1.8 and to bound the rate of correlation decay in the infinite-volume
configuration $\sigma^{\eta,\operatorname{Dob}}$. We record these in our final
result.

######  Theorem 1.10.

There exist $C,c>0$ such that the following holds under the assumptions of
Theorem 1.7 (for a sufficiently small $c_{0}>0$). Let

$c(\nu^{\parallel},\nu^{\perp},d):=\frac{c}{\kappa
d^{2}}\left(\min\left\\{\frac{\alpha^{\parallel}}{\alpha^{\perp}},1\right\\}\right)^{\frac{d-2}{d-1}}$
(1.24)

using the notation (1.12) (with
$\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d))$ and (1.13). Let also
$\Lambda(k):=\\{-k,\dots,k\\}^{d}$ for integer $k\geq 0$.

  1. (1)

Let $L_{1}>L_{0}\geq 0$ integer. Let $\Lambda\subset\mathbb{Z}^{d}$ be a
finite subset containing $\Lambda(L_{1})$. Then

$\mathbb{P}\left(\sigma^{\eta,\operatorname{Dob}}|_{\Lambda(L_{0})\times\mathbb{Z}}\not\equiv\sigma^{\eta,\Lambda,\operatorname{Dob}}|_{\Lambda(L_{0})\times\mathbb{Z}}\right)\leq
C\exp\left(-c(\nu^{\parallel},\nu^{\perp},d)\left(L_{1}-L_{0}\right)^{\frac{d-2}{d}}\right).$
(1.25)

  2. (2)

Let $u,v\in\mathbb{Z}^{d}$ and $L\geq 0$ integer, and suppose
$\|u-v\|_{\infty}>2L$. Let
$f,g:\\{-1,1\\}^{\Lambda(L)\times\mathbb{Z}}\to[-1,1]$ be measurable. Then

$\operatorname{Cov}(f(\sigma^{\eta,\operatorname{Dob}}|_{(u+\Lambda(L))\times\mathbb{Z}}),g(\sigma^{\eta,\operatorname{Dob}}|_{(v+\Lambda(L))\times\mathbb{Z}}))\\\
\leq
C\exp\left(-c(\nu^{\parallel},\nu^{\perp},d)\left(\|u-v\|_{\infty}-2L\right)^{\frac{d-2}{d}}\right)$
(1.26)

where $\operatorname{Cov}(X,Y)$ denotes the covariance of the random variables
$X,Y$.

  3. (3)

The process $(\eta,\sigma^{\eta,\operatorname{Dob}})$ is $G^{d}$-invariant.
Moreover, define the $\mathbb{Z}^{d}$-tail sigma algebra of
$(\eta,\sigma^{\eta,\operatorname{Dob}})$ as the intersection of the sigma
algebras $(\mathcal{T}_{n})$, where $\mathcal{T}_{n}$ is generated by
$\sigma^{\eta,\operatorname{Dob}}|_{(\mathbb{Z}^{d}\setminus\Lambda(n))\times\mathbb{Z}}$
and by $(\eta_{e})$ for the edges $e=\\{(u,k),(v,\ell)\\}$ with
$\\{u,v\\}\cap(\mathbb{Z}^{d}\setminus\Lambda(n))\neq\emptyset$. Then the
$\mathbb{Z}^{d}$-tail sigma algebra of
$(\eta,\sigma^{\eta,\operatorname{Dob}})$ is trivial. In particular,
$(\eta,\sigma^{\eta,\operatorname{Dob}})$ is ergodic with respect to the group
of translations in the first $d$ coordinates.

Theorem 1.7 is proved in Section 4.3, Theorem 1.8, Corollary 1.9 and Theorem
1.10 are proved in Section 8 and Lemma 1.5 is proved in Appendix B.

###  1.1. Background

####  1.1.1. Localization predictions

The domain walls of the disordered Ising ferromagnet were studied by
Huse–Henley [HH85], Bovier–Fröhlich–Glaus [BFG86] and Fisher [F86] using
methods which are not mathematically rigorous. They predicted that the
interface with Dobrushin boundary conditions is rough in dimensions $2\leq
D\leq 3$, is localized in dimensions $D\geq 5$, and, for sufficiently
concentrated disorder, is also localized in dimension $D=4$.

####  1.1.2. Disordered Solid-On-Solid model

The following simplified model for the interface under Dobrushin boundary
conditions is considered by [HH85, BFG86]: The interface is described by a
(height) function $\varphi:\mathbb{Z}^{d}\to\mathbb{Z}$, whose energy is given
by the formal “disordered Solid-On-Solid (SOS)” Hamiltonian

$H^{\text{SOS},\zeta}(\varphi):=\sum_{\\{u,v\\}\in
E(\mathbb{Z}^{d})}|\varphi_{u}-\varphi_{v}|+\sum_{v\in\mathbb{Z}^{d}}\zeta_{v,\varphi_{v}}$
(1.27)

where $\zeta:\mathbb{Z}^{d}\times\mathbb{Z}\to\mathbb{R}$ is an environment
describing the quenched disorder. This model is obtained from the disordered
ferromagnet with two approximations: (i) It is assumed that the interface in
$D=d+1$ dimensions has _no overhangs_ , i.e., it may be described by a height
function above a $d$-dimensional base, (ii) all the coupling constants
corresponding to perpendicular plaquettes (i.e., all the $\eta_{\\{u,v\\}}$
for $\\{u,v\\}\in E^{\perp}(\mathbb{Z}^{D})$) are set equal (with the
normalization of (1.27) they are set equal to $1/2$ while
$\zeta_{v,k}:=2\eta_{\\{v,k\\},\\{v,k+1\\}}$).

######  Remark 1.11.

In fact, as part of the analysis of this paper, we prove the (possibly
surprising) fact that at zero temperature the no overhangs approximation (i)
is actually a consequence of the equal perpendicular couplings approximation
(ii) (see Lemma 3.1). Thus, our main results for the anistropic disordered
ferromagnet cover also the disordered SOS model (1.27), as the special case in
which the disorder distribution $\nu^{\perp}$ is a delta measure at $1/2$
(however, in this special case our proof may be greatly simplified due to the
no overhangs property).

A mathematically-rigorous study of the disordered SOS model (1.27) was carried
out by Bovier–Külske [BK94, BK96], following an earlier analysis by
Bovier–Picco [BP91] of a hierarchical version of the model (see also [BK92,
BK93]). It was shown in [BK94] that in each dimension $d\geq 3$, at low
temperature (including zero temperature), when the
$(\zeta_{v,\cdot})_{v\in\mathbb{Z}^{d}}$ are independent and identically
distributed, the sequence $k\mapsto\zeta_{v,k}$ is stationary for each $v$
(this is more general than being independent!) and the $\zeta_{v,k}$ are
sufficiently concentrated, the finite-volume Gibbs measures of the Hamiltonian
(1.27) converge, on a non-random sequence of volumes, to a limiting infinite-
volume Gibbs measure, $\zeta$-almost-surely. Some control of the fluctuations
of the infinite-volume Gibbs measure, at least at zero temperature, is also
provided [BK94, Proposition 3.6]. These results for the disordered SOS model
(1.27) thus have the flavor of our Theorem 1.7 and Theorem 1.8, though they,
on the one hand, apply also at low positive temperature and allow for more
general disorder distributions and, on the other hand, do not quantify the
dependence on the dimension $d$ (i.e., their sufficient concentration
requirement may become more stringent as $d$ increases). The work [BK94]
further discusses alternative assumptions on $\zeta$ relevant to the interface
in the _random-field_ Ising model (see also Section 9.2).

The behavior of the disordered SOS model (1.27) in the low dimensions $d=1,2$
was studied in [BK96] (using a method of Aizenman–Wehr [AW90]), who proved a
result showing a form of delocalization in these dimensions. Specifically,
they prove that, at all finite non-zero temperatures, when the $(\zeta_{v,k})$
are independently sampled from a distribution with positive variance which
either has no isolated atoms or has compact support, the model does not admit
translation-covariant and coupling-covariant metastates. Here, a metastate is
a measurable mapping from $\zeta$ to probability distributions over (infinite-
volume) Gibbs measures of the model, and the coupling covariance requirement
is that, for each finite $\Lambda\subset\mathbb{Z}^{d}$, the metastate changes
in a natural way under modification of
$(\zeta_{v,k})_{v\in\Lambda,k\in\mathbb{Z}}$.

####  1.1.3. Long-range order in the random-field Ising model

The localization proof of [BK94] in dimensions $d\geq 3$ is closely tied to
earlier developments on the problem of long-range order in the random-field
Ising model (see (1.28) below). Imry–Ma [IM75] predicted that at low
temperatures and weak disorder in dimensions $d\geq 3$, the random-field Ising
model retains the ferromagnetic ordered phase of the pure Ising model (and
that this does not occur when $d=2$). The prediction for $d=3$ was initially
challenged in the physics literature (e.g., [PS79]), but received support in
works of Chalker [C83] and Fisher–Fröhlich–Spencer [FFS84] and was finally
confirmed in the breakthrough works of Imbrie [I84, I85] and
Bricmont–Kupiainen [BK87, BK88]. The proof of [BK94] adapts the proof
technique of [BK88]. Recently, a short proof of the existence of an ordered
phase in the random-field Ising model was found by Ding–Zhuang [DZ23]. In this
paper, we use an adaptation of the Ding–Zhuang argument as one of the
ingredients in our proof of localization of the Dobrushin interface in the
disordered Ising ferromagnet (see Section 1.2 below).

####  1.1.4. Law of large numbers and large deviations of the ground energy

In dimensions $D>2$, following initial work by Kesten [K87] (and [ACCFR83,
CC86] in the case of $\\{0,1\\}$ coupling constants), there has been
significant advances in the understanding of the law of large numbers and
large deviations of the ground energy of the disordered ferromagnet (or the
maximal flow in a dual network) in various settings [Z00, Z18, T07, T08, W09,
RT10, CT11a, CT11b, CT11c, T14, CT14, D20, DT20, DT21].

####  1.1.5. Ground energy fluctuations

Dembin–Garban [DG23] studied the fluctuations of the ground energy of the
disordered ferromagnet in the cylinder
$\\{0,1,\ldots,L\\}^{d}\times\\{-H,{-H+1},\ldots,H\\}$, with boundary
conditions $-$ ($+$) imposed on the bottom face
$\\{0,1,\ldots,L\\}^{d}\times\\{-H\\}$ (on the top face
$\\{0,1,\ldots,L\\}^{d}\times\\{H\\}$) and free boundary conditions elsewhere
(equivalently, they studied minimal weight cutsets separating the top and
bottom faces of the cylinder). They obtain the first “sub-surface-order”
bounds (superconcentration) on the variance of the ground energy, proving that
when $H\geq CL$ then the variance is at most $C\frac{L^{d}}{\log L}$. The
result is proved for coupling fields sampled independently from a disorder
distribution supported on exactly two positive atoms. They mention that while
their result applies in any dimension $d\geq 1$, this may be due to the free
boundary conditions on the sides of the cylinder which allow a translation
freedom for the cutset. Indeed, in [DG23, Open question 1] they ask to prove
that the variance is of order $L^{d}$ when imposing Dobrushin boundary
conditions on the cylinder and $d$ is sufficiently high. Their Proposition 5.1
implies that this is the case if the corresponding Dobrushin interface is
localized. Our localization result, Theorem 1.7, thus answers a version of
[DG23, Open question 1] in the following sense: We prove that for any non-
atomic disorder distribution $\nu$ satisfying
$\min(\operatorname{supp}(\nu))>0$ and $\operatorname{wid}(\nu)<\infty$ there
exists $d_{0}(\nu)\geq 3$ such that the Dobrushin interface in the infinite
cylinder ($H=\infty$) is localized in dimensions $d\geq d_{0}(\nu)$ (with
$d_{0}(\nu)$ being $D-1$ for the minimal $D\geq 4$ for which (1.7), with $c>0$
a small universal constant, is satisfied). Moreover, there exists $C(\nu,d)>0$
(depending on the disorder distribution $\nu$ and the dimension $d$) such that
the same conclusion holds with any $H\geq
C(\nu,d)(\log(L))^{\frac{d-1}{d-2}}$, as the Dobrushin interface of the
infinite cylinder coincides with the interface in such a finite cylinder with
high probability, by the probability estimate (1.19).

####  1.1.6. Concentrated disorder distributions

Our results apply for sufficiently concentrated disorder distributions (see
conditions (1.7) and (1.14)). Related assumptions were also useful in the
study of the limit shape in first-passage percolation, by
Basdevant–Gouéré–Théret [BGT22] (for $\\{0,1\\}$ passage times) and by
Dembin–Elboim–Peled [DEP22, Theorem 1.5].

####  1.1.7. Number of ground configurations

Wehr [W97] proved that the number of ground configurations of the disordered
ferromagnet is two or infinity. The result applies for coupling fields sampled
independently from a non-atomic disorder distribution with finite mean. It
thus follows from our main result, Theorem 1.2, that there are infinitely many
ground configurations under the assumptions there (see also Section 9.3).

####  1.1.8. Translation-covariant ground metastates

As previously mentioned, Wehr–Wasielak [WW16] proved that
$\mathbb{Z}^{D}$-translation-covariant ground metastates must be supported on
the constant configurations when the disorder distribution $\nu$ is non-atomic
and has finite mean (or, more generally, has sufficiently light tail). This
result is applied in the discussion in Section 9.3.

####  1.1.9. The Dobrushin interface in other settings

Our localization result, Theorem 1.7, extends the seminal work of Dobrushin
[D72] to the setting of the zero-temperature disordered ferromagnet.
Dobrushin’s result has previously been extended to various (non-disordered)
settings, of which we mention the Widom–Rowlinson model [BLPO79, BLP79],
lattice Gauge theories [B84] (see also Section 9.5), the Falicov–Kimball model
[DMN00], percolation and the random-cluster model [GG02, CK03] and in studying
fine properties of the Dobrushin interface of the Ising model [GL20].
Alternative approaches for showing the existence of non-translation-invariant
Gibbs states include the correlation inequality approach of van Beijeren
[vB75] and the restricted reflection positivity approach of Shlosman–Vignaud
[SV07]. These alternative approaches do not seem to be applicable in our
disordered setting.

###  1.2. Overview of the proof

In this section we overview the proof of the localization of the Dobrushin
interface stated in Theorem 1.7.

The basic idea is to synthesize Dobrushin’s approach [D72] for proving the
localization of the Dobrushin interface in the _pure_ (i.e., non-disordered)
Ising model with the simple method for proving long-range order in the random-
field Ising model presented by Ding–Zhuang [DZ23]. As it turns out,
difficulties arise in this synthesis which necessitate the development of
additional tools.

