# Continuity of the extremal decomposition of the free state for finite-spin
models on Cayley trees

Loren Coquille111Université Grenoble Alpes, CNRS, Institut Fourier, F-38000
Grenoble, France<EMAIL_ADDRESS>Christof
Külske222Ruhr-Universität Bochum, Fakultät für Mathematik, D44801 Bochum,
Germany<EMAIL_ADDRESS>Arnaud Le Ny333Laboratoire
d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, UPEC, 61 Avenue du
Général de Gaulle, 94010 Créteil cedex, France<EMAIL_ADDRESS>
###### Abstract

We prove the continuity of the extremal decomposition measure of the free
state of low temperature Potts models, and more generally of ferromagnetic
finite-spin models, on a regular tree, including general clock models. The
decomposition is supported on uncountably many inhomogeneous extremal states,
that we call glassy states. The method of proof provides explicit
concentration bounds on branch overlaps, which play the role of an order
parameter for typical extremals. The result extends to the counterpart of the
free state (called central state) in a wide range of models which have no
symmetry, allowing also the presence of sufficiently small field terms. Our
work shows in particular that the decomposition of central states into
uncountably many glassy states in finite-spin models on trees at low
temperature is a generic phenomenon, and does not rely on symmetries of the
Hamiltonian.

AMS 2000 subject classification: 60K35, 82B20, 82B26

Key words: Gibbs measures, DLR formalism, spin models on trees, disordered
systems, extremal decompositions, Markov fields, Markov chains.

###### Contents

  1. 1 Introduction
  2. 2 Definitions and tools
    1. 2.1 Models
    2. 2.2 Choquet simplex of DLR measures
    3. 2.3 Tree-indexed Markov chains
  3. 3 Results
    1. 3.1 Reconstruction bounds for central states
    2. 3.2 Positivity of the Edwards-Anderson parameter and non-extremality
    3. 3.3 Continuity of the extremal decomposition of the free state
    4. 3.4 Concentration of branch-overlap for typical extremals
    5. 3.5 Structure of the proofs
  4. 4 Proofs
    1. 4.1 Markov chains on trees and boundary laws.
      1. 4.1.1 Boundary laws
      2. 4.1.2 Free states of clock models
      3. 4.1.3 Central states of perturbed clock models
      4. 4.1.4 Lazyness assumption
    2. 4.2 Contours and Excess Energy Lemma
    3. 4.3 Key lemma
    4. 4.4 Positivity of the Edwards-Anderson parameter
    5. 4.5 Overlap control for typical extremals
      1. 4.5.1 Exponential decorrelation of bad sites
      2. 4.5.2 Almost sure convergence of empirical means of bad sites
      3. 4.5.3 Concentration of spin-overlaps under typical extremals
      4. 4.5.4 Proof of Theorem 3.4
    6. 4.6 From overlap control to almost sure singularity of extremals

##  1 Introduction

Models of statistical mechanics supported on trees and more generally non-
amenable graphs, are known to show rather interesting behavior with multiple
critical values, which displays complex phenomenology and challenges for a
rigorous understanding.

The free state of the Ising model in zero external magnetic field on a regular
tree, obtained as the infinite-volume Gibbs measure with free (open) boundary
conditions, is such an example. This homogenous (tree-automorphism invariant)
Gibbs measure indeed displays a complex structure at low temperature, which
took a long history to understand.

First recall [5] that the free state $\mu^{{\mathbb{Z}}^{d}}_{\beta}$ of the
Ising model on the lattice ${\mathbb{Z}}^{d}$ at inverse temperature $\beta$
decomposes into the symmetric convex combination

$\begin{split}\mu^{{\mathbb{Z}}^{d}}_{\beta}=(\mu^{+,{\mathbb{Z}}^{d}}_{\beta}+\mu^{-,{\mathbb{Z}}^{d}}_{\beta})/2,\quad\text{for
all }\beta>0.\end{split}$ (1)

where $\mu^{+,{\mathbb{Z}}^{d}}_{\beta}$ (respectively
$\mu^{-,{\mathbb{Z}}^{d}}_{\beta}$) is obtained as the infinite-volume Gibbs
measure with with $+$ (respectively $-$)-boundary conditions. The extremal
decomposition of this form becomes non-trivial below the critical temperature
of the model, i.e. for all
$\beta>\beta_{c}^{{\mathbb{Z}}^{d}}:=\inf\\{\beta>0,\mu^{+,{\mathbb{Z}}^{d}}_{\beta}\neq\mu^{-,{\mathbb{Z}}^{d}}_{\beta}\\}$.
By abstract Gibbs theory [19] for any spin-model on countable graphs, a unique
decomposition into extremal (pure, non-decomposable) infinite-volume Gibbs
measures always holds.

On the tree $\mathcal{T}^{d}$ with $d+1>2$ nearest neighbors however,

$\begin{split}\mu^{\mathcal{T}^{d}}_{\beta}\neq(\mu^{+,\mathcal{T}^{d}}_{\beta}+\mu^{-,\mathcal{T}^{d}}_{\beta})/2\end{split}$
(2)

in the whole non-uniqueness region $d(\tanh\beta)>1$ corresponding to
$\beta>\beta_{c}^{\mathcal{T}^{d}}$ [29]. Moreover, the free state is non-
extremal in the even lower temperature region $d(\tanh\beta)^{2}>1$, which was
shown in the sequence of works [20, 2, 4, 22, 21].

Note that Ising models on a branching plane
$\mathcal{T}^{d}\times{\mathbb{Z}}$ were investigated in [28] and display
three parameter regimes corresponding to: (i) uniqueness, (ii) non-uniqueness
with tree-like structure (2) and (iii) non-uniqueness with lattice-like
structure (1).

Understanding the extremal decomposition of the Ising free state on a tree is
known to be particularly complex as it involves uncountably many Gibbs
measures on which the extremal decomposition measure is supported, see the
much more recent work of [15]. These states have a broken translational
symmetry, and show characteristics of glassy behavior.

To the best of our knowledge there are no result concerning the extremal
decomposition measure of the Ising free state in an external field, nor of the
one of the Potts free state without a field or in a field, and how sensitive
the decomposition measure depends on particularities and symmetries of the
model. Our purpose in the present work is to investigate this complexity in
the general framework of ferromagnetic finite-spin models with nearest
neighbor interactions.