####  1.2.1. The random-field Ising model

The random-field Ising model (RFIM) is the model on
$\sigma:\mathbb{Z}^{d}\to\\{-1,1\\}$ given by the formal Hamiltonian

$H^{\text{RFIM},\zeta}(\sigma):=-\sum_{\\{u,v\\}\in
E(\mathbb{Z}^{d})}\sigma_{u}\sigma_{v}-\lambda\sum_{v}\zeta_{v}\sigma_{v}$
(1.28)

where $(\zeta_{v})$ are independently sampled from, say, the standard Gaussian
distribution and $\lambda>0$ denotes the random-field strength. Imbrie [I84,
I85] established long-range order in the RFIM in dimensions $d\geq 3$ at zero
temperature and small $\lambda$ (and Bricmont–Kupiainen [BK87, BK88] proved
the analogous fact at low, positive temperatures). It is instructive to begin
our overview by describing Ding–Zhuang’s [DZ23] short approach to this. Let
$\sigma^{\zeta,L}$ be the ground configuration of the RFIM in
$\\{-L,\ldots,L\\}^{d}$ with $+1$ boundary conditions. Let us show that it is
unlikely that there exists some $A\subset\mathbb{Z}^{d}$, connected with
connected complement and containing the origin, such that
$\sigma^{\zeta,L}\equiv-1$ ($\sigma^{\zeta,L}\equiv+1$) on the interior
(exterior) vertex boundary of $A$. Suppose $A$ is such a set. Define a new
configuration and random field by

$\begin{split}&\sigma^{\zeta,L,A}_{v}:=\begin{cases}-\sigma^{\zeta,L}_{v}&v\in
A\\\ \sigma^{\zeta,L}_{v}&v\notin A\end{cases},\\\
&\zeta^{A}_{v}:=\begin{cases}-\zeta_{v}&v\in A\\\ \zeta_{v}&v\notin
A\end{cases}.\end{split}$ (1.29)

The discrete $\pm 1$ symmetry of the RFIM then leads to the energy gap

$H^{\text{RFIM},\zeta}(\sigma^{\zeta,L})-H^{\text{RFIM},\zeta^{A}}(\sigma^{\zeta,L,A})\geq
2|\partial A|$ (1.30)

where $\partial A$ is the edge boundary of $A$. This implies that also

$\operatorname{GE}^{\text{RFIM},\zeta,L}-\operatorname{GE}^{\text{RFIM},\zeta^{A},L}\geq
2|\partial A|$ (1.31)

where
$\operatorname{GE}^{\text{RFIM},\zeta,L}:=H^{\text{RFIM},\zeta}(\sigma^{\zeta,L})$
denotes the energy of the ground configuration in the random field $\zeta$.
The argument will be (eventually) concluded by proving that

$\mathbb{P}\left(\begin{subarray}{c}\exists A\subset\mathbb{Z}^{d}\text{
connected with connected complement, $0\in A$ and $|\partial A|=\ell$},\\\
|\operatorname{GE}^{\text{RFIM},\zeta,L}-\operatorname{GE}^{\text{RFIM},\zeta^{A},L}|\geq
2|\partial A|\end{subarray}\right)\leq
C_{d}\exp\left(-c_{d}\frac{\ell^{\frac{d-2}{d-1}}}{\lambda^{2}}\right)$ (1.32)

(with $C_{d},c_{d}>0$ depending only on $d$). Let us explain how (1.32) is
derived. First, observe that $\operatorname{GE}^{\text{RFIM},\zeta,L}$
satisfies the conditional concentration inequality (see Theorem 2.1 below)

$\mathbb{P}\left(\big{|}\operatorname{GE}^{\text{RFIM},\zeta,L}-\mathbb{E}(\operatorname{GE}^{\text{RFIM},\zeta,L}\,|\,\zeta|_{A^{c}})\big{|}\geq
t\,|\,\zeta|_{A^{c}}\right)\leq
C\exp\left(-c\frac{t^{2}}{\lambda^{2}|A|}\right)$ (1.33)

(with $C,c>0$ absolute constants). Next, note that $\zeta^{A}$ has the same
distribution as $\zeta$, even conditioned on $\zeta|_{A^{c}}$. Consequently,
the _difference of ground energies_ satisfies the same concentration
inequality (with different constants),

$\mathbb{P}\left(\big{|}\operatorname{GE}^{\text{RFIM},\zeta,L}-\operatorname{GE}^{\text{RFIM},\zeta^{A},L}\big{|}\geq
t\right)\leq C\exp\left(-c\frac{t^{2}}{\lambda^{2}|A|}\right).$ (1.34)

In particular, if $A$ was a _non-random_ set then we would have (using the
isoperimetric inequality $|A|\leq C_{d}|\partial A|^{d/(d-1)}$)

$\mathbb{P}(\operatorname{GE}^{\text{RFIM},\zeta,L}-\operatorname{GE}^{\text{RFIM},\zeta^{A},L}\geq
2|\partial A|)\leq C\exp\left(-c\frac{|\partial
A|^{2}}{\lambda^{2}|A|}\right)\leq C\exp\left(-c_{d}\frac{|\partial
A|^{\frac{d-2}{d-1}}}{\lambda^{2}}\right).$ (1.35)

Such an estimate, however, does not suffice to establish (1.32) via a union
bound, since the number of subsets $0\in A\subset\mathbb{Z}^{d}$, connected
with connected complement, which have $|\partial A|=\ell$ is at least
$c_{d}\exp(C_{d}\ell)$ (see also (6.27) for the converse bound). Instead, the
estimate (1.32) is derived from the concentration bound (1.34) using a coarse-
graining technique (or chaining argument) introduced by
Fisher–Fröhlich–Spencer [FFS84] in a closely-related, but different, context.
To this end one defines $A_{N}$, the $N$-coarse-grained version of
$A\subset\mathbb{Z}^{d}$, as the union of all cubes $B\subset\mathbb{Z}^{d}$,
of the form $v+\\{0,1,\ldots,N-1\\}^{d}$ with $v\in N\mathbb{Z}^{d}$, which
satisfy $|A\cap B|\geq\frac{1}{2}|B|$. Then, one writes the chaining expansion

$\operatorname{GE}^{\text{RFIM},\zeta,L}-\operatorname{GE}^{\text{RFIM},\zeta^{A},L}=\sum_{k=0}^{K-1}\left(\operatorname{GE}^{\text{RFIM},\zeta^{A_{2^{k+1}}},L}-\operatorname{GE}^{\text{RFIM},\zeta^{A_{2^{k}}},L}\right)$
(1.36)

where $K$ is chosen sufficiently large that $A_{2^{K}}=\emptyset$ (so that
$\zeta^{A_{2^{K}}}=\zeta$), and noting that $A_{2^{0}}=A_{1}=A$. A version of
the concentration inequality (1.34) is available (with the same proof) for any
two finite $A^{\prime},A^{\prime\prime}\subset\mathbb{Z}^{d}$,

$\mathbb{P}\left(\big{|}\operatorname{GE}^{\text{RFIM},\zeta^{A^{\prime}},L}-\operatorname{GE}^{\text{RFIM},\zeta^{A^{\prime\prime}},L}\big{|}\geq
t\right)\leq C\exp\left(-c\frac{t^{2}}{\lambda^{2}|A^{\prime}\Delta
A^{\prime\prime}|}\right).$ (1.37)

The idea of the coarse-graining technique is to apply the concentration bound
(1.37) to each of the terms on the right-hand side of (1.36) (with suitable
$t_{k}$ summing to $2\partial A$), using a union bound over the possible
$A_{2^{k}}$ and bounds for $|A_{2^{k}}\Delta A_{2^{k+1}}|$, for $0\leq k\leq
K-1$. The gain over the direct application (1.35) of (1.34) lies in the
smaller denominator in the right-hand side of the concentration inequality
(1.37) compared to (1.34), and the fact that the number of possibilities for
$A_{2^{k}}$ is greatly reduced as $k$ increases (roughly, $|\partial
A_{N}|\approx|\partial A|$ so that $A_{N}$ may be regarded as a set with
surface volume $|\partial A|/N^{d-1}$ after shrinking the lattice
$\mathbb{Z}^{d}$ by a factor $N$. This is complicated, however, by the fact
that $A_{N}$ need not be connected or have connected complement).

####  1.2.2. The disordered Solid-On-Solid model

It is instructive to first try and adapt the above approach to the disordered
SOS model (1.27), before discussing the disordered ferromagnet. The goal there
is to recover a version of the result of [BK94], showing that in dimensions
$d\geq 3$ when, say, the disorder $(\zeta_{v,k})$ is given by independent
Gaussians with _small variance_ , then there is localization of the ground
configuration $\varphi^{\zeta,L}$ in $\\{-L,\ldots,L\\}^{d}$ with zero
boundary values. To this end, it suffices to show that it is unlikely that
there exists an integer $m\geq 0$ and some $A\subset\mathbb{Z}^{d}$, connected
with connected complement and containing the origin, such that $\varphi\geq
m+1$ ($\varphi\leq m$) on the interior (exterior) vertex boundary of $A$. We
have checked (but do not provide full details here) that a proof may be
carried out very similarly to the RFIM case with the main difference being
that the discrete $\pm 1$ symmetry of the RFIM is now replaced by the discrete
translation symmetry of adding an integer constant to $\varphi$. Thus, instead
of (1.29), a new configuration and disorder are defined by

$\begin{split}&\varphi^{\zeta,L,A}:=\varphi^{\zeta,L}-1_{A},\\\
&\zeta^{A}_{(v,k)}:=\begin{cases}\zeta_{(v,k+1)}&v\in A\\\
\zeta_{(v,k)}&v\notin A\end{cases}\end{split}$ (1.38)

(where $1_{A}$ is the indicator function of $A$), leading to the energy gap

$H^{\text{SOS},\zeta}(\varphi^{\zeta,L})-H^{\text{SOS},\zeta^{A}}(\varphi^{\zeta,L,A})\geq|\partial
A|.$ (1.39)

While we do not enter into further detail, we remind that the disordered SOS
model may be seen as a special case of the anisotropic disordered ferromagnet;
see Remark 1.11.

####  1.2.3. The disordered ferromagnet

We proceed to overview our approach to proving Theorem 1.7 \- localization of
the Dobrushin interface in the disordered ferromagnet. While the approach
adapts several of the ideas appearing above, it is significantly more
complicated, essentialy due to the fact that the Dobrushin interface may have
overhangs (i.e., have several parallel interface plaquettes in the same
“column”). Below we describe the obstacles that arise and our methods for
overcoming them.

We work in dimension $D=d+1\geq 4$ under the assumptions of Theorem 1.7. For
finite $\Lambda\subset\mathbb{Z}^{d}$, we write
$\sigma^{\eta,\Lambda,\operatorname{Dob}}$ for the ground configuration in
$\Omega^{\Lambda\times{\mathbb{Z}},\rho^{\operatorname{Dob}}}$ as given by
Lemma 1.5, and we let $\operatorname{GE}^{\Lambda}(\eta)$ be its energy (i.e.,
the ground energy) in the coupling field $\eta$ (see (4.6) below for a precise
definition). Our goal is to show that, for $(v_{0},k_{0})\in\mathbb{Z}^{D}$,
the event

$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(v_{0},k_{0})}\neq\rho^{\operatorname{Dob}}_{(v_{0},k_{0})}$
(1.40)

is unlikely (with $\rho^{\operatorname{Dob}}$ defined in (1.17)).

#####  1.2.3.1 _Shifts and energy gap_

We aim to obtain an energy gap after a transformation of the configuration and
the disorder based on a discrete symmetry, in a similar way to (1.38) and
(1.39). The symmetry used is the translation of the $\mathbb{Z}^{D}$ lattice
along its last coordinate, but its use is more complicated than in the
disordered SOS model. The amount to translate by is encoded by a function
$\tau:\mathbb{Z}^{d}\to\mathbb{Z}$ having finite
$\operatorname{supp}(\tau):=\\{v\in\mathbb{Z}^{d}\colon\tau(v)\neq 0\\}$; we
call any such function a _shift_.

The shifted disorder $\eta^{\tau}$ is defined as follows: We fix, once and for
all, an arbitrary function $\iota:E(\mathbb{Z}^{d})\to\mathbb{Z}^{d}$ that
chooses an endpoint for each edge (i.e., $\iota(e)\in e$). Then

$\eta^{\tau}_{e}:=\begin{cases}\eta_{e+(0,\tau(u))}&e=\\{(u,k),(u,k+1)\\}\in
E^{\parallel}(\mathbb{Z}^{D}),\\\
\eta_{e+(0,\tau(\iota(\\{u,v\\})))}&e=\\{(u,k),(v,\ell)\\}\in
E^{\perp}(\mathbb{Z}^{D}),\end{cases}$ (1.41)

where $\\{x,y\\}+z=\\{x+z,y+z\\}$ for $x,y,z\in\mathbb{Z}^{D}$ (i.e., the
“column of disorders” above a base vertex $u$ is shifted by $\tau(u)$, and the
“column” above a base edge $\\{u,v\\}\in E(\mathbb{Z}^{d})$ is shifted by
$\tau(\iota(\\{u,v\\}))$; see also (4.7)) and (4.8)). Two useful features of
this definition are that $\iota(\\{u,v\\})$ is unimportant when
$\tau(u)=\tau(v)$ and that
$(\eta^{\tau_{1}})^{\tau_{2}}=\eta^{\tau_{1}+\tau_{2}}$ for
$\tau_{1},\tau_{2}$ shifts.

The action on the configuration $\sigma^{\eta,\Lambda,\operatorname{Dob}}$ is
more complicated, since a simple shift would not suffice to eliminate
overhangs. Instead, our definition involves an additional subset
$\tilde{A}\subset\mathbb{Z}^{d}$ which should be thought of as the projection
to $\mathbb{Z}^{d}$ of the overhangs and “interface walls” that we would like
to remove. We thus define, for $(u,k)\in\mathbb{Z}^{D}$,

$\sigma^{\eta,\Lambda,\operatorname{Dob},\tau,\tilde{A}}_{(u,k)}:=\begin{cases}\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(u,k+\tau(u))}&u\notin\tilde{A},\\\
\rho^{\operatorname{Dob}}_{(u,k)}&u\in\tilde{A}.\end{cases}$ (1.42)

The energy gap obtained from this definition is the difference

$\operatorname{GE}^{\Lambda}(\eta)-\operatorname{GE}^{\Lambda}(\eta^{\tau})\geq
H^{\eta}(\sigma^{\eta,\Lambda,\operatorname{Dob}})-H^{\eta^{\tau}}(\sigma^{\eta,\Lambda,\operatorname{Dob},\tau,\tilde{A}})$
(1.43)

and we aim for the right-hand side to consist exactly of (twice) the coupling
constants corresponding to the overhangs and walls of
$\sigma^{\eta,\Lambda,\operatorname{Dob}}$ above $\tilde{A}$ (more precisely
regarding the overhangs, for each $u\in\tilde{A}$ such that
$\\{u\\}\times\mathbb{Z}$ has multiple parallel interface plaquettes we gain
the coupling constants of all these plaquettes except the one between
$(u,\tau(u))$ and $(u,\tau(u)+1)$). To this end, we ensure the following
compatibility relations between $\tau$ and $\tilde{A}$,

If $\\{u,v\\}\in E(\mathbb{Z}^{d})$ and $\\{u,v\\}\not\subset\tilde{A}$ then
$\tau(u)=\tau(v)$. (1.44) If $u\in\tilde{A}$, $v\notin\tilde{A}$ and
$\\{u,v\\}\in E(\mathbb{Z}^{d})$ then
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(u,k+\tau(u))}=\rho^{\operatorname{Dob}}_{u,k}$
for $k\in\mathbb{Z}$. (1.45) If $u\in\tilde{A}$ then
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(u,\tau(u))}=-\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(u,\tau(u)+1)}$
(our construction also gives
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(u,\tau(u))}=-1$). (1.46)

A key role in our proof of Theorem 1.7 is thus played by defining $\tau$ and
$\tilde{A}$ as above so that: (i) a sufficient energy gap is generated when
(1.40) holds, and (ii) the shift $\tau$ is taken from a small enough class
(that we call _admissible shifts_ ; see Section 1.2.3.5 below) for which we
may develop suitable enumeration theorems for the required union bounds (there
is no need to also enumerate over $\tilde{A}$ as it does not appear on the
left-hand side of (1.43)).