In the paper [10], we prove the existence of an uncountable family of extremal
inhomogeneous states for a large class of models on regular trees, including
finite-spin models. More precisely, we give an explicit description of a
family of local ground state configurations $\Omega_{GS}$, which have a sparse
enough set of broken bonds, and which give rise to extremal low temperature
states, in the sense that for any $\omega\in\Omega_{GS}$, the weak limit of
finite volume measures with boundary condition $\omega$ exists and is an
extremal state. Note that many other extremal states exist, see discussions in
[10, 24, 17] and references therein.

The purpose of the current paper is to show that uncountably many states,
which can be seen as perturbations of the above inhomogeneous states, enter
the extremal decomposition of the free state. The result extends to the
counterpart of the free state (called central state), in a wide range of
asymmetric models e.g. where small inhomogenous field terms are added. Our
approach has been inspired by the work of Gandolfo, Maes, Ruiz and Shlosman
[15], who discuss these questions in the case of the Ising model in zero
external field. Our approach is nevertheless essentially different as we
develop a new method based on concentration of branch overlaps, see below.

In the Ising case, soon after the characterization of the homogeneous Gibbs
measures as Markov chains by Spitzer [31], Higuchi [20] proved the non-
extremality of the ’third Markov chain’ (the free state) for
$d(\tanh\beta)^{2}>1$. These works were complemented by Bleher et al. [4] and
Ioffe [22] who proved extremality of the free state for $d(\tanh\beta)^{2}<1$.
In the latter, and already in [9, 13], the fluctuations leading to non
extremality have been related to fluctuations of the spin-glass order
paramater on trees used as a discriminating tail observable, the so-called
Edwards-Anderson parameter, which is nothing but the variance of the root
magnetization. Note that non-extremality of the free state was also proved by
Markov chains Kesten-Stigum techniques in [27], who also conjectured
independently the extremality at intermediate temperatures.

For general finite spin models, including $q$-clock models, we introduce a
generalised version of the Edwards-Anderson parameter measuring the non-
degeneracy of the law of the root-spin when the boundary condition at infinity
is sampled according to the free/central state $\mu$:

$q_{EA}=\frac{1}{q}\sum_{a\in\\{1,\ldots,q\\}}{\rm
Var}_{\mu}\bigl{(}{\pi({\sigma_{0}=a}|\cdot})\bigr{)}$

were the $\pi$-kernel $\pi(\cdot|\omega)$ is the Gibbs-measure with boundary-
condition $\omega$ at infinity, which exists and is extremal for $\mu$-almost
every $\omega$ (for precise definitions, see Section 2.2 below). We prove
$q_{EA}$ to be strictly positive at large enough $\beta$ (see Theorem 3.2),
implying that the free/central state is not extremal at low temperature. From
an information-theoretic view, this means the model is reconstruction-solvable
[26] as the $\pi$-kernel is able to restore information which was sent from
the root to infinity by means of the measure $\mu$.

Our main results concern the extremal decomposition of the free/central state
$\mu$. The DLR equations imply

$\displaystyle\mu(\cdot)=\int_{\Omega}\pi(\cdot|\omega)d\mu(\omega)=\int_{{{\rm
ex}\mathcal{G}(\gamma)}}\nu(\cdot){\rm d}\alpha_{\mu}(\nu)$

where $\alpha_{\mu}$ is the extremal decomposition measure of the free state.
In Theorem 3.3, we prove that if we sample independently at random two
boundary conditions $\omega$ and $\omega^{\prime}$ under the free state $\mu$,
then the states $\pi(\cdot|\omega)$ and $\pi(\cdot|\omega^{\prime})$ are
almost surely singular with respect to each other:

$\displaystyle\mu\otimes\mu(\\{(\omega,\omega^{\prime}):\pi(\cdot|\omega)\perp\pi(\cdot|\omega^{\prime})\\})=1.$

which implies that the measure $\alpha_{\mu}$ has no atom. The extremal
decomposition of the free state $\mu$ is thus supported on an uncountable
family of extremal states.
To this purpose, we introduce the following tail measurable observable

$\begin{split}&\underline{\phi}^{\omega}=\liminf_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{\sigma_{v}=\omega_{v}}\cr\end{split}$

which we call branch overlap, and which measures how much a configuration
$\sigma$ agrees with $\omega$ on a (sparse enough) sequence of vertices lying
on a branch of the tree. We prove that $\underline{\phi}^{\omega}$ has
different expectations under $\pi(\cdot|\omega)$ and
$\pi(\cdot|\omega^{\prime})$, which is enough to imply almost sure
singularity. More precisely, we prove that $\underline{\phi}^{\omega}$ has an
expectation tending to 1 under $\pi(\cdot|\omega)$, and tending to
$\sum_{a}\mu(\sigma_{0}=a)^{2}$ (which equals $1/q$ in the case of the $q$
state Potts model) under $\pi(\cdot|\omega^{\prime})$, as $\beta\to\infty$,
with explicit bounds. This can be viewed as a quantitative statement of some
”boundary condition resampling chaos”.

The paper is organized as follows: In Section 2 we first recall the necessary
backgrounds on Gibbs measures, extremal decompositions and Markov chains on
Cayley trees. We then state our results in Section 3 and provide the proofs in
Section 4.

##  2 Definitions and tools

###  2.1 Models

Let $\mathcal{T}^{d}=(V,E)$ denote the Cayley tree of order $d$, on which any
vertex $v\in V$ has exactly $d+1$ neighbors. Pairs $\\{v,w\\}\in E$ of
nearest-neighbors are written $v\sim w$. To any vertex $v\in V$, we attach a
spin, which is a random variable $\sigma_{v}$ taking values in
$\Omega_{0}={{\mathbb{Z}}_{q}=\\{0,\dots,q-1\\}}$, for $q\in\\{2,3,\ldots\\}$.
Let $\Omega_{0}$ be equipped with the $\sigma$-algebra
$\mathcal{E}=\mathcal{P}(\Omega_{0})$ of all subsets of $\Omega_{0}$. We are
interested in probability measures on the product space
$(\Omega,\mathcal{F})=(\Omega_{0}^{V},\mathcal{E}^{\otimes V})$. For any
subset $\Lambda\subset V$ and $\omega\in\Omega$ we define the finite-volume
configuration (or projection)
${\sigma_{\Lambda}(\omega)}=\omega_{\Lambda}=(\omega_{v})_{v\in\Lambda}$. If
$\Lambda\subset V$ is a finite subset, we also write $\Lambda\Subset V$. We
write $\partial\Lambda=\\{v\in V{\backslash\Lambda}:\exists w\in\Lambda,v\sim
w\\}$. We also denote by $\mathcal{F}_{\Lambda}$ the $\sigma$-algebra
generated by the variables $\sigma_{\Lambda}=(\sigma_{v})_{v\in\Lambda}$, or
equivalently by the cylinders denoted
$\\{\sigma_{{\Lambda}}=\omega_{\Lambda}\\}$. Another important
$\sigma$-algebra will be the tail $\sigma$-algebra of asymptotic events
$\mathcal{F}_{\infty}=\cap_{\Lambda\Subset V}\mathcal{F}_{\Lambda^{c}}$ where
the intersection runs over finite subsets.