#####  1.2.3.2 _Definition of $\tau$ and $\tilde{A}$_

Let $E\subset\Lambda$ (initially $E=\\{v_{0}\\}$ for the $v_{0}$ of (1.40).
However, later parts in our argument necessitate consideration of more general
$E$). We aim to define $\tilde{A}$ as the “projection to $\mathbb{Z}^{d}$ of
the places with overhangs and interface walls which surround $E$” and to
define $\tau$ in a compatible and admissible manner. Our definitions are
motivated by the ideas of Dobrushin [D72], to which we add a new result (Lemma
3.1) in order to define $\tau$ as an admissible shift. In fact, the absence of
bubbles (_finite_ connected components of spins of one sign) in our zero-
temperature setup allows us to simplify the approach of [D72] and we present a
self-contained treatment in Section 7 (with some inspiration from [PS23,
PS20]). This also yields an improved dependence on the dimension $d$. A brief
description of our construction follows.

First, we define a function
$I:\mathbb{Z}^{d}\to\mathbb{Z}\cup\\{{\rm``layered"}\\}$, which partitions
$\mathbb{Z}^{d}$ into different regions according to the height of the
Dobrushin interface, as follows: (i) $I(v)=k$ if
$\sigma^{\eta,\Lambda,\operatorname{Dob}}$ has a _unique_ sign change in
$\\{v\\}\times\mathbb{Z}$, with
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(v,k)}=-1$ and
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(v,k+1)}=1$, (ii)
$I(v)={\rm``layered"}$ if $\sigma^{\eta,\Lambda,\operatorname{Dob}}$ has
_multiple_ sign changes in $\\{v\\}\times\mathbb{Z}$. Define the set
$V_{\sigma^{\eta,\Lambda,\operatorname{Dob}}}\subset Z^{d}$ (the “projected
interface vertices”) as those $v$ satisfying that there exists an edge
$\\{u,v\\}\in E(\mathbb{Z}^{d})$ with either $I(u)\neq I(v)$ or
$I(u)=I(v)=``\text{layered}^{\prime\prime}$ (i.e., all layered vertices and
their neighbors and all non-layered vertices having a neighbor with a
different value of $I$). We then define $\tilde{A}$ to be the union of those
connected components of $V_{\sigma^{\eta,\Lambda,\operatorname{Dob}}}$ which
surround $E$ (i.e., those connected components $C$ for which some vertex of
$E$ lies in a finite connected component of $\mathbb{Z}^{d}\setminus C$).

Second, we define a “pre-shift”
$\tau_{0}:\mathbb{Z}^{d}\to\mathbb{Z}\cup\\{\text{``layered''}\\}$ as follows:
For $v\in\tilde{A}$ we set $\tau_{0}(v)=I(v)$. For $v\notin\tilde{A}$, we let
$B_{v}$ be the connected component of $v$ in
$\mathbb{Z}^{d}\setminus\tilde{A}$ and observe that $I$ is necessarily some
constant integer on the external vertex boundary of $B_{v}$; then we set
$\tau_{0}(v)$ equal to this constant (necessarily $\tau_{0}(v)=0$ if $B_{v}$
is infinite).

Third, the requisite shift $\tau$ is formed from $\tau_{0}$ by setting
$\tau(v)=\tau_{0}(v)$ whenever $\tau_{0}(v)\in\mathbb{Z}$ and choosing values
$\tau(v)\in\mathbb{Z}$ at those $v$ where $\tau_{0}(v)=\text{``layered''}$
(such $v$ are necessarily in $\tilde{A}$). While our choice is limited by the
compatibility relation (1.46), this still leaves it significant freedom; the
main limiting factor is our requirement that $\tau$ be an admissible shift. To
choose the values we use our Lemma 3.1, which gives a mechanism for modifying
the configuration $\sigma^{\eta,\Lambda,\operatorname{Dob}}$ on each connected
component of layered vertices into a configuration $\sigma^{\prime}$ with the
properties: (i) $\sigma^{\prime}$ has no overhangs, (ii) if
$\sigma^{\prime}_{(v,k)}=-\sigma^{\prime}_{(v,k+1)}$ at some $(v,k)$ then the
same holds for $\sigma^{\eta,\Lambda,\operatorname{Dob}}$ at $(v,k)$, and
(iii) $\sigma^{\prime}$ has a fewer or equal number of perpendicular interface
plaquettes than $\sigma^{\eta,\Lambda,\operatorname{Dob}}$. Choosing $\tau$ on
layered vertices to be the height of the unique sign change in
$\sigma^{\prime}$ is shown to yield the requisite admissible shift.

#####  1.2.3.3 _Chaining and concentration_

We have shown that on the event (1.40) there exists an admissible shift $\tau$
inducing an energy gap in (1.43) (and this gap is large if $k_{0}$ of (1.40)
is large). Consequently, it remains to prove Theorem 4.3 below, which states
that it is unlikely that there exists any admissible shift producing an energy
gap which is large in absolute value. To this end, motivated by the chaining
expansion (1.36) of the RFIM, our first step is to write

$\operatorname{GE}^{\Lambda}(\eta)-\operatorname{GE}^{\Lambda}(\eta^{\tau})=\sum_{k=0}^{K-1}\left(\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k+1}}})-\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})\right)$
(1.47)

where $\tau_{N}$ represents some notion of $N$-coarse-graining of $\tau$, with
$\tau_{2^{0}}=\tau_{1}=\tau$ and with $K$ large enough that
$\tau_{2^{K}}\equiv 0$ (so that $\eta^{\tau_{2^{K}}}=\eta$). We choose to
define $\tau_{N}:\mathbb{Z}^{d}\to\mathbb{Z}$ as a function which is constant
on cubes of the form $v+\\{0,1,\ldots,N-1\\}^{d}$ with $v\in N\mathbb{Z}^{d}$,
and equal on each such cube $B$ to the average of $\tau$ on $B$ rounded to the
closest integer (arbitrarily rounding $k+1/2$ to $k$ for integer $k$).
Significant effort is then devoted in Section 6.2 and Section 6.3 to develop
an enumeration theory (reminiscent of [FFS84]) for the number of possibilities
for $\tau_{N}$ according to the complexity of $\tau$ (complexity is discussed
in Section 1.2.3.5 below). The proof also introduces an extra “fine grained”
shift $\tau_{I}$, for $I\subset[d]=\\{1,\ldots,d\\}$, which “lies between”
$\tau$ and $\tau_{2}$ and is obtained by averaging and rounding $\tau$ on
boxes of the form $v+\\{0,1\\}^{I}\times\\{0\\}^{[d]\setminus I}$. This extra
ingredient allows our assumptions on the disorder distributions ((1.7) and
(1.14)) to become less restrictive as the dimension $d$ increases.

The next step following (1.47) is to obtain a concentration inequality for the
ground energy differences appearing in its right-hand side, similar to the
concentration inequality (1.37) of the RFIM. Here, however, lies a major
hurdle in our analysis, as the available inequality is significantly weaker
than the one available for the RFIM or the disordered SOS model. Let us
describe the inequality that we have. Let $\tau_{1},\tau_{2}$ be shift
functions. We introduce a version of the ground energy in which we minimize
over a restricted set of configurations: For $A\subset\mathbb{Z}^{d}$ and
$b^{\parallel},b^{\perp}\geq 0$, let
$\operatorname{GE}^{\Lambda,A,(b^{\parallel},b^{\perp})}(\eta)$ be the minimal
energy in the coupling field $\eta$ among configurations in
$\Omega^{\Lambda\times\mathbb{Z},\rho^{\operatorname{Dob}}}$ which have at
most $b^{\parallel}$ parallel plaquettes and at most $b^{\perp}$ perpendicular
plaquettes above $A$ in the Dobrushin interface (see (6.3) for a precise
definition). Then (see Lemma 6.1)

$\mathbb{P}\left(\big{|}\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{1}-\tau_{2}),(b^{\parallel},b^{\perp})}(\eta^{\tau_{1}})-\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{1}-\tau_{2}),(b^{\parallel},b^{\perp})}(\eta^{\tau_{2}})\big{|}\geq
t\right)\\\ \leq
C\exp\left(-c\frac{t^{2}}{\operatorname{wid}(\nu^{\parallel})^{2}b^{\parallel}+\operatorname{wid}(\nu^{\perp})^{2}b^{\perp}}\right),$
(1.48)

so that the concentration estimate deteriorates as $b^{\parallel}$ and
$b^{\perp}$ grow. Thus, in order to apply (1.48) to the $k$th term in (1.47)
(and successfully use a union bound over the possible $\eta^{\tau_{2^{k}}}$
and $\eta^{\tau_{2^{k+1}}}$) we need that for sufficiently small
$b_{k}^{\parallel}(s)$ and $b_{k}^{\perp}(s)$ (depending on $k$ and the energy
gap $s$; see Lemma 6.2),

$\begin{split}&\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{2^{k+1}}-\tau_{2^{k}}),(b_{k}^{\parallel}(s),b_{k}^{\perp}(s))}(\eta^{\tau_{2^{k}}}),\\\
&\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k+1}}})=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{2^{k+1}}-\tau_{2^{k}}),(b_{k}^{\parallel}(s),b_{k}^{\perp}(s))}(\eta^{\tau_{2^{k+1}}}).\end{split}$
(1.49)

However, we are not guaranteed that these equalities hold!

#####  1.2.3.4 _The maximal energy gap._

It remains to deal with the case that one of the inequalities (1.49) is
violated. Our strategy is to show that in this case there is a new admissible
shift $\tau^{\prime}$ inducing a significantly larger absolute energy gap
$|\operatorname{GE}^{\Lambda}(\eta)-\operatorname{GE}^{\Lambda}(\eta^{\tau^{\prime}})|$
than the shift $\tau$. The argument then proceeds by focusing on the
admissible shift with the _maximal_ energy gap and deducing that for that
shift all the equalities (1.49) hold.

To this end, suppose, e.g., that the first equality in (1.49) is violated. Set
$E:=\operatorname{supp}(\tau_{2^{k+1}}-\tau_{2^{k}})$. By definition, this
means that $\sigma^{\eta^{\tau_{2^{k}}},\Lambda,\operatorname{Dob}}$ either
has more than $b_{k}^{\parallel}(s)$ parallel interface plaquettes above $E$
or has more than $b_{k}^{\perp}(s)$ perpendicular interface plaquettes above
$E$. We may thus use the construction of Section 1.2.3.2, with
$\sigma^{\eta^{\tau_{2^{k}}},\Lambda,\operatorname{Dob}}$ and $E$, to define
$\tau^{\prime}$ and $\tilde{A}^{\prime}$ inducing a large energy gap
$\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})-\operatorname{GE}^{\Lambda}((\eta^{\tau_{2_{k}}})^{\tau^{\prime}})$.
If $b_{k}^{\parallel}(s)$ and $b_{k}^{\perp}(s)$ are not too small (see Lemma
6.2 for their value) then the new gap will indeed be much greater than the old
one, as we require. One difficulty, however, is that the new gap is induced
for the shifted disorder $\eta^{\tau_{2^{k}}}$ rather than for the original
disorder $\eta$. This is simply resolved though, since

$\begin{split}&\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})-\operatorname{GE}^{\Lambda}((\eta^{\tau_{2^{k}}})^{\tau^{\prime}})=\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})-\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}+\tau^{\prime}})\\\
&=\left(\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})-\operatorname{GE}^{\Lambda}(\eta)\right)-\left(\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}+\tau^{\prime}})-\operatorname{GE}^{\Lambda}(\eta)\right)\end{split}$
(1.50)

so that a large energy gap induced for the shifted disorder
$\eta^{\tau_{2^{k}}}$ implies a large energy gap in absolute value for the
original disorder (induced either by the shift $\tau_{2^{k}}$ or by the shift
$\tau_{2^{k}}+\tau^{\prime}$).

#####  1.2.3.5 _Admissible shifts._

The above argument may give rise to shift functions with a complicated
structure. Initially, given the input (1.40), we construct a relatively simple
shift $\tau$ (and set $\tilde{A}$) in order to remove the interface walls
surrounding the vertex $v_{0}$. However, as explained in Section 1.2.3.4
above, we may need to replace the shift $\tau$ by the shifts $\tau_{2^{k}}$ or
$\tau_{2^{k}}+\tau^{\prime}$ appearing in (1.50), and upon iterating this
procedure the shifts may become more and more complicated. We thus need to
define a class of shifts which, on the one hand, is broad enough to be closed
under such operations and, on the other hand, is narrow enough to enable
efficient enumeration (of the number of possibilities for the shift and its
coarse grainings), allowing the union bounds in the chaining argument to go
through. This is our motivation for defining in Section 4.1.3 the class of
_admissible_ shifts, which depends on the coupling field $\eta$. We measure
the complexity of a shift $\tau$ by its total variation
$\operatorname{TV}(\tau)$ (i.e., the $\ell_{1}$-norm of its gradient) and by a
quantity $R(\tau)$ that we call _trip entropy_ , which is the minimal length
of a path visiting all level components of $\tau$ (i.e., visiting all
connected components of level sets of $\tau$). Admissible shifts are then
defined as those that induce an energy gap for the coupling field $\eta$ that
is sufficiently large compared to the complexity of the shift. This definition
turns out to strike the requisite balance between broadness and narrowness.

##  2\. Notation, conventions and concentration results

We use the convention $\mathbb{N}:=\\{1,2,\ldots\\}$. For $k\in\mathbb{N}$, we
let $[k]:=\\{1,2,\ldots,k\\}$ and for any set $A$, let

$\binom{A}{k}:=\\{I\subseteq A\colon|I|=k\\}$

be the family of subsets of size $k$ of $A$.

For $x\in\mathbb{R}^{m}$ and $p\geq 1$ we let $\lVert
x\rVert_{p}=(\sum_{i=1}^{m}|x_{i}|^{p})^{1/p}$ be the standard $p$-norm.

Unless explicitly stated otherwise, all “geometric” notions in
${\mathbb{Z}}^{d}$ are with respect to the $\ell_{1}$ metric. In particular,
the (closed) ball of radius $r\geq 0$ around $a\in{\mathbb{Z}}^{d}$ is

${\mathcal{B}}_{r}(a):=\\{v\in{\mathbb{Z}}^{d}\colon\lVert v-a\rVert_{1}\leq
r\\},$

the diameter of a bounded set $A\subset{\mathbb{Z}}^{d}$ is ${\rm
diam}(A)=\max_{u_{1},u_{2}}\lVert u_{1}-u_{2}\rVert_{1}$, the distance from
$\omega\in{\mathbb{Z}}^{d}$ to a non-empty set $A\subset{\mathbb{Z}}^{d}$ is
${\rm dist}(\omega,A)=\min_{u\in A}\lVert\omega-u\rVert_{1}$ and the distance
between two non-empty sets $A,B\subset{\mathbb{Z}}^{d}$ is ${\rm
dist}(A,B)=\min_{u\in A,\,v\in B}\lVert u-v\rVert_{1}$; we say that
$u,v\in{\mathbb{Z}}^{d}$ are adjacent, and denote it by $u\sim v$, if $\lVert
u-v\rVert_{1}=1$; let
$E({\mathbb{Z}}^{d}):=\\{\\{u,v\\}\in\binom{{\mathbb{Z}}^{d}}{2}\colon u\sim
v\\}$; the _edge boundary_ of a set $A\subset{\mathbb{Z}}^{d}$ is

$\partial A:=\\{(u,v)\colon u\in A,\,v\in{\mathbb{Z}}^{d}\setminus A,\,u\sim
v\\}$

its _inner vertex boundary_ is

$\partial^{\rm in}A:=\\{u\in A\colon\exists v\in{\mathbb{Z}}^{d}\setminus
A\text{ such that }u\sim v\\},$

and its _outer vertex boundary_ is

$\partial^{\rm out}A:=\\{v\in\mathbb{Z}^{d}\setminus A\colon\exists u\in
A\text{ such that }u\sim v\\}.$

Denote by $\pi$ the projection from $\mathbb{Z}^{d+1}$ to $\mathbb{Z}^{d}$
defined by $\pi(x_{1},\dots,x_{d},x_{d+1})=(x_{1},\dots,x_{d})$.