We introduce an interaction potential $\Phi$ and consider equilibrium states
to be Gibbs measures built with the DLR framework, see e.g. [19]: they are the
probability measures $\mu$ consistent with the Gibbsian specification
$\gamma^{\Phi}$ in the sense that a version of their conditional probabilities
w.r.t. the outside of any finite set $\Lambda$ of the tree is provided by the
DLR equations,

$\displaystyle\forall\Lambda\Subset
V,\;\forall\omega_{\Lambda}\in\Omega_{\Lambda},\;\mu[\sigma_{\Lambda}=\omega_{\Lambda}\mid\mathcal{F}_{\Lambda^{c}}](\cdot)=\gamma_{\Lambda}^{\Phi}(\omega_{\Lambda}\mid\cdot),\;\mu-a.s.$
(3)

where the elements of the Gibbs specification
$\gamma^{\Phi}=\gamma^{\Phi}(\beta)$ are the probability kernels
$\gamma_{\Lambda}^{\Phi}$ from $\Omega_{\Lambda^{c}}$ to
$\mathcal{F}_{\Lambda}$ defined for all $\Lambda\Subset V$ as

$\displaystyle\gamma_{\Lambda}^{\Phi}(\omega_{\Lambda}\mid\omega_{\Lambda^{c}})=\frac{1}{Z_{\Lambda}^{\omega}}e^{-\beta
H_{\Lambda}(\omega_{\Lambda}\omega_{\Lambda^{c}})}.$ (4)

The partition function $Z_{\Lambda}^{\omega}$ is the normalization constant
for a boundary condition $\omega_{\Lambda^{c}}$, at finite volume $\Lambda$,
and the corresponding Hamiltonian $H_{\Lambda}^{\Phi}$ with b.c.
$\omega_{\Lambda^{c}}$ is provided by

$\displaystyle
H_{\Lambda}(\omega_{\Lambda}\omega_{\Lambda^{c}})=\sum_{A\cap\Lambda\neq\emptyset}\Phi_{A}(\omega_{\Lambda}\omega_{\Lambda^{c}})$
(5)

where $\omega_{\Lambda}\omega_{\Lambda^{c}}$ denotes the concatenation of
$\omega_{\Lambda}$ and $\omega_{\Lambda^{c}}$. We write $H$ for the
Hamiltonian with free boundary conditions:

$H_{\Lambda}(\omega)=\sum_{A\subset\Lambda}\Phi_{A}(\omega_{\Lambda}).$ (6)

and define the free state $\mu$ to be the Gibbs measure obtained by taking the
infinite-volume limit with free boundary conditions. The limit exists in
particular, when all finite volume measures are consistent, which is e.g. the
case for the Ising model or the Potts model in zero external field. In
general, in a non-zero field one could think to consider sub-sequences with
free boundary conditions, but it is necessary to put suitable non-
deterministic boundary conditions generalizing the free ones, for which we
will need the theory of boundary laws, see Section 4.1 and in particular
Theorem 4.1.

Throughout the paper we consider ferromagnetic nearest-neighbor (n.n.)
potentials of the form

$\displaystyle\Phi_{\\{v,w\\}}(\omega)$
$\displaystyle=\sum_{i,j=1}^{q}u_{i,j}\cdot\mathbf{1}_{\\{(\omega_{v},\omega_{w})=(i,j)\\}}{\cdot}{\mathbf{1}_{v\sim
w}}$ $\displaystyle\text{where }u_{i,i}$ $\displaystyle=0\text{ and
}0<u:=\min_{i\neq j}u_{i,j}\leq\max_{i,j}u_{i,j}=:U$ (7)

The strict positivity of the lower bound expresses that homogeneous spin
configurations are energetically favored by the pair interaction, while the
energies of excitations w.r.t. different ground states may depend in general
on the ground state. The latter includes the $q$-state Potts model (the Ising
model corresponding to $q=2$), for which

$\Phi^{\text{Potts}}_{\\{v,w\\}}(\omega)=\mathbf{1}_{\omega_{v}\neq\omega_{w}}\cdot\mathbf{1}_{v\sim
w}.$ (8)

More generally than the Potts model, we consider $q$-state clock models as an
important sub-class. In these models the pair potential has a discrete
rotational symmetry which means that there is a positive function $\bar{u}$
for which

$u_{i,j}=\bar{u}(|i-j|)$ (9)

where $|i-j|$ is the distance between $i$ and $j\in{\mathbb{Z}}_{q}$. As the
finite-volume Gibbs measures with free boundary conditions are consistent
measures in clock-models, they immediately yield a well-defined free state in
infinite volume.

Finally we extend our framework from the free state in clock models to central
states (see the definition in Section 4.1.3), which are constructed as (small)
deformations of a free state. Here we do not assume the strict discrete clock-
symmetry of the interaction. Observe that the simplest example of such a
central state which is not a free state, is obtained for the low-temperature
Ising model in a small field.

As our proofs do not rely on symmetry considerations, we are indeed able to
extend them to Hamiltonians of the following type

$\begin{split}H(\omega)=\sum_{v\sim w}\Phi_{\\{v,w\\}}(\omega)+\sum_{v\in
V}\Psi(\omega_{v}).\end{split}$ (10)

with a pair potential $\Phi_{\\{v,w\\}}$ fulfilling (2.1) uniformly in
$\\{v,w\\}$ and a homogeneous single site potential
$\Psi:{\mathbb{Z}}_{q}\rightarrow{\mathbb{R}}$ such that

$\|\Psi\|_{\infty}\leq u(d-1)/8.$ (11)

We moreover need a suitable assumption on the pair potential and the single-
site potential imposed by the ’lazyness’ condition (see Definition 2.2).