The proofs of our main results require a concentration inequality for the
minimal energy of configurations of the disordered ferromagnet. According to
whether the disorder distributions have compact support or are Lipschitz
functions of a Gaussian, one of the following two inequalities will be used.

A function $f:D\to\mathbb{R}$, defined on a subset $D\subset\mathbb{R}^{n}$,
is said to be Lipschitz with constant $L>0$ if

$|f(x)-f(y)|\leq L\|x-y\|_{2},\qquad x,y\in D.$ (2.1)

######  Theorem 2.1 (Gaussian concentration inequality; see, e.g.
[BLM13]*Theorem 5.6).

Let $g_{1},\ldots,g_{n}$ be independent standard Gaussian random variables.
Suppose $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is Lipschitz with constant $L$.
Set $X:=f(g_{1},\ldots,g_{n})$. Then $\mathbb{E}(|X|)<\infty$ and for each
$t>0$,

$\mathbb{P}(|X-\mathbb{E}(X)|\geq t)\leq 2e^{-\frac{t^{2}}{2L^{2}}}.$

The theorem is part of the Gaussian concentration phenomenon as initiated by
Paul Lévy, Christer Borell, Tsirelson–Ibragimov–Sudakov and Maurey–Pisier.

A function $f:\mathbb{R}^{n}\to\mathbb{R}$ is called _quasi-convex_ if
$\\{x\in\mathbb{R}^{n}\colon f(x)\leq s\\}$ is a convex set for every
$s\in\mathbb{R}$.

######  Theorem 2.2 ([BLM13]*Theorem 7.12, going back to Johnson–Schechtman
[JS91], following Talagrand [T88]).

Let $z_{1},...,z_{n}$ be independent random variables taking values in the
interval $[0,1]$ and let $f:[0,1]^{n}\to\mathbb{R}$ be a quasi-convex function
which is also Lipschitz with constant $1$. Set $X:=f(z_{1},\ldots,z_{n})$.
Then, for each $t>0$,

$\mathbb{P}(|X-\operatorname{med}(X)|\geq t)\leq 4e^{-\frac{t^{2}}{4}}$ (2.2)

where $\operatorname{med}(X)$ is any median of $X$.

We remark that it is standard (and simple; see [MS86]*p. 142) that (2.2)
implies the same conclusion with the median replaced by the (necessarily
finite) expectation, in the form

$\mathbb{P}(|X-\mathbb{E}(X)|\geq t)\leq Ce^{-ct^{2}}$ (2.3)

for some universal constants $C,c>0$.

For our later use, it is convenient to deduce a unified result from the
previous two theorems, applicable to distributions of finite width. For a
random variable $W$ we set

$\operatorname{wid}(W):=\operatorname{wid}(\mathcal{L}(W))$ (2.4)

where $\mathcal{L}(W)$ is the distribution of $W$ (and
$\operatorname{wid}(\mathcal{L}(W))$ is defined by (1.5)).

######  Corollary 2.3.

There exist $C,c>0$ such that the following holds. Let $W_{1},\ldots,W_{n}$ be
independent random variables with $0<\operatorname{wid}(W_{i})<\infty$ for all
$i$. Suppose $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is a quasi-convex function
which is Lipschitz with constant $L>0$ in the sense of (2.1). Set

$X:=f\left(\frac{W_{1}}{\operatorname{wid}(W_{1})},\ldots,\frac{W_{n}}{\operatorname{wid}(W_{n})}\right).$
(2.5)

Then $\mathbb{E}(|X|)<\infty$ and for each $t>0$,

$\mathbb{P}(|X-\mathbb{E}(X)|\geq t)\leq Ce^{-c\frac{t^{2}}{L^{2}}}.$ (2.6)

We remark regarding the restriction $\operatorname{wid}(W_{i})>0$ that a
distribution $\nu$ with $\operatorname{wid}(\nu)=0$ is supported on a single
point. Indeed, this is clear if
$\operatorname{wid}(\nu)=\operatorname{diam}(\nu)$, while if
$\operatorname{wid}(\nu)=\operatorname{Lip}(\nu)$ then one may either argue
directly or deduce the fact from Theorem 2.1.

###### Proof of Corollary 2.3.

Let us assume, without loss of generality, that $0\leq k\leq n$ is such that
$\operatorname{wid}(W_{i})=\operatorname{Lip}(\mathcal{L}(W_{i}))$ for $1\leq
i\leq k$ while
$\operatorname{wid}(W_{i})=\operatorname{diam}(\mathcal{L}(W_{i}))$ for
$k+1\leq i\leq n$. By subtracting suitable constants from the $W_{i}$ with
$k+1\leq i\leq n$ we may further assume, without loss of generality, that each
such $W_{i}$ is supported on an interval of the form $[0,a_{i}]$ with
$\operatorname{diam}(W_{i})=a_{i}$. This implies that
$W_{i}/\operatorname{wid}(W_{i})\in[0,1]$ for $k+1\leq i\leq n$, as will be
required for using Theorem 2.2.

It suffices to prove that for any $t>0$ we have

$\mathbb{P}(|X-\mathbb{E}(X\,|\,W_{1},\ldots,W_{k})|\geq
t\,|\,W_{1},\ldots,W_{k})\leq Ce^{-c\frac{t^{2}}{L^{2}}}$ (2.7)

almost surely, and

$\mathbb{P}(|\mathbb{E}(X|W_{1},\ldots,W_{k})-\mathbb{E}(X)|\geq t)\leq
Ce^{-c\frac{t^{2}}{L^{2}}}.$ (2.8)

Inequality (2.7) follows Theorem 2.2, in the form (2.3). To see this, first
note that $f/L$ is a quasi-convex function which is Lipschitz with constant
$1$. Conclude that, for each fixed values of
$x_{1},\ldots,x_{k}\in\mathbb{R}$, the restricted function
$x_{k+1},\ldots,x_{n}\mapsto f(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{n})/L$
satisfies the same properties, and finally recall that
$W_{i}/\operatorname{wid}(W_{i})\in[0,1]$ for $k+1\leq i\leq n$.

We proceed to deduce inequality (2.8). Observe first that the average of a
Lipschitz function with respect to some of its variables is still a Lipschitz
function, with the same constant, of the remaining variables. In particular,
the function

$\tilde{f}(x_{1},\ldots,x_{k}):=\mathbb{E}\left(f\left(x_{1},\ldots,x_{k},\frac{W_{k+1}}{\operatorname{wid}(W_{k+1})},\ldots,\frac{W_{n}}{\operatorname{wid}(W_{n})}\right)\right)$
(2.9)

is Lipschitz with constant $L$. Fix $\varepsilon>0$. Let $g_{1},\ldots,g_{k}$
be a independent standard Gaussian random variables. Write, for $1\leq i\leq
k$, $W_{i}=h_{i}(g_{i})$ where $h_{i}:\mathbb{R}\to\mathbb{R}$ satisfies
$\operatorname{Lip}(h_{i})\leq\operatorname{Lip}(W_{i})(1+\varepsilon)$. It
follows that
$(y_{1},\ldots,y_{k})\mapsto\tilde{f}\left(\frac{h_{1}(y_{1})}{\operatorname{wid}(W_{1})},\ldots,\frac{h_{k}(y_{k})}{\operatorname{wid}(W_{k})}\right)$
is a Lipschitz function with constant $L(1+\varepsilon)$. Inequality (2.8)
then follows from Theorem 2.1, taking into account that $\varepsilon$ is
arbitrary. ∎

##  3\. Disorders which are constant on perpendicular plaquettes

Say that an Ising configuration $\sigma\colon\mathbb{Z}^{d+1}\to\\{-1,1\\}$ is
_interfacial_ if for every $v\in\mathbb{Z}^{d}$

$\text{$\lim_{k\to-\infty}\sigma_{(v,k)}=-1$ and
$\lim_{k\to\infty}\sigma_{(v,k)}=1$}.$ (3.1)

A configuration $\sigma$ is said to have _no overhangs_ if it is interfacial
and for every $v\in\mathbb{Z}^{d}$, there is a _unique_ $k$ for which
$\sigma_{(v,k)}=-\sigma_{(v,k+1)}$.

Recall the definition of $\Omega^{\Delta,\rho}$ and the definition of a ground
configuration in $\Omega^{\Delta,\rho}$ from (1.15). We use these here with
$\Delta=\Lambda\times\mathbb{Z}$ for a finite $\Lambda\subset\mathbb{Z}^{d}$
and a $\rho$ with no overhangs. Note that a ground configuration in
$\Omega^{\Lambda\times\mathbb{Z},\rho}$ may not exist for a general coupling
field $\eta$. However, such a ground configuration, which is moreover
interfacial, will exist if $\inf_{e\in E(\mathbb{Z}^{D})}\eta_{e}>0$ (see a
related discussion after Observation 4.1).

######  Lemma 3.1.

Let $\Lambda\subset\mathbb{Z}^{d}$ be finite and let
$\rho\colon\mathbb{Z}^{d+1}\to\\{-1,1\\}$ have no overhangs. Suppose the
coupling field $\eta\colon E(\mathbb{Z}^{d+1})\to[0,\infty)$ satisfies that
$\eta$ is _constant_ on $E^{\perp}(\mathbb{Z}^{d+1})$. Then for each
interfacial configuration $\sigma\in\Omega^{\Lambda\times\mathbb{Z},\rho}$
there exists $\sigma^{\prime}\in\Omega^{\Lambda\times\mathbb{Z},\rho}$ _with
no overhangs_ such that $H^{\eta}(\sigma^{\prime})\leq H^{\eta}(\sigma)$ and
whenever $\\{x,x+e_{d+1}\\}\in E(\mathbb{Z}^{d+1})$ is such that
$\sigma^{\prime}_{x}=-1$ and $\sigma^{\prime}_{x+e_{d+1}}=1$ then also
$\sigma_{x}=-1$ and $\sigma_{x+e_{d+1}}=1$.

Consequently, if $\eta$ is such that there exists an interfacial ground
configuration in $\Omega^{\Lambda\times\mathbb{Z},\rho}$, then there also
exists a ground configuration in $\Omega^{\Lambda\times\mathbb{Z},\rho}$ which
has no overhangs.

We note that in the terminology of (3.2) below, the lemma asserts that the
sign changes of $\sigma^{\prime}$ (having no overhangs) are contained in the
odd sign changes of $\sigma$.

The proof of the lemma uses the following preliminary definitions and
proposition.

Fix $\Lambda\subset\mathbb{Z}^{d}$ finite and a configuration $\rho$ with no
overhangs. We make the following definitions for an interfacial configuration
$\sigma\in\Omega^{\Lambda\times\mathbb{Z},\rho}$:

  1. (1)

The next definitions capture a notion of “odd” and “even” sign changes in
$\sigma$,

$\begin{split}&\text{OSC}(\sigma):=\\{\\{x,x+e_{d+1}\\}\in
E(\mathbb{Z}^{d+1})\colon\sigma_{x}=-1,\sigma_{x+e_{d+1}}=1\\},\\\
&\text{ESC}(\sigma):=\\{\\{x,x+e_{d+1}\\}\in
E(\mathbb{Z}^{d+1})\colon\sigma_{x}=1,\sigma_{x+e_{d+1}}=-1\\}.\end{split}$
(3.2)

Note that as $\rho$ has no overhangs and $\sigma$ is interfacial, then

     * •

for each $v\in\mathbb{Z}^{d}$ there are finitely many $k$ for which
$\\{(v,k),(v,k+1)\\}\in\text{OSC}(\sigma)$, with a unique such $k$ when
$v\in\mathbb{Z}^{d}\setminus\Lambda$.

     * •

for each $v\in\mathbb{Z}^{d}$, the number of
$\\{(v,k),(v,k+1)\\}\in\text{ESC}(\sigma)$ equals the number of
$\\{(v,k),(v,k+1)\\}\in\text{OSC}(\sigma)$ minus $1$. In particular, if
$\\{(v,k),(v,k+1)\\}\in\text{ESC}(\sigma)$ then $v\in\Lambda$.

  2. (2)

Let $\text{NESC}(\sigma)$ be the number of “adjacent even sign changes” in
$\sigma$, defined as the number of pairs
$\\{\\{(u,k),(u,k+1)\\},\\{(v,\ell),(v,\ell+1)\\}\\}\subset\text{ESC}(\sigma)$
satisfying that $\\{u,v\\}\in E(\mathbb{Z}^{d})$ and $k=\ell$.

  3. (3)

Define the number of perpendicular domain wall plaquettes above $\Lambda$ to
be

$D^{\Lambda}(\sigma):=|\\{\\{x,y\\}\in
E^{\perp}(\mathbb{Z}^{d+1})\colon\\{x,y\\}\cap(\Lambda\times\mathbb{Z})\neq\emptyset,\,\sigma_{x}\neq\sigma_{y}\\}|.$

Finally, we define a partial order on interfacial configurations in
$\Omega^{\Lambda\times\mathbb{Z},\rho}$ as follows: Say that
$\sigma^{\prime}<\sigma$ if

$D^{\Lambda}(\sigma^{\prime})\leq D^{\Lambda}(\sigma)$

and, either

$\text{OSC}(\sigma^{\prime})\subsetneq\text{OSC}(\sigma)$

or

$\text{OSC}(\sigma^{\prime})=\text{OSC}(\sigma)\text{ and
}\text{NESC}(\sigma^{\prime})>\text{NESC}(\sigma).$

The following proposition is the key step in proving Lemma 3.1.

######  Proposition 3.2.

Let $\sigma\in\Omega^{\Lambda\times\mathbb{Z},\rho}$ be interfacial. If
$\text{ESC}(\sigma)\neq\emptyset$ then there exists
$\sigma^{\prime}\in\Omega^{\Lambda\times\mathbb{Z},\rho}$ such that
$\sigma^{\prime}<\sigma$ (in particular, $\sigma^{\prime}$ is interfacial).

###### Proof.

Fix a $\sigma\in\Omega^{\Lambda\times\mathbb{Z},\rho}$ with
$\text{ESC}(\sigma)\neq\emptyset$. Fix some
$\\{(v_{0},k_{0}),(v_{0},k_{0}+1)\\}\in\text{ESC}(\sigma)$. Consider the set
of all positions which are directly below even sign changes at height $k_{0}$,

$\Delta:=\\{v\in\Lambda\colon\\{(v,k_{0}),(v,k_{0}+1)\\}\in\text{ESC}(\sigma)\\}.$

For a given height $k$, define the sum of the configuration surrounding
$\Delta$ at height $k$,

$S(k):=\sum_{v\in\partial^{\rm out}\Delta}\sigma_{(v,k)}.$

The definition of $\Delta$ implies that $S(k_{0})\leq S(k_{0}+1)$. Thus,
either $S(k_{0})\leq 0$ or $S(k_{0}+1)\geq 0$ (or both). Let us assume without
loss of generality that $S(k_{0})\leq 0$ as the other case can be treated
analogously.

Define $k_{1}\leq k_{0}$ to be the smallest integer with the following
property: For all $k_{1}\leq k\leq k_{0}$ it holds that

$\begin{split}&\sigma_{(v,k)}=\sigma_{(v,k_{0})}=1\quad\text{for
$v\in\Delta$},\\\ &\sigma_{(v,k)}\leq\sigma_{(v,k_{0})}\quad\text{for
$v\in\partial^{\rm out}\Delta$}.\end{split}$

The definition implies, in particular, that

$S(k)\leq S(k_{0})\leq 0$ (3.3)

for all $k_{1}\leq k\leq k_{0}$. Finally, define a configuration
$\sigma^{\prime}$ as follows

$\sigma^{\prime}_{(v,k)}=\begin{cases}-1&v\in\Delta,k_{1}\leq k\leq k_{0},\\\
\sigma_{(v,k)}&\text{otherwise}.\end{cases}$

The inequality (3.3) implies that $D^{\Lambda}(\sigma^{\prime})\leq
D(\sigma)$. Moreover, the definition of $k_{1}$ implies that either
$\text{OSC}(\sigma^{\prime})\subsetneq\text{OSC}(\sigma)$ or
$\text{OSC}(\sigma^{\prime})=\text{OSC}(\sigma)$ and
$\text{NESC}(\sigma^{\prime})>\text{NESC}(\sigma)$. Thus,
$\sigma^{\prime}<\sigma$, as we wanted to prove. ∎

A repeated use of Proposition 3.2 yields the following corollary.