###  2.2 Choquet simplex of DLR measures

We denote by $\mathcal{G}(\gamma)$ the set of DLR measures satisfying DLR
equations (3) for a general specification $\gamma$, see [30, 19]. Note that by
compactness of $\Omega_{0}$, $\mathcal{G}(\gamma)\neq\emptyset$. This convex
set has in general a particularly interesting structure of being a Choquet
simplex, that is a compact convex set possessing a subset of extremal elements
${\rm ex}\mathcal{G}(\gamma)$ such that any $\mu\in\mathcal{G}(\gamma)$ has a
unique convex decomposition onto ${\rm ex}\mathcal{G}(\gamma)$. These extremal
elements are mutually singular and considered to be the physical states of the
system, see the discussions in [19, 14, 25]. Note that spatially homogeneous
states may decompose into extremal but non-homogeneous states. It is the
precisely the purpose of this work, to describe how this happens for free and
central states on trees.

To briefly formalize this, denote $\mathcal{M}_{1}^{+}(\Omega)$ to be the set
of probability measures on $(\Omega,\mathcal{F})$ and recall the tail
$\sigma$-algebra of asymptotic events

$\mathcal{F}_{\infty}=\cap_{\Lambda\Subset V}\mathcal{F}_{\Lambda^{c}}.$

######  Definition 2.1 (see [19]).

Any $\mu\in{\rm ex}\mathcal{G}(\gamma)$ is characterized by the equivalent
items:

  1. 1.

Tail-triviality: $A\in\mathcal{F}_{\infty}\Longrightarrow\mu(A)\in\\{0,1\\}.$

  2. 2.

Short-range correlations

$\lim_{\Lambda\uparrow V}\sup_{B\in\mathcal{F}_{\Lambda^{c}}}\big{|}\mu(A\cap
B)-\mu(A)\mu(B)\big{|}=0.$ (12)

By consistency, for any $\Lambda\Subset V$, the kernel $\gamma_{\Lambda}$ is a
regular version of conditional probability of $\mu$ given
$\mathcal{F}_{\Lambda^{c}}$:

$\gamma_{\Lambda}(A|\cdot)=\mu[A|\mathcal{F}_{\Lambda^{c}}](\cdot),\;\mu-{\rm
a.s.},\;\forall A\in\mathcal{F}$ (13)

which means in particular that for any sequence $\Lambda_{n}\uparrow V$,

$\forall\mu\in\mathcal{G}(\gamma),\;\mu(\cdot)=\int_{\Omega}\gamma_{\Lambda_{n}}(\cdot|\omega)\;d\mu(\omega).$

One eventually gets the starting point of the simplicial decomposition

$\forall\mu\in\mathcal{M}_{1}^{+}(\Omega),\;\mu(\cdot)=\int_{\Omega}\mu[\cdot\,|\mathcal{F}_{\infty}](\omega)\;d\mu(\omega)$
(14)

obtained from the DLR equations by using standard Backward Martingale Limit
theorem with respect to the filtration
$(\mathcal{F}_{{\Lambda_{n}^{c}}})_{{n}}$. Define now, for any $\omega$, the
so-called boundary condition at infinity kernel $\pi(\cdot|\omega)$ to be the
tail-measurable kernel

$\displaystyle\pi(\cdot|\omega)=\lim_{\Lambda\uparrow\mathcal{T}^{d}}\gamma_{\Lambda}(\cdot|\omega)$
(15)

if the limits exists for all cylinder events (generating $\mathcal{F}$), and
$\pi(\cdot|\omega):=$$\nu$ to be a fixed arbitrary probability measure else.

######  Theorem 2.1 (see [19]).

The limit (15) exists and is an extremal Gibbs measure for $\mu$-almost every
$\omega$.

Equation (14) thus yields that for any $\mu$ satisfying (3),

$\displaystyle\mu(\cdot)=\int\pi(\cdot|\omega)d\mu(\omega).$ (16)

To work with probability measures on spaces $\mathcal{M}$ of (probability)
measures, we first endow such spaces with a canonical measurable structure.
For any subset of probability measures
$\mathcal{M}\subset\mathcal{M}_{1}^{+}(\Omega)$, the natural way to do so is
to evaluate any $\mu\in\mathcal{M}$ via the numbers
$\big{\\{}\mu(A),\;A\in\mathcal{F}\big{\\}}$. One introduces then the
evaluation maps on $\mathcal{M}$ defined for all $A\in\mathcal{F}$ by

$e_{A}:\mathcal{M}\longrightarrow[0,1];\mu\longmapsto e_{A}(\mu)=\mu(A).$ (17)

The evaluation $\sigma$-algebra $e(\mathcal{M})$ is then the $\sigma$-algebra
generated by the sets $\\{e_{A}\leq c\\}$ for all $A\in\mathcal{F},\;0\leq
c\leq 1$, or equivalently the smallest $\sigma$-algebra on $\mathcal{M}$ that
makes these evaluation maps measurable. For any bounded measurable function
$f\in\mathcal{F}$, the map

$e_{f}:\mathcal{M}\longrightarrow\mathbb{R};\mu\longmapsto e_{f}(\mu):=\mu[f]$

is then $e(\mathcal{M})$-measurable. It is direct by standard measure-
theoretic techniques that all the events and maps we consider in this work are
measurable in this sense.

######  Theorem 2.2 ([12, 19, 14, 25]).

Assume that $\mathcal{G}(\gamma)\neq\emptyset$. Then $\mathcal{G}(\gamma)$ is
a convex subset of $\mathcal{M}_{1}^{+}(\Omega)$ whose extreme boundary is
denoted ${\rm ex}\mathcal{G}(\gamma)$, and satisfies the following properties:

$\forall\mu\in\mathcal{G}(\gamma),\;\mu=\int_{{\rm
ex}\mathcal{G}(\gamma)}\;\nu\cdot\alpha_{\mu}(d\nu)$ (18)

where $\alpha_{\mu}\in\mathcal{M}_{1}^{+}\Big{(}{\rm
ex}\mathcal{G}(\gamma),e({\rm ex}\mathcal{G}(\gamma))\Big{)}$ is defined for
all $M\in e({\rm ex}\mathcal{G}(\gamma))$ by

$\alpha_{\mu}(M)=\mu\Big{[}\big{\\{}\omega\in\Omega:\pi(\cdot|\omega)\in
M\big{\\}}\Big{]}.$ (19)

Our goal in this work is to rigorously establish that for the low temperature
free state $\mu$ (and for central states) the measure $\alpha_{\mu}$ has no
atom on ${\rm ex}\mathcal{G}(\gamma)$.