######  Corollary 3.3.

For every interfacial $\sigma\in\Omega^{\Lambda\times\mathbb{Z},\rho}$, there
is an interfacial configuration $\sigma^{\prime}$ that has a unique sign
change above every vertex (i.e., $\sigma^{\prime}$ has no overhangs), with
$\sigma$ having the same sign change at the same height, and
$D^{\Lambda}(\sigma^{\prime})\leq D^{\Lambda}(\sigma)$, i.e.,

$\lvert\\{\\{x,y\\}\in
E^{\perp}(\mathbb{Z}^{d+1})\colon\\{x,y\\}\cap(\Lambda\times\mathbb{Z})\neq\emptyset,\,\sigma^{\prime}_{x}\neq\sigma^{\prime}_{y}\\}\rvert\\\
\leq\lvert\\{\\{x,y\\}\in
E^{\perp}(\mathbb{Z}^{d+1})\colon\\{x,y\\}\cap(\Lambda\times\mathbb{Z})\neq\emptyset,\,\sigma_{x}\neq\sigma_{y}\\}\rvert.$
(3.4)

###### Proof.

If $\text{ESC}(\sigma)=\emptyset$ then $\sigma$ has no overhangs and we are
done. Otherwise, apply Proposition 3.2 iteratively to produce a sequence
$\sigma_{m}<\sigma_{m-1}<\cdots<\sigma_{0}=\sigma$ of configurations in
$\Omega^{\Lambda\times\mathbb{Z},\rho}$, with
$\text{ESC}(\sigma_{m})=\emptyset$ (the iterations necessarily terminate at
some finite $m\geq 1$ since the number of odd sign changes above each
$v\in\mathbb{Z}^{d}$ cannot increase and the number of even sign changes above
each $v\in\Lambda$ is no larger than the number of odd sign changes above
$v$). Then, $\sigma_{m}$ has no overhangs, and by the definition of the
partial order, $\text{OSC}(\sigma_{m})\subset\text{OSC}(\sigma)$ and
$D^{\Lambda}(\sigma_{m})\leq D^{\Lambda}(\sigma)$. ∎

Lemma 3.1 immediately follows from Corollary 3.3.

###### Proof of Lemma 3.1.

Let $\eta\colon E(\mathbb{Z}^{d+1})\to[0,\infty)$ satisfy the properties in
the lemma. Let $\sigma$ be an interfacial configuration in
$\Omega^{\Lambda\times\mathbb{Z},\rho}$. Let $\sigma^{\prime}$ be the
configuration guaranteed by Corollary 3.3. Since $\eta$ is constant on
$E^{\perp}(\mathbb{Z}^{d+1})$, it follows from (3.4) that
$H^{\eta}(\sigma^{\prime})\leq H^{\eta}(\sigma)$. ∎

##  4\. Stability of the ground energy under shifts of the disorder and a
deduction of Theorem 1.7

In this section we present our main technical result, Theorem 4.3 below, which
bounds the probability that certain “admissible shifts of the disorder” lead
to a significant change in the energy of the ground configuration under
Dobrushin boundary conditions. Our main localization theorem, Theorem 1.7,
follows by combining Theorem 4.3 with the fact, stated in Lemma 4.4 below,
that admissible shifts inducing large energy changes necessarily exist
whenever the interface in the ground configuration deviates (in prescribed
locations) from the flat interface. Theorem 4.3 will also be instrumental in
the proof of Theorem 1.8 (presented in Section 8) on the convergence of the
semi-infinite-volume ground configurations in the infinite-volume limit.

We begin in Section 4.1 with required definitions, continue in Section 4.2
with the statement of our main technical result and finally deduce Theorem 1.7
in Section 4.3.

###  4.1. Preliminaries

This section contains the required definitions of ground energies, shifts and
their action on the disorder, and admissibility of shifts.

####  4.1.1. Coupling fields, energies and ground configurations

Generic coupling fields. We often work with coupling fields $\eta$ whose
values on all edges are uniformly bounded from $0$. In addition, in order to
ensure uniqueness of finite-volume ground configurations we ask that the
coupling field $\eta$ satisfies the assumption

$\sum_{i=1}^{k}s_{i}\eta_{f_{i}}\neq 0,\quad
k\in\mathbb{N},\\{s_{i}\\}_{i=1}^{k}\subseteq\\{-1,1\\},\\{f_{i}\\}_{i=1}^{k}\subset
E(\mathbb{Z}^{d+1})\text{ and }\\{f_{i}\\}_{i=1}^{k}\nsubseteq
E^{\perp}(\mathbb{Z}^{d+1}).$ (4.1)

This is captured with the following notation: Given
$\alpha^{\parallel},\alpha^{\perp}\in(0,\infty)$ let

$\mathcal{D}(\alpha^{\parallel},\alpha^{\perp}):=\left\\{\eta\colon
E(\mathbb{Z}^{d+1})\to(0,\infty)\colon\begin{subarray}{c}\eta_{e}\in(\alpha^{\parallel},\infty)\text{
for }e\in E^{\parallel}(\mathbb{Z}^{d+1}),\\\
\eta_{e}\in(\alpha^{\perp},\infty)\text{ for }e\in
E^{\perp}(\mathbb{Z}^{d+1}),\\\ \eta\text{ satisfies \eqref{eq:no finite
binary zero combination}}\end{subarray}\right\\}.$ (4.2)

In addition, we set

$\mathcal{D}:=\bigcup_{\alpha^{\parallel},\alpha^{\perp}\in(0,\infty)}\mathcal{D}(\alpha^{\parallel},\alpha^{\perp}).$
(4.3)

Such disorders arise, almost surely, under our assumption that
$\nu^{\parallel}$ and $\nu^{\perp}$ satisfy (1.11).

Semi-infinite volumes and an equivalent Hamiltonian. Most of our analysis will
be done on the semi-infinite volume $\Lambda\times\mathbb{Z}$ for a finite
$\Lambda\subset\mathbb{Z}^{d}$. Recalling the definition of the Dobrushin
boundary conditions $\rho^{\operatorname{Dob}}$ from (1.17) and of
$\Omega^{\Delta,\rho}$ from (1.15), we work on the following configuration
space

$\Omega^{\Lambda,\operatorname{Dob}}:=\left\\{\sigma\in\Omega^{\Lambda\times{\mathbb{Z}},\rho^{\operatorname{Dob}}}\colon\sigma=\rho^{\operatorname{Dob}}\
\text{at all but finitely many points of $\Lambda\times\mathbb{Z}$}\right\\}.$
(4.4)

Now, given also a coupling field $\eta\colon E(\mathbb{Z}^{d+1})\to[0,\infty]$
and a finite $\Lambda\subset\mathbb{Z}^{d}$ define the Hamiltonian

$\mathcal{H}^{\eta,\Lambda}(\sigma):=\sum_{\begin{subarray}{c}\\{x,y\\}\in
E(\mathbb{Z}^{d+1})\\\
\\{x,y\\}\cap(\Lambda\times\mathbb{Z})\neq\emptyset\end{subarray}}\eta_{\\{x,y\\}}(1-\sigma_{x}\sigma_{y})=2\sum_{\begin{subarray}{c}\\{x,y\\}\in
E(\mathbb{Z}^{d+1})\\\
\\{x,y\\}\cap(\Lambda\times\mathbb{Z})\neq\emptyset\end{subarray}}\eta_{\\{x,y\\}}1_{\sigma_{x}\neq\sigma_{y}}$
(4.5)

and note that it is well defined on $\Omega^{\Lambda,\operatorname{Dob}}$.

From the following observation it follows that the minimizers of the
Hamiltonian $\mathcal{H}^{\eta,\Lambda}$ in
$\Omega^{\Lambda,\operatorname{Dob}}$ coincide with the ground configurations
discussed in Lemma 1.5. It is proved in appendix B for completeness.

######  Observation 4.1.

Let $\sigma,\sigma^{\prime}\in\Omega^{\Lambda,\operatorname{Dob}}$, and
$\eta:E(\mathbb{Z}^{D})\rightarrow\left[0,\infty\right)$. The following holds

$H^{\eta}(\sigma)-H^{\eta}(\sigma^{\prime})=\mathcal{H}^{\eta,\Lambda}(\sigma)-\mathcal{H}^{\eta,\Lambda}(\sigma^{\prime}).$

We note that when $\eta\in\mathcal{D}$ then there is a unique minimizer of
$\mathcal{H}^{\eta,\Lambda}$ in $\Omega^{\Lambda,\operatorname{Dob}}$. Indeed,
there are only finitely many configurations in
$\Omega^{\Lambda,\operatorname{Dob}}$ whose energy is lower than
$\mathcal{H}^{\eta,\Lambda}(\rho^{\operatorname{Dob}})$, and no two of them
have equal energy by (4.1). With a slight abuse of notation, we will denote
this unique minimizer by $\sigma^{\eta,\Lambda,\operatorname{Dob}}$, noting
that it coincides with the minimizer of Lemma 1.5 under the assumptions there.
We will use the terminology _ground energy_ to refer to the energy of the
minimizing configuration. We thus define, for each $\eta\in\mathcal{D}$ and
finite $\Lambda\subset\mathbb{Z}^{d}$,

$\operatorname{GE}^{\Lambda}(\eta):=\mathcal{H}^{\eta,\Lambda}(\sigma^{\eta,\Lambda,\operatorname{Dob}}).$
(4.6)

####  4.1.2. Shifts of the coupling field

Shifts and shifted coupling fields. We use the term _shift_ to denote any
function $\tau\colon\mathbb{Z}^{d}\to\mathbb{Z}$ which equals zero except at
finitely many vertices. We denote the (finite) support of $\tau$ by

$\operatorname{supp}(\tau):=\\{v\in\mathbb{Z}^{d}\colon\tau(v)\neq 0\\}.$

We occasionally refer to the $\ell_{1}$ norm of a shift,
$\|\tau\|_{1}:=\sum_{v\in\mathbb{Z}^{d}}|\tau_{v}|$. The set of all shifts
will be denoted by $\mathcal{S}$.

We define an operation of shifts on coupling fields $\eta$: first fix an
arbitrary choice function $\iota:E({\mathbb{Z}}^{d})\to{\mathbb{Z}}^{d}$ that
chooses for each edge one of its endpoints, i.e., $\iota(e)\in e$ for every
$e\in E({\mathbb{Z}}^{d})$; the shifted coupling field $\eta^{\tau}$ is
defined by shifting the “column of disorders” above a base vertex $u$ by
$\tau(u)$, and a similar shift up for “columns” above any base edge
$\\{u,v\\}$ such that $\iota(\\{u,v\\})=u$. Precisely, given a shift $\tau$
and a disorder $\eta\colon E({\mathbb{Z}^{d+1}})\to[0,\infty)$, define
$\eta^{\tau}\colon E({\mathbb{Z}^{d+1}})\to[0,\infty)$ as follows: for every
$u\in\mathbb{Z}^{d}$ and $k\in{\mathbb{Z}}$,

$\eta^{\tau}_{\\{(u,k),(u,k+1)\\}}:=\eta_{\\{(u,k+\tau(u)),(u,k+1+\tau(u))\\}},$
(4.7)

and for every $\\{u,v\\}\in E(\mathbb{Z}^{d})$ and $k\in{\mathbb{Z}}$,

$\eta^{\tau}_{\\{(u,k),(v,k)\\}}:=\eta_{\\{(u,k+\tau(\iota(\\{u,v\\}))),(v,k+\tau(\iota(\\{u,v\\})))\\}}.$
(4.8)

Note that if $\tau(u)=\tau(v)$ for adjacent $u,v\in{\mathbb{Z}}^{d}$, then for
every $k\in{\mathbb{Z}}$,

$\eta^{\tau}_{\\{(u,k),(v,k)\\}}=\eta_{\\{(u,k+\tau(u)),(v,k+\tau(u))\\}}=\eta_{\\{(u,k+\tau(v)),(v,k+\tau(v))\\}}.$
(4.9)

Changes to the ground energy. Of central importance in our arguments will be
the change in ground energy induced by shifts of the coupling field. This is
captured by the following definition. For each $\eta\in\mathcal{D}$, finite
$\Lambda\subset\mathbb{Z}^{d}$ and shifts $\tau,\tau^{\prime}$ we set

$G^{\eta,\Lambda}(\tau,\tau^{\prime}):=\operatorname{GE}^{\Lambda}(\eta^{\tau^{\prime}})-\operatorname{GE}^{\Lambda}(\eta^{\tau}).$
(4.10)

We also abbreviate

$G^{\eta,\Lambda}(\tau):=G^{\eta,\Lambda}(\tau,0)=\operatorname{GE}^{\Lambda}(\eta)-\operatorname{GE}^{\Lambda}(\eta^{\tau}).$
(4.11)

With these definitions, for any shifts $\tau_{1},\ldots,\tau_{k}$ we have the
telescopic sum

$G^{\eta,\Lambda}(\tau_{1})=\sum_{i=1}^{k-1}G^{\eta,\Lambda}(\tau_{i},\tau_{i+1})+G(\tau_{k}).$
(4.12)

####  4.1.3. Enumeration of shifts, admissible shifts and the maximal
energetic change

The counting of various classes of shifts plays an important role in our
arguments (the shifts play a role somewhat analogous to that of contours in
the classical Peierls argument). To facilitate it, we need a way to succinctly
describe shifts. To this end, the following notations regarding a shift $\tau$
are handy:

  * •

The _total variation_ of $\tau$ is defined as

$\operatorname{TV}{(\tau}):=\sum_{\\{u,v\\}\in
E(\mathbb{Z}^{d})}\lvert\tau(u)-\tau(v)\rvert.$

  * •

A _level component_ of a shift $\tau$ is a connected set on which $\tau$ is
constant and which is not strictly contained in another set with this property
(i.e., a connected component of $\tau^{-1}(k)$ for some $k$). Denote the
collection of all level components of $\tau$ by $\mathcal{LC}(\tau)$.

  * •

A finite sequence $(v_{i})_{i\geq 0}$ of points in $\mathbb{Z}^{d}$ with
$v_{0}=0$ is a _root sequence_ for a collection $\mathcal{F}$ of sets in
${\mathbb{Z}}^{d}$ if there is a point of $\\{v_{i}\\}_{i\geq 0}$ in every set
in $\mathcal{F}$. We further define the _trip entropy_ $R(\tau)$ of the shift
$\tau$ as

$R(\tau):=\min\left\\{\sum_{i\geq 1}\|v_{i}-v_{i-1}\|_{1}\colon(v_{i})_{i\geq
0}\text{ is a root sequence for }\mathcal{LC}(\tau)\right\\}.$

Similarly, define the trip entropy $R(E)$ of a set
$E\subseteq{\mathbb{Z}}^{d}$ as

$R(E):=\min\left\\{\sum_{i\geq
1}\|v_{i}-v_{i-1}\|_{1}\colon\begin{subarray}{c}(v_{i})_{i\geq 0}\text{ is a
root sequence for the collection}\\\ \text{of connected components of
}E\end{subarray}\right\\}.$

These definitions are put to use in estimating the number of shifts in
Proposition 6.3.

We next define a restricted class of shifts, depending on $\eta$, that we term
_admissible shifts_ (while restricted, the set of admissible shifts is still
defined in a broad enough fashion to contain all the shifts arising in our
proof). Very roughly, the class is defined as those shifts whose action on the
coupling field induces a sufficiently large energetic change to the ground
energy (as defined in (4.11)) to compensate for the number of shifts in the
class. Here, the first notion one has for the number of shifts with given
parameters is that coming from our later Proposition 6.3. However, we will see
later that this notion will need to be further refined in our argument, where
we will also need to take care of the number of coarse grainings (and fine
grainings) of our shifts. The need to account also for these more refined
counting problems lies at the heart of our choice for the definition of root
sequence above and the precise definition of admissible shifts below.

Given a coupling field $\eta\colon E(\mathbb{Z}^{d+1})\to[0,\infty]$, finite
$\Lambda\subset\mathbb{Z}^{d}$ and positive
$\alpha^{\parallel},\alpha^{\perp}$, the class of
$(\alpha^{\parallel},\alpha^{\perp})$-admissible shifts is defined by

$\operatorname{\mathcal{AS}}^{\eta,\Lambda}(\alpha^{\parallel},\alpha^{\perp}):=\left\\{\tau\in{\mathcal{S}}\colon|G^{\eta,\Lambda}(\tau)|\geq\max\left\\{\frac{\alpha^{\perp}}{2}\operatorname{TV}(\tau),\min\\{\alpha^{\parallel},\alpha^{\perp}\\}\frac{d}{200}R(\tau)\right\\}\right\\}.$

Lastly, we give a notation to the maximal change in the ground energy that is
induced by an $(\alpha^{\parallel},\alpha^{\perp})$-admissible shift,

$\operatorname{MG}^{\eta,\Lambda}(\alpha^{\parallel},\alpha^{\perp}):=\sup_{\tau\in\operatorname{\mathcal{AS}}^{\eta,\Lambda}(\alpha^{\parallel},\alpha^{\perp})}|G^{\eta,\Lambda}(\tau)|.$
(4.13)

Our proof will make use of the fact that
$\operatorname{MG}^{\eta,\Lambda}(\alpha^{\parallel},\alpha^{\perp})<\infty$,
almost surely, under suitable assumptions on the disorder distributions. This
is implied by the following lemma, proved in appendix B.

######  Lemma 4.2.

In the anisotropic disordered ferromagnet, suppose the disorder distributions
$\nu^{\parallel}$ and $\nu^{\perp}$ satisfy (1.11). Then, for any finite
$\Lambda\subset\mathbb{Z}^{d}$ and positive
$\alpha^{\parallel},\alpha^{\perp}$ we have that
$|\operatorname{\mathcal{AS}}^{\eta,\Lambda}(\alpha^{\parallel},\alpha^{\perp})|<\infty$
almost surely.

###  4.2. Stability of the ground energy

We proceed to state our main technical result, Theorem 4.3 below. It gives a
quantitative bound on the probability that there exists an admissible shift
whose action on the disorder yields a large change in the ground energy.

######  Theorem 4.3.

There exist constants $c_{0},c,C>0$ such that the following holds. In the
anisotropic disordered ferromagnet, suppose the disorder distributions
$\nu^{\parallel}$ and $\nu^{\perp}$ satisfy (1.11). Let
$\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d)$ be as in definition (1.12). Let
$\underline{\alpha}^{\parallel}$ and $\underline{\alpha}^{\perp}$ be the
minimums of the supports of $\nu^{\parallel}$ and $\nu^{\perp}$, as in (1.13).
Let $D=d+1\geq 4$ and suppose that condition (1.14) holds (with the constant
$c_{0}$). Then the following holds for all finite
$\Lambda\subset\mathbb{Z}^{d}$ and $t>0$,

$\mathbb{P}\left(\operatorname{MG}^{\eta,\Lambda}(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})\geq
t\right)\leq C\exp\left(-\frac{c}{\kappa
d^{2}}\left(\frac{t}{\underline{\alpha}^{\perp}}\right)^{\frac{d-2}{d-1}}\right).$
(4.14)

Moreover, for small $t$ we have the improved dependence on dimension: if
$t<\underline{\alpha}^{\perp}\frac{4^{d-1}}{d}$ then

$\mathbb{P}\left(\operatorname{MG}^{\eta,\Lambda}(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})\geq
t\right)\leq C\exp\left(-\frac{ct}{\kappa\underline{\alpha}^{\perp}}\right).$
(4.15)

The theorem will be proven at the end of subsection 5.2.

###  4.3. Deduction of Theorem 1.7

The following deterministic lemma shows that if the interface of the ground
configuration is not flat around the origin then there necessarily exists an
admissible shift whose action on the coupling field induces a large change in
the ground energy.

######  Lemma 4.4.

Let $\eta\in\mathcal{D}(\alpha^{\parallel},\alpha^{\perp})$ for some
$\alpha^{\parallel},\alpha^{\perp}>0$ and let $\Lambda\subset\mathbb{Z}^{d}$
be a finite subset. If
$\sigma^{\eta,\Lambda,\operatorname{Dob}}_{(0,k)}\neq\rho^{\operatorname{Dob}}_{(0,k)}$
then there exists a shift
$\tau\in\operatorname{\mathcal{AS}}^{\eta,\Lambda}({\alpha}^{\parallel},{\alpha}^{\perp})$
for which

$G^{\eta,\Lambda}(\tau)\geq 2|k|{\alpha}^{\perp}.$

The lemma is proved in Section 7.5.

Theorem 1.7 follows as a direct consequence of Lemma 4.4 and Theorem 4.3.
First, it suffices to establish the inequality (1.19) at the vertex $v=0$,
since the choice of $\Lambda$ in Theorem 1.7 is arbitrary. Then, inequality
(1.19) with $v=0$ follows directly by combining Lemma 4.4 with (4.14).

##  5\. Coarse and fine grainings of shifts and their use in proving the
stability of the ground energy

In this section we take the first step towards proving Theorem 4.3, describing
a form of “chaining” argument on the set of admissible shifts which is used to
control their effect on the ground energy. The notion of coarse grainings of
shifts which lies at the heart of our chaining argument is modelled after a
similar graining method for sets which was introduced by
Fisher–Fröhlich–Spencer [FFS84] in their discussion of the domain walls in the
random-field Ising model.

###  5.1. Coarse and fine grainings of shifts

The chaining argument is based on the notions of coarse and fine grainings of
shifts that we now describe.

Given a partition $\mathcal{P}$ of $\mathbb{Z}^{d}$ into finite sets and a
shift $\tau$, we write $\tau_{\mathcal{P}}$ for the shift obtained by
averaging the value of $\tau$ on each partition element of $\mathcal{P}$ and
rounding to the closest integer. Precisely, we set

$\tau_{\mathcal{P}}(v):=\left[\frac{1}{|P(v)|}\sum_{u\in P(v)}\tau(u)\right]$
(5.1)

where we write $P(v)$ for the unique partition element of $\mathcal{P}$
containing $v$, and where $\left[a\right]$ is the rounding of $a$ to the
nearest integer, with the convention $\left[k+\frac{1}{2}\right]=k$ for
$k\in\mathbb{Z}$.

We make use of two special cases of the above definition:

  * •

Coarse graining: Given an integer $N\geq 1$, we use the notation
$\tau_{N}:=\tau_{\mathcal{P}_{N}}$ (as in (5.1)), with $\mathcal{P}_{N}$ is
the following partition into discrete cubes of side length $N$,

$\mathcal{P}_{N}=\\{Q_{N}(v)\\}_{v\in N\mathbb{Z}^{d}}\quad\text{where}\quad
Q_{N}(v):=v+\\{0,1,\ldots,N-1\\}^{d}.$

  * •

Fine graining: Given a subset of the coordinates $I\subset[d]$, we use the
notation $\tau_{I}:=\tau_{\mathcal{P}_{I}}$ (as in (5.1)), with
$\mathcal{P}_{I}$ is the following partition into discrete boxes with side
length $2$ in the directions in $I$ and side length $1$ in the directions in
$[d]\setminus I$,

$\mathcal{P}_{I}=\\{Q_{I}(v)\\}_{v\in(2\mathbb{Z})^{I}\times\mathbb{Z}^{[d]\setminus
I}}\quad\text{where}\quad Q_{I}(v):=v+\\{0,1\\}^{I}\times\\{0\\}^{[d]\setminus
I}.$

###  5.2. The chaining argument

We work under the assumptions of Theorem 4.3. Precisely, let the disorder
$\eta$ be sampled as in the anisotropic disordered ferromagnet, with the
disorder distributions $\nu^{\parallel}$ and $\nu^{\perp}$ satisfying (1.11).
Let $D\geq 4$ and suppose that condition (1.14) holds with a constant $c>0$
chosen sufficiently small for the arguments below.

Fix a finite $\Lambda\subset\mathbb{Z}^{d}$. For brevity, we will write $G$
for $G^{\eta,\Lambda}$, $\operatorname{\mathcal{AS}}$ for
$\operatorname{\mathcal{AS}}^{\eta,\Lambda}(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})$
(recall (1.13)), _admissible_ for
$(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})$-_admissible_ and
$\operatorname{MG}$ for
$\operatorname{MG}^{\eta,\Lambda}(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})$.
First, since $\operatorname{MG}<\infty$ almost surely by Lemma 4.2, we have

$\mathbb{P}(\operatorname{MG}\geq
t)=\sum_{k=0}^{\infty}\mathbb{P}\left(\operatorname{MG}\in[t2^{k},t2^{k+1})\right).$

Next, for any $s>0$, integer $K\geq 1$, integer $1\leq r\leq d$, any positive
$(\gamma_{j})_{j\in[K]\cup\\{(0,r),(r,1)\\}}$ with
$\gamma_{(0,r)}+\gamma_{(r,1)}+\sum_{1\leq k\leq K}\gamma_{k}\leq 1$ and any
function $I_{\tau}$ which assigns a subset of $[d]$ of size $r$ to each shift
$\tau$, we have the chaining argument (noting that the supremum is realized in
(4.13) due to Lemma 4.2, and also recalling (4.12))

$\displaystyle\mathbb{P}($
$\displaystyle\operatorname{MG}\in[s,2s))=\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\left\\{\max_{\tau\in\operatorname{\mathcal{AS}}}|G(\tau)|\geq
s\right\\}\right)$ $\displaystyle=$
$\displaystyle\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\left\\{\max_{\tau\in\operatorname{\mathcal{AS}}}|G(\tau,\tau_{I_{\tau}})+G(\tau_{I_{\tau}},\tau_{2})+\sum_{k=1}^{K-1}G(\tau_{2^{k}},\tau_{2^{k+1}})+G(\tau_{2^{K}})|\geq
s\right\\}\right)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\left\\{\max_{\tau\in\operatorname{\mathcal{AS}}}|G(\tau,\tau_{I_{\tau}})|\geq\gamma_{(0,|I_{\tau}|)}s\right\\}\right)$
(5.2a) $\displaystyle+\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\left\\{\max_{\tau\in\operatorname{\mathcal{AS}}}|G(\tau_{I_{\tau}},\tau_{2})|\geq\gamma_{(|I_{\tau}|,1)}s\right\\}\right)$
(5.2b) $\displaystyle+\sum_{k=1}^{K-1}\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\left\\{\max_{\tau\in\operatorname{\mathcal{AS}}}|G(\tau_{2^{k}},\tau_{2^{k+1}})|\geq\gamma_{k}s\right\\}\right)$
(5.2c) $\displaystyle+\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\left\\{\max_{\tau\in\operatorname{\mathcal{AS}}}|G(\tau_{2^{K}})|\geq\gamma_{K}s\right\\}\right).$
(5.2d)

The following notion will become useful for bounding the terms (5.2a) and
(5.2b) from above. A set of indices $I\subseteq[d]$ will be called
_compatible_ with a shift $\tau$ if it holds that

$\operatorname{TV}(\tau_{I})\leq 20(2\lvert
I\rvert+1)\operatorname{TV}(\tau)\qquad\text{and}\qquad\|\tau_{I}-\tau\|_{1}\leq\frac{4\lvert
I\rvert}{d}\operatorname{TV}(\tau).$

Denote

$\operatorname{comp}(\tau):=\left\\{I\subset[d]\colon I\text{ is compatible
with }\tau\right\\}.$

The following proposition is proved in Section 6.2.2.

######  Proposition 5.1.

Let $\tau$ be a shift. For each $0\leq r\leq d$ there exists
$I\in\operatorname{comp}(\tau)$ with $|I|=r$.

It is clear that sufficiently coarse grainings of a shift will yield the
identity (all zero) shift. The following proposition, proved in Section 6.2.1,
quantifies this statement.

######  Proposition 5.2.

Let $\tau$ be a shift. For each integer
$N>\sqrt[d]{2}\left(\frac{\operatorname{TV}\left(\tau\right)}{2d}\right)^{\frac{1}{d-1}}$
it holds that $\tau_{N}\equiv 0$.

The next lemma, whose proof will be the focus of section 6 allows to estimate
the expressions (5.2a), (5.2b) and (5.2c).

######  Lemma 5.3.

Define $\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d)$ as in (1.12):

$\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d):=\left(\frac{1}{\underline{\alpha}^{\parallel}\underline{\alpha}^{\perp}}+\frac{1}{d(\underline{\alpha}^{\perp})^{2}}\right)\operatorname{wid}(\nu^{\parallel})^{2}+\frac{1}{(\underline{\alpha}^{\perp})^{2}}\operatorname{wid}(\nu^{\perp})^{2}.$

There exist universal constants $C,c>0$ such that the following hold for every
$s>0$.

  1. (1)

For any $1\leq r\leq d$, any map $\tau\mapsto I_{\tau}$ assigning to each
shift $\tau$ a compatible set $I_{\tau}\in\operatorname{comp}(\tau)$ with
$|I|=r$, and $Cr\kappa\frac{\log
d}{d}\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)\leq\Gamma\leq
1$,

$\mathbb{P}\left(\left\\{\operatorname{MG}\leq
2s\right\\}\cap\left\\{\exists\tau\in\operatorname{\mathcal{AS}}\colon|G(\tau,\tau_{I_{\tau}})|>\sqrt{\Gamma}s\right\\}\right)\leq
C\exp\left(-c\frac{\Gamma}{\kappa\underline{\alpha}^{\perp}r}s\right).$

  2. (2)

For any $1\leq r\leq d$, any map $\tau\mapsto I_{\tau}$ map assigning to each
shift $\tau$ a compatible set $I_{\tau}\in\operatorname{comp}(\tau)$ with
$|I|=r$, and
$C\kappa\frac{dr}{2^{r}}\left(r+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}dr}+1\right)\right)\leq\Gamma\leq
1$,

$\mathbb{P}\left(\left\\{\operatorname{MG}\leq
2s\right\\}\cap\left\\{\exists\tau\in\operatorname{\mathcal{AS}}\colon|G(\tau_{I_{\tau}},\tau_{2})|>\sqrt{\Gamma}s\right\\}\right)\leq
C\exp\left(-c\frac{\Gamma}{\kappa\underline{\alpha}^{\perp}d}s\right).$

  3. (3)

For any $k\geq 1$ and
$C\kappa\frac{d^{3}}{2^{k(d-2)}}\left(dk+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}d^{2}}+1\right)\right)\leq\Gamma\leq
1$,

$\mathbb{P}\left(\left\\{\operatorname{MG}\leq
2s\right\\}\cap\left\\{\exists\tau\in\operatorname{\mathcal{AS}}\colon|G(\tau_{2^{k}},\tau_{2^{k+1}})|\geq\sqrt{\Gamma}s\right\\}\right)\leq
C\exp\left(-c\frac{\Gamma}{\kappa\underline{\alpha}^{\perp}d^{2}2^{k}}s\right).$

For small $s$, we may obtain an improved dependence on the dimension $d$,
using the following lemma.