###  2.3 Tree-indexed Markov chains

A probability measure $\mu$ on $({\mathbb{Z}}_{q})^{V}$ is a homogeneous tree-
indexed Markov chain if and only if it allows the following iterative
construction:

  1. 1.

Sample $\sigma_{0}$ at (arbitrary) root $0$ according to single-site marginal
of $\mu$.

  2. 2.

Sample $\sigma_{w}$ via some transition matrix $P(\omega_{v},\omega_{w})$ from
inside to outside.

The abstract general definition of tree-indexed Markov chain which does not
assume any invariance under any tree-automorphism requires that

$\mu(\sigma_{w}=\cdot|{\mathcal{F}}_{\text{past of
}(v,w)})=\mu(\sigma_{w}=\cdot|{\mathcal{F}}_{v})$ (20)

holds for all oriented edges $(v,w)$, and the past of an oriented edge is the
set of vertices which are closer to $v$ than to $w$.

We note that in the trivial case $d=1$ (that is excluded for the most part in
the analysis of this paper) $\mu$ is a homogeneous Markov chain if it is
shift-invariant and reversible. This is clear, as time-reversal corresponds to
a reflection of ${\mathbb{Z}}$. We note the important background theorem:

######  Theorem 2.3 ([19] Theorem 12.6).

__
If $\mu$ is an extremal Gibbs measure then $\mu$ is a tree-indexed Markov
chain.

The converse is not true, as the famous example of the free state for the
Ising model in zero external field shows (see Higuchi [20], Remark p342).

In the whole article all we need to suppose is that $\mu$ is a homogeneous
tree-indexed Markov chain for a Hamiltonian in the class (10), with a certain
lazyness property in the following sense.

######  Definition 2.2 (Lazyness parameter).

Let $\mu$ be a homogeneous tree-indexed Markov chain. Denote by $P$ its
transition matrix (which is then homogeneous, too) and define the
corresponding maximal jump probability to be

$\displaystyle p_{1}=p_{1}(\mu):=\max_{i\in\mathbb{Z}_{q}}\sum_{j\neq
i}P(i,j).$ (21)

Equivalently, for all $v\sim 0$ we have

$\displaystyle p_{1}=\max_{i\in\mathbb{Z}_{q}}\mu(\sigma_{v}\neq
i|\sigma_{0}=i).$ (22)

In the course of the proof of our main theorem in Section 3, which asserts the
continuity of the extremal decomposition measure of a Markov chain $\mu$, we
ask for sufficient smallness of the quantity $p_{1}(\mu)$. This requirement
means that all states are sufficiently lazy, i.e. with large probability the
chain stays in each of the states.

Below we will see that the examples of central states obtained by small
perturbations of low temperature free states of clock models keep the Markov
chain property and the small $p_{1}$ property. The reason is that $p_{1}$
deforms smoothly under perturbations, and so lazyness carries over from the
unperturbed model.

##  3 Results

###  3.1 Reconstruction bounds for central states

We have the following reconstruction statement, for the central states in any
model with Hamiltonian of the class (10).

######  Theorem 3.1.

Consider any central state $\mu$ in the class of models with Hamiltonian of
the form (10), fulfilling the bounds (2.1) and (11). Then, there exists
$\beta_{0}>0$ large enough such that for any $\beta>\beta_{0}$, for any
$a\in{\mathbb{Z}}_{q}$,

$\displaystyle\mu\Bigl{(}\\{\omega:\pi(\sigma_{0}=a|\omega)\leq
1-\epsilon_{1}(\beta)\\}\Bigl{|}\sigma_{0}=a\Bigr{)}\leq\epsilon_{2}(p_{1})$
(23)

where $\epsilon_{1}(\beta)=\epsilon_{1}(\beta,u,U,d)\downarrow 0$ as
$\beta\uparrow\infty$ and
$\epsilon_{2}(p_{1})=\epsilon_{2}(p_{1},u,U,d)\downarrow 0$ as
$p_{1}\downarrow 0$.
In particular, if $\mu$ is the free state of a clock model, the reconstruction
bound

$\displaystyle\int
d\mu(\omega|\sigma_{0}=a)\pi(\sigma_{0}=a|\omega)>\frac{1}{q}=\mu(\sigma_{0}=a)$
(24)

holds at large enough $\beta$.

From a signal recontruction point of view (see [26]), the statement says that
a signal $a$ which is sent from the origin to infinity through noisy canals
can be restored by the best tail-measurable predictor
$\pi(\sigma_{0}=a|\omega)$ up to thermal fluctuations $\epsilon_{1}(\beta)$ up
to an error probability $\epsilon_{2}(p_{1})$.

The proof is a direct consequence of the Key Lemma 4.6 proved in Section 4.3
which relies on a good site/bad site decomposition for boundary conditions
$\omega$, together with Peierls bounds under disorder.

###  3.2 Positivity of the Edwards-Anderson parameter and non-extremality

The so-called Edwards-Anderson parameter is usually defined in spin glass
models as a quantity measuring the degree of randomness of the (random) spin
magnetisation at the origin (see e.g. [9]). In our context, the analogue
should be a quantity measuring the degree of randomness of the probability
vector $(\pi(\sigma_{0}=a|\omega))_{a\in{\mathbb{Z}}_{q}}$, when the boundary
condition at infinity $\omega$ is distributed according to $\mu$. Let us thus
define the following quantity.

######  Definition 3.1 (Central state Edwards-Anderson parameter).

$\begin{split}q_{\rm EA}^{\mu}&:=\frac{1}{q}\sum_{a\in{\mathbb{Z}}_{q}}{\rm
Var}_{\mu}\bigl{(}{\pi({\sigma_{0}=a}|\cdot})\bigr{)}.\cr\end{split}$ (25)

Note that when $\mu$ is the free state of a clock model, by symmetry the above
definition boils down to

$\begin{split}q_{\rm EA}^{\text{Clock}}&:={\rm
Var}_{\mu}\bigl{(}{\pi({\sigma_{0}=a}|\cdot})\\\
&=\mu({\pi({\sigma_{0}=a}|\cdot})^{2})-\frac{1}{q^{2}}\end{split}$ (26)

for any fixed $a\in{\mathbb{Z}}_{q}$, whereas for the free state of the Ising
model in zero field, Definition 3.1 writes

$\begin{split}&q_{\rm EA}^{\text{Ising}}=\frac{1}{4}{\rm
Var}\bigl{(}{\pi({\sigma_{0}}|\cdot)}\bigr{)}.\end{split}$ (27)

Clearly, having $q_{\rm EA}>0$ implies that there exists some
$a\in{\mathbb{Z}}_{q}$ such that ${\rm Var}_{\mu}(\pi(\sigma_{0}=a|\cdot))>0$
and thus, recalling (14), the tail-measurable random variable
$\mu(\sigma_{0}=a|{\cal{F}_{\infty}})(\cdot)$ is not $\mu$-a.s. constant. Thus
$\mu$ cannot be tail-trivial. We have the following quantitative lower bound.