######  Lemma 5.4.

There exists universal constants $C,c>0$ such that the following holds. Assume
$0<s\leq\underline{\alpha}^{\perp}4^{d-1}d$, and let
$\kappa=\kappa(\nu^{\parallel},\nu^{\perp},d)$ be as in (1.12), then

$\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}}\colon|G(\tau)|>s\\}\right)\leq
C\exp\left(-\frac{c}{\kappa\underline{\alpha}^{\perp}}s\right).$ (5.3)

###### Proof of Theorem 4.3.

Throughout the proof we will use $C$ and $c$ to denote positive absolute
constants; the values of these constants will be allowed to change from line
to line, even within the same calculation, with the value of $C$ increasing
and the value of $c$ decreasing.

Set
$K:=\lceil\frac{1}{d-1}\log_{2}\left(\frac{4s}{\underline{\alpha}^{\perp}d}\right)\rceil+1$.
By Proposition 5.2 and the definition of admissibility, the term (5.2d)
vanishes for any choice of $\gamma_{K}$.

Set $\gamma_{(0,|I|)}=\gamma_{(|I|,1)}:=\frac{1}{4}$ and
$\gamma_{k}:=\gamma\,2^{-\frac{1}{4}\min\\{k,K-k\\}}$ for any $1\leq k\leq
K-1$, where
$\gamma=\left(2\sum_{k=1}^{K-1}2^{-\frac{1}{4}\min\\{k,K-k\\}}\right)^{-1}$.
Set $r:=\lceil\min\\{10\log_{2}d,\frac{d}{2}\\}\rceil$ and a map $\tau\mapsto
I_{\tau}$ assigning to each shift $\tau$ a compatible set
$I_{\tau}\in\operatorname{comp}(\tau)$ of size $r$, existing by Proposition
5.1. Notice that $\gamma_{(0,r)}+\gamma_{(r,1)}+\sum_{k=1}^{K-1}\gamma_{k}=1$
and that $1\leq r\leq d$ so one may use the chaining argument (5.2).

Recall that
$\kappa\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)\leq
c_{0}\frac{d+1}{\log^{2}(d+1)}$, by assumption (1.14) on the noise
distributions. It is easy to verify that for $c_{0}$ sufficiently small, this
enables us to use the first part of Lemma 5.3 to bound the term (5.2a), the
second part of Lemma 5.3 to bound the term (5.2b), and the third part of Lemma
5.3 to bound the term (5.2c) (recall that the term (5.2d) vanishes).

This yields that for every positive $s$,

$\displaystyle\mathbb{P}(\operatorname{MG}\in[s,2s))\leq$ $\displaystyle
C\exp\left(-c\frac{s}{\kappa\underline{\alpha}^{\perp}r}\right)+C\exp\left(-c\frac{s}{\kappa\underline{\alpha}^{\perp}d}\right)+C\sum_{k=1}^{\lceil\frac{K}{2}\rceil}\exp\left(-c\frac{s}{\kappa\underline{\alpha}^{\perp}d^{2}2^{\frac{3}{2}k}}\right)$
$\displaystyle+C\sum_{k=\lceil\frac{K}{2}\rceil+1}^{K-1}\exp\left(-c\frac{s}{\kappa\underline{\alpha}^{\perp}d^{2}2^{K}}\right)$

Now, noticing that the $K-\lceil\frac{K}{2}\rceil-1$ last summands are
asymptotically dominant and that $2^{K}\leq
4\left(\frac{4s}{\underline{\alpha}^{\perp}d}\right)^{\frac{1}{d-1}}$, one
gets the bound

$\mathbb{P}(\operatorname{MG}\in[s,2s))\leq C\exp\left(-\frac{c}{\kappa
d^{2}}\left(\frac{s}{\underline{\alpha}^{\perp}}\right)^{\frac{d-2}{d-1}}\right)$

for every positive integer $s$. Hence,

$\displaystyle\mathbb{P}(\operatorname{MG}\geq t)$
$\displaystyle=\sum_{i=0}^{\infty}\mathbb{P}(\operatorname{MG}\in[2^{i}t,2^{i+1}t))\leq\sum_{i=0}^{\infty}C\exp\left(-\frac{c}{\kappa
d^{2}}\left(\frac{2^{i}t}{\underline{\alpha}^{\perp}}\right)^{\frac{d-2}{d-1}}\right)$
$\displaystyle\leq C\exp\left(-\frac{c}{\kappa
d^{2}}\left(\frac{t}{\underline{\alpha}^{\perp}}\right)^{\frac{d-2}{d-1}}\right).$

For $t\leq\underline{\alpha}^{\perp}\frac{4^{d-1}}{d}$, denoting by $k$ the
unique integer such that
$2^{k-1}t\leq\underline{\alpha}^{\perp}4^{d-1}d<2^{k}t$, the following holds,
by Lemma 5.4 and (4.14),

$\displaystyle\mathbb{P}(\operatorname{MG}\geq t)$
$\displaystyle=\sum_{i=0}^{k-1}\mathbb{P}\left(\operatorname{MG}\in\left[2^{i}t,2^{i+1}t\right)\right)+\mathbb{P}\left(\operatorname{MG}\geq
2^{k}t\right)$
$\displaystyle\leq\sum_{i=0}^{k-1}C\exp\left(-\frac{c}{\kappa\underline{\alpha}^{\perp}}2^{i}t\right)+C\exp\left(-\frac{c}{\kappa
d^{2}}\left(\frac{2^{k}t}{\underline{\alpha}^{\perp}}\right)^{\frac{d-2}{d-1}}\right)\leq
C\exp\left(-\frac{ct}{\kappa\underline{\alpha}^{\perp}}\right).\qed$

##  6\. Concentration of ground-energy differences between consecutive
grainings

The goal of this section is to prove Proposition 5.1, Proposition 5.2, Lemma
5.3 and Lemma 5.4. The proofs of Lemma 5.3 and Lemma 5.4 are achieved via the
following pivotal statements.

Interface layering. In the following lemmas, the concept of interface layering
plays a significant role. By such layering, we mean the number of interface
plaquettes (in a ground configuration with Dobrushin boundary conditions)
lying above a given position in the base plane (i.e., having the same
projection). We use the following definitions: The _parallel layering_ of a
configuration $\sigma\in\Omega^{\Lambda,\operatorname{Dob}}$ over a set
$A\subset\Lambda\subset\mathbb{Z}^{d}$ is defined as

${\mathcal{L}}^{\parallel}_{A}(\sigma):=\left\lvert\left\\{\left\\{x,x+e_{d+1}\right\\}\in
E^{\parallel}(\mathbb{Z}^{d+1})\colon\pi\left(x\right)\in
A,\,\sigma_{x}\neq\sigma_{x+e_{d+1}}\right\\}\right\rvert.$ (6.1)

The _perpendicular layering_ of $\sigma$ over $A$ is defined as

${\mathcal{L}}^{\perp}_{A}(\sigma):=\left\lvert\left\\{\left\\{x,y\right\\}\in
E^{\perp}(\mathbb{Z}^{d+1})\colon\pi\left(x\right)\in A,\,\pi\left(y\right)\in
A,\,\sigma_{x}\neq\sigma_{y}\right\\}\right\rvert.$ (6.2)

With these definitions in mind one may think of
${\mathcal{L}}^{\perp}_{A}(\tau)+{\mathcal{L}}^{\parallel}_{A}(\tau)-|A|$ as
the number of excessive plaquettes in the interface created by the minimal
energy configuration above $A$, compared to the interface of the configuration
$\rho^{\operatorname{Dob}}$.

For $A\subset\Lambda\subset\mathbb{Z}^{d}$ and integer
$b^{\parallel},b^{\perp}\geq 0$, define:

$\Omega^{\Lambda,A,(b^{\parallel},b^{\perp})}:=\left\\{\sigma\in\Omega^{\Lambda,\operatorname{Dob}}\colon\sum\limits_{\genfrac{}{}{0.0pt}{}{\\{x,y\\}\in
E^{\theta}(\mathbb{Z}^{d+1})}{\\{\pi(x),\pi(y)\\}\cap
A\neq\emptyset}}\mathrm{1}_{\sigma_{x}\neq\sigma_{y}}\leq b^{\theta}\text{ for
}\theta\in\\{\parallel,\perp\\}\right\\},$

as well as

$\operatorname{GE}^{\Lambda,A,(b^{\parallel},b^{\perp})}(\eta):=\min\left\\{\mathcal{H}^{\eta,\Lambda}(\sigma)\colon\sigma\in\Omega^{\Lambda,A,(b^{\parallel},b^{\perp})}\right\\}.$
(6.3)

When $\Lambda$ is fixed, we will occasionally abbreviate by omitting it. Also
define

$G^{\eta,\Lambda,(b^{\parallel},b^{\perp})}(\tau,\tau^{\prime}):=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau-\tau^{\prime}),(b^{\parallel},b^{\perp})}(\eta^{\tau^{\prime}})-\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau-\tau^{\prime}),(b^{\parallel},b^{\perp})}(\eta^{\tau}),$

and abbreviate

$G^{\eta,\Lambda,(b^{\parallel},b^{\perp})}(\tau):=G^{\eta,\Lambda,(b^{\parallel},b^{\perp})}(\tau,0)=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau),(b^{\parallel},b^{\perp})}(\eta)-\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau),(b^{\parallel},b^{\perp})}(\eta^{\tau}).$

Concentration of ground energy differences. First, we provide a bound on the
probability of a given shift producing a large energetic gain, given some a
priori bound on the “number of excessive faces in the interface” above the
support of the shift.

######  Lemma 6.1.

There exist universal $C,c>0$ such that the following holds. Suppose the
disorder distributions $\nu^{\parallel},\nu^{\perp}$ satisfy (1.11). Then for
any two shifts $\tau,\tau^{\prime}$ and any non-negative
$b^{\parallel},b^{\perp}$ for which
$\rho^{\operatorname{Dob}}\in\Omega^{\Lambda,\operatorname{supp}(\tau-\tau^{\prime}),(b^{\parallel},b^{\perp})}$,

$\mathbb{P}\left(\lvert
G^{\eta,\Lambda,(b^{\parallel},b^{\perp})}(\tau,\tau^{\prime})\rvert\geq
t\right)\leq
C\exp\left(-c\frac{t^{2}}{\operatorname{wid}(\nu^{\parallel})^{2}b^{\parallel}+\operatorname{wid}(\nu^{\perp})^{2}b^{\perp}}\right).$
(6.4)

Layering bounds. Lemma 6.1 provides a concentration estimate for the ground
energy of a restricted set of configurations. In the following lemma, we show
that at each step of the graining, the non-restricted ground energy coincides
with an appropriate restricted ground energy.

######  Lemma 6.2.

There exists a universal $C>0$ such that the following holds. Let
$\eta\in\mathcal{D}(\alpha^{\parallel},\alpha^{\perp})$ for some
$\alpha^{\parallel},\alpha^{\perp}>0$. Let $\Lambda\subset\mathbb{Z}^{d}$ be a
finite subset. Let $s>0$ such that
$\operatorname{MG}(\alpha^{\parallel},\alpha^{\perp})\leq 2s$, and let $\tau$
be an $(\alpha^{\parallel},\alpha^{\perp})$-admissible shift.

  1. (1)

For any $\emptyset\neq I\in\operatorname{comp}(\tau)$,

$\displaystyle\operatorname{GE}^{\Lambda}(\\#)=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau-\tau_{I}),(b_{(0,|I|)}^{\parallel}(s),b_{(0,|I|)}^{\perp}(s))}(\\#)$
$\displaystyle\text{\emph{for } }\\#\in\\{\eta^{\tau},\eta^{\tau_{I}}\\},$
$\displaystyle\operatorname{GE}^{\Lambda}(\\#)=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{2}-\tau_{I}),(b_{(|I|,1)}^{\parallel}(s),b_{(|I|,1)}^{\perp}(s))}(\\#)$
$\displaystyle\text{\emph{for } }\\#\in\\{\eta^{\tau_{2}},\eta^{\tau_{I}}\\},$

where

$\displaystyle
b_{(0,|I|)}^{\parallel}(s):=C\left(\frac{1}{{\alpha}^{\parallel}}+\frac{1}{{\alpha}^{\perp}d}\right)|I|s,$
$\displaystyle\quad b_{(0,|I|)}^{\perp}(s):=\frac{C}{{\alpha}^{\perp}}|I|s,$
$\displaystyle
b_{(|I|,1)}^{\parallel}(s):=C\left(\frac{1}{{\alpha}^{\parallel}}+\frac{1}{{\alpha}^{\perp}d}\right)ds,$
$\displaystyle\quad b_{(|I|,1)}^{\perp}(s):=\frac{C}{{\alpha}^{\perp}}ds.$

  2. (2)

For any $k\geq 1$,

$\displaystyle\operatorname{GE}^{\Lambda}(\eta^{\tau_{2^{k}}})$
$\displaystyle=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{2^{k}}-\tau_{2^{k+1}}),(b_{k}^{\parallel}(s),b_{k}^{\perp}(s))}(\eta^{\tau_{2^{k}}})$
$\displaystyle\stackrel{{\scriptstyle(k\geq
2)}}{{=}}\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau_{2^{k-1}}-\tau_{2^{k}}),(b_{k-1}^{\parallel},b_{k-1}^{\perp})}(\eta^{\tau_{2^{k}}}),$

where

$b_{k}^{\parallel}(s):=C\left(\frac{1}{{\alpha}^{\parallel}}+\frac{1}{{\alpha}^{\perp}d}\right)d^{2}2^{k}s,\quad
b_{k}^{\perp}(s):=\frac{C}{{\alpha}^{\perp}}d^{2}2^{k}s.$

  3. (3)

For $s\leq{\alpha}^{\perp}4^{d-1}d$ it holds that

$\displaystyle\operatorname{GE}^{\Lambda}(\\#)=\operatorname{GE}^{\Lambda,\operatorname{supp}(\tau),(b^{\parallel},b^{\perp})}(\\#)$
$\displaystyle\text{\emph{for } }\\#\in\\{\eta^{\tau},\eta\\},$

where

$b^{\parallel}(s):=C\left(\frac{1}{{\alpha}^{\parallel}}+\frac{1}{{\alpha}^{\perp}d}\right)s,\quad
b^{\perp}(s):=\frac{C}{{\alpha}^{\perp}}s.$

Enumeration of shifts. Lastly, we provide a bound on the number of shifts and
their coarse/fine grainings, satisfying that their total variation and trip
entropy are bounded by given constants. This is done by the following
proposition and the corollary that follows it.

######  Proposition 6.3 (Counting shift functions).

There exists $C>0$ such that for each $\lambda,\rho>0$,

$\lvert\\{\tau\in{\mathcal{S}}\colon\operatorname{TV}(\tau)\leq\lambda,\,R(\tau)\leq\rho\\}\rvert\leq\exp\left(C\min\left\\{\lambda+\lambda\log\left(\frac{\rho}{\lambda}+1\right),\lambda\frac{\log
d}{d}+\rho\log d\right\\}\right),$

######  Corollary 6.4.

There exists a universal $C>0$ such that the following holds.