######  Theorem 3.2.

Consider any central state $\mu$ in the class of models with Hamiltonian of
the form (10) fulfilling the bounds (2.1) and (11). Then, there exists
$\beta_{0}>0$ large enough such that for any $\beta>\beta_{0}$,

$\displaystyle q_{\rm
EA}^{\mu}\geq(1-\epsilon_{1}(\beta))^{2}(1-\epsilon_{2}(p_{1}))-\max_{a\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=a)$
(28)

where $\epsilon_{1}(\beta)\downarrow 0$ as $\beta\uparrow\infty$ and
$\epsilon_{2}(p_{1})\downarrow 0$ as $p_{1}\downarrow 0$ are the same
quantities as in Theorem 3.1. In particular, for large enough $\beta$ and
small enough $p_{1}=p_{1}(\mu)$,

$q_{\rm EA}^{\mu}>0$

and thus $\mu$ is not extremal.

Note that for the free state of clock models
$\max_{a\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=a)$ is equal to $1/q$ by symmetry.
More generally, for a central state this term is close to $1/q$ (see (43) and
Definition 4.2 below).

Theorem 3.2 follows from Theorem 3.1. The proof is elementary and can be found
in Section 4.4.

###  3.3 Continuity of the extremal decomposition of the free state

We prove in the following theorem that uncountably many extremal Gibbs
measures enter the decomposition of the central state at low enough
temperature.

######  Theorem 3.3 (Almost-sure singularity of extremals).

Consider any central state $\mu$ in the class of models with Hamiltonian of
the form (10) fulfilling the bounds (2.1) and (11). Then, there exist
$\beta_{0}>0$ large enough and $p_{1}^{0}=p_{1}^{0}(\beta,u,U,d)>0$ small
enough such that for any $\beta>\beta_{0}$, $p_{1}(\mu)<p_{1}^{0}$ and for
$\mu\otimes\mu$-a.e. pair $(\omega,\omega^{\prime})$ the extremal Gibbs
measures $\pi(\cdot|\omega)$ and $\pi(\cdot|\omega^{\prime})$ are singular
with respect to each other, i.e.

$\displaystyle\mu\otimes\mu\big{(}\\{(\omega,\omega^{\prime})\in\Omega\times\Omega:\pi(\cdot|\omega)\perp\pi(\cdot|\omega^{\prime})\\}\big{)}=1.$
(29)

######  Corollary 3.1.

The decomposition measure $\alpha_{\mu}$ has no atoms, i.e.
$\alpha_{\mu}(\\{\nu\\})=0$ for all $\nu\in{\rm ex}\mathcal{G}(\gamma)$. In
particular there are uncountably many extremal states which enter in the
extremal decomposition of $\mu$.

###### Proof.

Suppose the opposite, namely that $\mu(\pi(\cdot|\omega)=\mu_{0})>0$, for some
atom $\mu_{0}$. Then

$\displaystyle\mu\otimes\mu(\\{(\omega,\omega^{\prime}):\pi(\cdot|\omega)\perp\pi(\cdot|\omega^{\prime})\\})$
$\displaystyle=1-\mu\otimes\mu(\\{(\omega,\omega^{\prime}):\pi(\cdot|\omega)=\pi(\cdot|\omega^{\prime})\\})$
$\displaystyle\leq 1-\mu(\pi(\cdot|\omega)=\mu_{0})^{2}<1.$

which is a contradiction. ∎

The idea to prove almost sure singularity of typical extremals taken from the
product measure is to produce a tail-measurable order parameter, which carries
enough information to distinguish to two different typical extremal Gibbs
measures. Indeed, as the infinite-volume kernel $\pi(\cdot|\omega)$ is
supported on the extremals, and extremals are uniquely described by their
restriction to the tail-sigma algebra, it suffices to find a tail-measurable
observable $\phi$ on which the expectations $\pi(\phi|\omega)$ and
$\pi(\phi|\omega^{\prime})$ differ.

For this purpose we construct $\phi$ by looking at empirical sequences of
overlaps of the spin variables $\sigma$ with $\omega$, and show that its
expectation becomes big in $\pi(\phi|\omega)$ on the one hand, and small in
$\pi(\phi|\omega^{\prime})$ on the other hand, for typical choices of
$(\omega,\omega^{\prime})$. Theorem 3.3 will thus follow from a control of so-
called branch overlaps defined in the section below.

###  3.4 Concentration of branch-overlap for typical extremals

Let $n\in{\mathbb{N}}$ and ${\rm r}=(r_{1},r_{2},\ldots)$ be an increasing
sequence of positive integers. Let

$\begin{split}&{\Lambda_{n}=\Lambda_{n}^{\rm
r}}=\\{v_{1},v_{2},\dots,v_{n^{2}}\\}\subset V\end{split}$ (30)

so that $|\Lambda_{n}|=n^{2}$, and the vertices $v_{i}$ are chosen along a
branch of the tree, in such a way that their spacing in graph distance is
given by the sequence ${\rm r}$, i.e. $|v_{i+1}-v_{i}|=:r_{i}$ for all
$i\in\\{1,\ldots,n^{2}\\}$.

######  Definition 3.2 (Thinned branch overlaps).

Define the tail-measurable function, called thinned branch-overlap, measuring
how much the configuration $\sigma$ matches with $\omega$ on the increasing
sparse volumes $\Lambda_{n}$, as

$\begin{split}&\underline{\phi}^{\omega}=\liminf_{n\uparrow\infty}\frac{1}{|\Lambda^{\rm
r}_{n}|}\sum_{v\in\Lambda^{\rm
r}_{n}}1_{\sigma_{v}=\omega_{v}}.\cr\end{split}$ (31)

For any fixed configuration $\omega$, this is a tail-measurable observable
w.r.t. the dependence on the spins $\sigma$, which takes values on the
interval $[0,1]$. Analogous tail-measurability also holds w.r.t. to the
parameter $\omega$.

Note that there is thinning in two ways: the volume becomes increasingly
sparse, and the liminf is taken along volumes of $n^{2}$ sites. The following
theorem can be viewed as a quantitative statement of the glassiness of the
states $\pi(\cdot|\omega)$, for $\mu$-almost every $\omega$: it describes
quantitatively how much typical configurations $\sigma$ sampled from
$\pi(\cdot|\omega)$ are $\omega$-like.

######  Theorem 3.4 (Branch overlap).

Consider any central state $\mu$ in the class of models with Hamiltonian of
the form (10) fulfilling the bounds (2.1) and (11). Then there are two
functions $\epsilon_{1}(\beta)\downarrow 0$ as $\beta\uparrow\infty$ and
$\epsilon_{2}(p_{1})\downarrow 0$ as $p_{1}\downarrow 0$ such that for sparse
enough sets $\Lambda_{n}=\Lambda_{n}^{\rm r}$ the following holds:
For $\mu$-a.e. $\omega$, the thinned branch-overlap
$\underline{\phi}^{\omega}$ is $\pi(\cdot|\omega)$-a.s. lower bounded by

$\begin{split}\underline{\phi}^{\omega}=&\liminf_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{\sigma_{v}=\omega_{v}}\geq
1-\epsilon_{1}(\beta)-\epsilon_{2}(p_{1}).\end{split}$ (32)

The quantities $\epsilon_{1}(\beta)$ and $\epsilon_{2}(p_{1})$ are the same
quantities as in Theorem 3.1. The proof can be found in Section 4.5.4. From
Theorem 3.4, we deduce that the tail-measurable observable
$\underline{\phi}^{\omega}$ has different expectations under
$\pi(\cdot|\omega)$ and $\pi(\cdot|\omega^{\prime})$ if
$\omega\neq\omega^{\prime}$, through the following corollary.

######  Corollary 3.2.

Let $\beta$ be large enough and $p_{1}$ be small enough such that

$\epsilon_{1}(\beta)+\epsilon_{2}(p_{1})<\frac{1}{2}(1-{\sum_{a\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=a)^{2}}).$
(33)

Then there is a sequence of integers ${\rm r}=(r_{i})_{i\in{\mathbb{N}}}$ such
that for the correspondingly defined tail-measurable observable
$\underline{\phi}^{\omega}$ we have the strict inequality

$\begin{split}&\pi(\underline{\phi}^{\omega}|\omega)>\pi(\underline{\phi}^{\omega}|\omega^{\prime})\end{split}$
(34)

for $\mu\otimes\mu$-a.e. $(\omega,\omega^{\prime})$.

The quantities $\epsilon_{1}(\beta)$ and $\epsilon_{2}(p_{1})$ are the same
quantities as in Theorem 3.1. The proof can be found in Section 4.6. From
Corollary 3.2 the statement of Theorem 3.3 is immediate.

###  3.5 Structure of the proofs

To prove the concentration bound of Theorem 3.4 it is useful to adopt a
quenched-disordered systems view (we refer to [7] for a general introduction
to statistical mechanics of disordered systems). A configuration $\omega$
drawn from the free/central state $\mu$ corresponds to a boundary condition at
infinity, but also plays the role of disorder. The intuition behind is that
the extremal measure $\pi(\cdot|\omega)$, which is in general inhomogeneous
and plays the role of a quenched disordered state, mostly resembles $\omega$
locally, and for $\mu$-typical $\omega$.

To make this precise, we will need contours, as introduced in [10], and
Peierls bounds relative to the reference configuration $\omega$. Typically
$\omega$ will contain a small density of broken bonds (along which $\omega$
changes). However the latter will not be uniformly sparse, as needed in [10]
and [16] to ensure the excess energy estimate leading to Peierls bounds. To
treat the rare but arbitrarily large regions where the Peierls bound locally
fails, we introduce a notion of ”bad site” (see Definition 4.5), which is an
essential tool. Our good/bad site decomposition is somewhat reminiscent to
that invented by Chayes-Chayes-Fröhlich in [8], to treat lattice Ising models
with i.i.d. random bonds, which are mostly but not strictly ferromagnetic.
However, we work on the tree, and in a regime where the ”disorder-measure”
$\mu$ is not tail-trivial, it is far from an i.i.d. disorder measure, making
things more intricate.

Nevertheless, as we shall see, there is one-dimensional correlation decay,
conditionally on the state of the root, along a branch of the tree. This will
be exploited to prove exponential decorrelation of bad sites in their
distance, see Lemma 4.7, leading to concentration of thinned branch overlaps
around their means, under the ”quenched measures” $\pi(\cdot|\omega)$, for
$\mu$-almost every $\omega$, see Lemma 4.9.

##  4 Proofs

We first recall some tools about tree-indexed Markov chains and boundary laws.

###  4.1 Markov chains on trees and boundary laws.

Being nearest neighbor (n.n.), our class of models (10) lead to Gibbs measures
that are (spatially) Markov fields. As we work on trees, there is an important
class of Gibbs measures including the extremal Gibbs measures but not equal to
them, which has a direct transcription in terms of Markov chains on trees.
These are described via so-called boundary laws introduced by Zachary [32].
Boundary laws are (non-normalized) positive measures which are invariant under
an interaction-dependent non-linear map along the tree. Moreover they are
closely related (but not equal) to the invariant single-site probability
measures for the associated one-step Markov chain transition matrix, see
below, and see Georgii [19, Chapter 12] for details.