  1. (1)

For each integer $N\geq 2$ and $\lambda,\rho>0$,

$\displaystyle\lvert\\{\tau_{N}\colon\tau\in{\mathcal{S}},\,\operatorname{TV}(\tau)\leq\lambda,\,$
$\displaystyle R(\tau)\leq\rho\\}\rvert$
$\displaystyle\leq\exp\left(C\frac{d\lambda}{N^{d-1}}\left(d\log
N+\log\left(\frac{\rho}{d\lambda}+1\right)\right)\right).$ (6.5)

  2. (2)

For each integer $1\leq r\leq d$, a mapping $\tau\mapsto I_{\tau}$ such that
$I_{\tau}\in\operatorname{comp}(\tau)$ and $|I_{\tau}|=r$, and
$\lambda,\rho>0$,

$\lvert\\{\tau_{I_{\tau}}\colon\tau\in{\mathcal{S}},\,\operatorname{TV}(\tau)\leq\lambda,\,R(\tau)\leq\rho\\}\rvert\leq\exp\left(C\frac{r\lambda}{2^{r}}\left(r+\log\left(\frac{\rho}{r\lambda}+1\right)\right)\right).$
(6.6)

Lemma 6.1 will be proved in subsection 6.4, using a concentration estimate.
Lemma 6.2 will be proved in section 6.5, requiring both basic properties of
grainings to be established as well as using Lemmas inspired by Dobrushin’s
work. Proposition 6.3 and Corollary 6.4 will be proved in section 6.3 using
the work of Bollobás–Balister [BB07] (continuing on Lebowitz–Mazel [LM98]).

###  6.1. Proof of Lemmas 5.3 and 5.4

In this section we show how Lemma 6.1, Lemma 6.2, Proposition 6.3 and
Corollary 6.4 imply Lemmas 5.3 and 5.4. We will continue to use the
abbreviations of section 5, specifically $G$ for $G^{\eta,\Lambda}$,
$\operatorname{\mathcal{AS}}$ for
$\operatorname{\mathcal{AS}}^{\eta,\Lambda}(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})$
and $\operatorname{MG}$ for
$\operatorname{MG}^{\eta,\Lambda}(\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp})$.
Throughout this section we will use $C$ and $c$ to denote positive absolute
constants; the values of these constants will be allowed to change from line
to line, even within the same calculation, with the value of $C$ increasing
and the value of $c$ decreasing.

In the proofs of Lemmas 5.3 and 5.4 we will use the following corollary of
Proposition 6.3 and Corollary 6.4.

######  Corollary 6.5.

The following bounds hold.

  1. (1)

For every $t>0$,

$\lvert\\{\tau\in\mathcal{AS}\colon G(\tau)\leq
t\\}\rvert\leq\exp\left(C\frac{(\log
d)t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right).$
(6.7)

  2. (2)

For every integer $N\geq 2$ and $t>0$,

$\lvert\\{\tau_{N}\colon\tau\in\mathcal{AS},\,G(\tau)\leq
t\\}\rvert\leq\exp\left(C\frac{2d\,t}{\underline{\alpha}^{\perp}N^{d-1}}\left(d\log
N+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}d^{2}}+1\right)\right)\right).$
(6.8)

  3. (3)

For every integer $1\leq r\leq d$, a mapping $\tau\mapsto I_{\tau}$ such that
$I_{\tau}\in\operatorname{comp}(\tau)$ and $|I_{\tau}|=r$, and $t>0$,

$\lvert\\{\tau_{I_{\tau}}\colon\tau\in\mathcal{AS},\,G(\tau)\leq
t\\}\rvert\leq\exp\left(C\frac{2rt}{\underline{\alpha}^{\perp}2^{r}}\left(r+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}dr}+1\right)\right)\right).$
(6.9)

###### Proof.

For every $t>0$,

$\\{\tau\in\mathcal{AS}\colon G(\tau)\leq
t\\}\subseteq\left\\{\tau\in{\mathcal{S}}\colon\operatorname{TV}(\tau)\leq\frac{2t}{\underline{\alpha}^{\perp}},\,R(\tau)\leq\frac{200t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right\\}.$

Hence, by Proposition 6.3,

$\displaystyle\lvert\\{$ $\displaystyle\tau\in\mathcal{AS}\colon G(\tau)\leq
t\\}\rvert\leq\left\lvert\left\\{\tau\in{\mathcal{S}}\colon\operatorname{TV}(\tau)\leq\frac{2t}{\underline{\alpha}^{\perp}},\,R(\tau)\leq\frac{200t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right\\}\right\rvert$
$\displaystyle\leq\exp\left(C\min\left\\{\frac{2t}{\underline{\alpha}^{\perp}}+\frac{2t}{\underline{\alpha}^{\perp}}\log\left(\frac{100\underline{\alpha}^{\perp}}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}+1\right),\frac{2t}{\underline{\alpha}^{\perp}d}\log{d}+\frac{200t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\log
d,\right\\}\right)$ $\displaystyle\leq\exp\left(C\frac{(\log
d)t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right),$

for every integer $N\geq 2$, by (1),

$\displaystyle\lvert\\{\tau_{N}\colon\tau\in\mathcal{AS},\,G(\tau)\leq
t\\}\rvert$
$\displaystyle\leq\left\lvert\left\\{\tau_{N}\colon\tau\in{\mathcal{S}},\,\operatorname{TV}(\tau)\leq\frac{2t}{\underline{\alpha}^{\perp}},\,R(\tau)\leq\frac{200t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}}d\right\\}\right\rvert$
$\displaystyle\leq\exp\left(C\frac{2d\,t}{\underline{\alpha}^{\perp}N^{d-1}}\left(d\log
N+\log\left(\frac{100}{d^{2}}\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)+1\right)\right)\right)$
$\displaystyle\leq\exp\left(C\frac{d\,t}{\underline{\alpha}^{\perp}N^{d-1}}\left(d\log
N+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}d^{2}}+1\right)\right)\right),$

and for every integer $1\leq r\leq d$ and mapping $\tau\mapsto I_{\tau}$ such
that $I_{\tau}\in\operatorname{comp}(\tau)$ and $|I_{\tau}|=r$, by (6.6),

$\displaystyle\lvert\\{\tau_{I_{\tau}}\colon\tau\in\mathcal{AS},\,G(\tau)\leq
t\\}\rvert$
$\displaystyle\leq\left\lvert\left\\{\tau_{I_{\tau}}\colon\tau\in{\mathcal{S}},\,\operatorname{TV}(\tau)\leq\frac{2t}{\underline{\alpha}^{\perp}},\,R(\tau)\leq\frac{200t}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right\\}\right\rvert$
$\displaystyle\leq\exp\left(C\frac{2rt}{\underline{\alpha}^{\perp}2^{r}}\left(r+\log\left(\frac{100\underline{\alpha}^{\perp}}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}dr}+1\right)\right)\right)$
$\displaystyle\leq\exp\left(C\frac{rt}{\underline{\alpha}^{\perp}2^{r}}\left(r+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}dr}+1\right)\right)\right).\qed$

###### Proof of Lemma 5.3.

For the first part of the Lemma, let $1\leq r\leq d$, $\tau\mapsto I_{\tau}$ a
map assigning to each shift $\tau$ a compatible set
$I_{\tau}\in\operatorname{comp}(\tau)$ with $|I_{\tau}|=r$, and
$Cr\kappa\frac{\log
d}{d}\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)\leq\Gamma\leq
1$. It holds that

$\displaystyle\mathbb{P}$ $\displaystyle\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau,\tau_{I_{\tau}})|>\sqrt{\Gamma}s\\}\right)$
$\displaystyle=\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G^{\eta,\Lambda,(b^{\parallel}_{(0,r)}(t),b^{\perp}_{(0,r)}(t))}(\tau,\tau_{I_{\tau}})|>\sqrt{\Gamma}s\\}\right)$
$\displaystyle\leq C\exp\left(C\frac{(\log
d)s}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right)\exp\left(-c\frac{\Gamma
s^{2}}{\operatorname{Lip}(\nu^{\parallel})^{2}b^{\parallel}_{(0,r)}+\operatorname{Lip}(\nu^{\perp})^{2}b^{\perp}_{(0,r)}}\right)$
$\displaystyle=C\exp\left(\left(\frac{C\log
d}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}-c\frac{\Gamma}{\kappa\underline{\alpha}^{\perp}r}\right)s\right),$

with the first equality by Lemma 6.2, the first inequality by union bound,
(6.7), and Lemma 6.1, and the second equality by the definition of
$b^{\parallel}_{(0,r)},b^{\perp}_{(0,r)}$ and of $\kappa$. Now, for $C$
sufficiently large, if

$\Gamma\geq Cr\kappa\frac{\log
d}{d}\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)$

then the second term in the exponent is the asymptotically dominant one and
one gets

$\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau,\tau_{I_{\tau}})|>\sqrt{\Gamma}s\\}\right)\leq
C\exp\left(-c\frac{\Gamma s}{\kappa\underline{\alpha}^{\perp}r}\right).$

In an identical manner, it holds that

$\displaystyle\mathbb{P}$ $\displaystyle\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau_{I_{\tau}},\tau_{2})|>\sqrt{\Gamma}s\\}\right)$
$\displaystyle=\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G^{\eta,\Lambda,(b^{\parallel}_{(r,1)}(t),b^{\perp}_{(r,1)}(t))}(\tau_{I_{\tau}},\tau_{2})|>\sqrt{\Gamma}s\\}\right)$
$\displaystyle\leq
C\exp\left(C\frac{rs}{\underline{\alpha}^{\perp}2^{r}}\left(r+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}dr}+1\right)\right)\right)\exp\left(-c\frac{\Gamma
s^{2}}{\operatorname{Lip}(\nu^{\parallel})^{2}b^{\parallel}_{(r,1)}+\operatorname{Lip}(\nu^{\perp})^{2}b^{\perp}_{(r,1)}}\right)$
$\displaystyle=C\exp\left(\left(C\frac{r}{\underline{\alpha}^{\perp}2^{r}}\left(r+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}dr}+1\right)\right)-c\frac{\Gamma}{\kappa\underline{\alpha}^{\perp}d}\right)s\right),$

with the first equality by Lemma 6.2, the first inequality by union bound,
(6.8), and Lemma 6.1, and the second equality by the definition of
$b^{\parallel}_{(r,1)},b^{\perp}_{(r,1)}$ and of $\kappa$. Now, for $C$
sufficiently large, if

$\Gamma\geq
C\kappa\frac{dr}{2^{r}}\left(r+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}dr}+1\right)\right)$

then the second term in the exponent is the asymptotically dominant one and
one gets

$\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau,\tau_{I_{\tau}})|>\sqrt{\Gamma}s\\}\right)\leq\exp
C\left(-c\frac{\Gamma s}{\kappa\underline{\alpha}^{\perp}d}\right).$

For the third bound, again in an identical manner

$\displaystyle\mathbb{P}$ $\displaystyle\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau_{2^{k}},\tau_{2^{k+1}})|>\sqrt{\Gamma}s\\}\right)$
$\displaystyle=\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G^{\eta,\Lambda,(b^{\parallel}_{k}(t),b^{\perp}_{k}(t))}(\tau_{2^{k}},\tau_{2^{k+1}})|>\sqrt{\Gamma}s\\}\right)$
$\displaystyle\leq
C\exp\left(C\frac{ds}{\underline{\alpha}^{\perp}2^{k(d-1)}}\left(dk+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}d^{2}}+1\right)\right)\right)\exp\left(-c\frac{\Gamma
s^{2}}{\operatorname{Lip}(\nu^{\parallel})^{2}b^{\parallel}_{k}+\operatorname{Lip}(\nu^{\perp})^{2}b^{\perp}_{k}}\right)$
$\displaystyle=C\exp\left(\left(C\frac{d}{\underline{\alpha}^{\perp}2^{k(d-1)}}\left(dk+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}d^{2}}+1\right)\right)-c\frac{\Gamma}{\kappa\underline{\alpha}^{\perp}d^{2}2^{k}}\right)s\right),$

with the first equality by Lemma 6.2, the first inequality by union bound,
(6.9), and Lemma 6.1, and the second equality by the definition of
$b^{\parallel}_{k},b^{\perp}_{k}$ and of $\kappa$. For $C>0$ sufficiently
large, if

$\Gamma\geq
C\kappa\frac{d^{3}}{2^{k(d-2)}}\left(dk+\log\left(\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}d^{2}}+1\right)\right)$

then the second term in the exponent is the asymptotically dominant one and
one gets

$\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau,\tau_{I_{\tau}})|>\sqrt{\Gamma}s\\}\right)\leq
C\exp\left(-c\frac{\Gamma
s}{\kappa\underline{\alpha}^{\perp}d^{2}2^{k}}\right).$

This concludes the proof. ∎

###### Proof of Lemma 5.4.

It holds that

$\displaystyle\mathbb{P}$ $\displaystyle\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau)|>s\\}\right)$
$\displaystyle=\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G^{\eta,\Lambda,(b^{\parallel},b^{\perp})}(\tau)|>s\\}\right)$
$\displaystyle\leq\exp\left(C\frac{(\log
d)s}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}\right)C\exp\left(-\frac{s^{2}}{\operatorname{Lip}(\nu^{\parallel})^{2}b^{\parallel}+\operatorname{Lip}(\nu^{\perp})^{2}b^{\perp}}\right)$
$\displaystyle=C\exp\left(\left(\frac{C\log
d}{\min\\{\underline{\alpha}^{\parallel},\underline{\alpha}^{\perp}\\}d}-\frac{c}{\kappa\underline{\alpha}^{\perp}}\right)s\right),$

with the first equality by Lemma 6.2, the first inequality by union bound,
(6.7), and Lemma 6.1, and the second equality by the definition of
$b^{\parallel},b^{\perp}$ and of $\kappa$. Then, for $c>0$ sufficiently small,
if
$\kappa\left(1+\frac{\underline{\alpha}^{\perp}}{\underline{\alpha}^{\parallel}}\right)\leq
c\frac{d}{\log d}$ which is always the case by condition (1.14), then the
second term in the exponent is the asymptotically dominant one and one gets

$\mathbb{P}\left(\\{\operatorname{MG}\leq
2s\\}\cap\\{\exists\tau\in\operatorname{\mathcal{AS}},|G(\tau)|>s\\}\right)\leq\exp\left(-\frac{c}{\kappa\underline{\alpha}^{\perp}}s\right).\qed$

###  6.2. Basic properties of grainings of shifts

This section provides estimates on basic parameters (total variation, support
size, trip-entropy, etc.) for coarse and fine grainings of shifts. These
estimates will be used in the proof of several of our preliminary statements
in the subsequent sections.

####  6.2.1. Bounding the total variation and weighted difference of coarse
grainings

In this section we will prove Proposition 5.2 as well as the following useful
bounds.

######  Proposition 6.6.

For every shift $\tau$ and every positive integer $N$,

$\operatorname{TV}(\tau_{N})\leq 10d\,\operatorname{TV}(\tau).$

######  Proposition 6.7.

For every shift $\tau$,

$\lvert\operatorname{supp}(\tau_{2}-\tau)\rvert\leq\lVert\tau_{2}-\tau\rVert_{1}\leq
4\,\operatorname{TV}(\tau)$

and moreover, for every positive integer $N$,

$\lvert\operatorname{supp}(\tau_{2N}-\tau_{N})\rvert\leq\lVert\tau_{2N}-\tau_{N}\rVert_{1}\leq(4d+9)N\,\operatorname{TV}(\tau).$

The proofs of Propositions 5.2, 6.6 and 6.7 will rely on several Lemmas.

######  Lemma 6.8.

Let $A$ be a finite set of points in ${\mathbb{Z}}^{d}$. Then,

$\lvert\partial A\rvert\geq 2d\lvert A\rvert^{1-\frac{1}{d}}.$ (6.10)

Moreover, for every $N\times N\times\cdots\times N$ $d$-dimensional cube $B$
in $\mathbb{Z}^{d}$,

$\lvert\partial A\cap(B\times B)\rvert\geq\frac{2}{3N}\min\left\\{\lvert A\cap
B\rvert,\lvert B\setminus A\rvert\right\\}.$ (6.11)

###### Proof.