In our case of a homogeneous n.n. interaction on the tree, the specification
(4) is equivalently described by a positive transfer matrix (or transfer
operator) $Q:{\mathbb{Z}}_{q}\times{\mathbb{Z}}_{q}\mapsto(0,\infty)$ via the
prescription

$\gamma_{\Lambda}(\omega_{\Lambda}\mid\omega_{\Lambda^{c}})=\frac{1}{Z^{\omega}_{\Lambda}}\prod_{{\\{v,w\\}\cap\Lambda\neq\emptyset}\atop{v\sim
w}}Q(\omega_{v},\omega_{w}).$

where, writing $b=\\{v,w\\}$, with pair potential $\Phi$, and single-site
potential $\Psi$ as in (10), and

$Q(\omega_{v},\omega_{w})=:Q_{b}(\omega)=e^{-\beta(\Phi_{b}(\omega)+(\Psi(\omega_{v})+\Psi(\omega_{w}))/(d+1))}.$

####  4.1.1 Boundary laws

######  Definition 4.1.

A boundary law $\lambda$ for a transfer matrix $Q$ is a family of row vectors
$\lambda_{vw}\in\;]0,\infty[^{\mathbb{Z}_{q}}$ which satisfy, for all oriented
pairs of nearest neighbors $v,w\in V$, the following consistency equation:
there exists some $c_{vw}>0$ such that for all $i\in\mathbb{Z}_{q}$,

$\lambda_{vw}(i)=c_{vw}\prod_{z\in\partial\\{v\\}\setminus\\{w\\}}\sum_{j\in{\mathbb{Z}}_{q}}Q(i,j)\lambda_{zv}(j).$

######  Theorem 4.1 ([32]).

There is a one-to-one relation between Gibbs measures $\mu$ which are also
tree-indexed Markov chains and boundary laws $\lambda$.
A Gibbs measure $\mu$ is then described via its finite-volume marginals in any
$\Lambda\Subset V$ in the form

$\mu(\omega_{\Lambda\cup\partial\Lambda})=(Z^{\omega}_{\Lambda})^{-1}\prod_{w\in\partial\Lambda}\lambda_{ww_{\Lambda}}(\omega_{w})\prod_{b\cap\Lambda\neq\emptyset}Q_{b}(\omega),$
(35)

where $w_{{\Lambda}}$ is the unique neighbor of $w\in\partial\Lambda$ which
lies in $\Lambda$. The Markov chain transition operator is given for all
$v,w\in V$ by the stochastic matrix

$P_{vw}(\omega_{v},\omega_{w})=\frac{Q(\omega_{v},\omega_{w})\lambda_{wv}(\omega_{w})}{\sum_{j\in{\mathbb{Z}}_{q}}Q(\omega_{v},j)\lambda_{wv}(j)}.$
(36)

We note that no homogeneity of the Gibbs measure, boundary law, and transition
operator are assumed. It is perfectly possible and very relevant that
homogeneous $Q$ allow for non-trivial non-homogeneous Gibbs measures $\mu$ (as
for example the Blekher-Ganikhodgaev states [3], which can be seen as some
analogues of the Dobrushin states [11] on the tree). On the other hand, the
theory presented above also allows for non-homogeneous interactions depending
on the edges, but we do not consider this case in the present work.

In the special case of homogeneous boundary laws $\lambda_{vw}(i)\equiv u(i)$
on regular trees of degree $d$, considered here, note that these must satisfy
the homogeneous equation

$u(i)=c\big{(}\sum_{j\in{\mathbb{Z}}_{q}}Q(i,j)u(j)\big{)}^{d},$ (37)

which we may write in short notation as $u=c(Qu)^{d}$. The constant $c$ can be
chosen to our convenience, a possible and often convenient choice is $c=1$.
The single-site marginal of the measure given in (35) then becomes444where the
$\ell^{p}$ norm of a function $f:{\mathbb{Z}}_{q}\to{\mathbb{R}}$ is defined
by $\|f\|_{p}:=(\sum_{i\in{\mathbb{Z}}_{q}}|f(i)|^{p})^{1/p}$.
${u^{\frac{d+1}{d}}}/{\|u^{\frac{d+1}{d}}\|_{1}}$ and the transition matrix
given in (36) becomes

$P^{u}(i,j)=\frac{Q(i,j)u(j)}{\sum_{k\in{\mathbb{Z}}_{q}}Q(i,k)u(k)},\text{
for }i,j\in{\mathbb{Z}}_{q}.$ (38)

In our non-hard core context, homogeneous boundary laws $u$ can also be
characterized by consistent effective boundary fields $h=(h_{1},\dots,h_{q})$,
defined as $u_{i}=e^{h_{i}}$, for $i\in\mathbb{Z}_{q}$, that themselves
satisfy a consistency mean-field equation.

In the Ising case, this equation involving hyperbolic tangent possibly has 3
independent solutions and gives rise to three homogeneous Markov chains
$\mu^{+},\mu^{-}$ and the free state $\mu$ (see the work of Spitzer [31], or
the one of Higuchi [20], who called $\mu$ the ’third Markov chain’, and
already noticed that it was not necessarily extremal). The case of the free
state $\mu$ is discussed in the next section.

For the Potts model, but also very generally for $q$-clock models in the case
of absence of an external field, this equation has a unique solution at small
$\beta$ and correspondingly uniqueness of the Gibbs measure. For large
$\beta$, the situation for general clock models in the class (9) is already
more interesting and looks as follows (see [24, 23] and [1]): For each subset
$A\subset{\mathbb{Z}}_{q}$ of the local spin-space, at sufficiently large
$\beta\geq\beta_{0}(|A|)$ there is a spatially homogeneous Markov chain
$\mu_{A}$ whose single-site marginals $\pi_{A}=\mu_{A}\circ\sigma_{0}^{-1}$
are concentrated on the spin-values in $A$. Moreover, the restriction of
$\pi_{A}$ to $A$ approximately equals the equidistribution on $A$, with
explicit $\beta$-dependent error bounds. It is important to realize that these
states contain the $q$ states with singleton-localization centers $|A|=1$,
constructed in [18], but at sufficiently low temperatures there are always
independent states with non-singleton localization centers $A$. More
precisely, in [1] an explicit threshold $\eta(d,N)>0$ was constructed such
that the condition $\|Q_{0}-1_{0}\|_{\frac{d+1}{2}}<\eta(d,|A|)$ implies
existence and concentration properties of $\mu_{A}$, where the function
$Q_{0}(i):=Q(i,0)$ describes the interaction of the clock model. It is
elementary that this smallness of the norm is implied by the explicit low-
temperature condition $(q-1)^{\frac{2}{d+1}}e^{-\beta u}<\eta(d,|A|)$ in our
class of models (9), as we shall see in the next section.

####  4.1.2 Free states of clock models

Clock models are defined by the requirement that the transfer matrix depends
only on the distance in ${\mathbb{Z}}_{q}$ between $i$ and $j$ and hence has
the form $Q(i,j)=Q_{0}(j-i)$ with an even function $Q_{0}$ on
${\mathbb{Z}}_{q}$ which we assume to be strictly positive.

In that case the homogeneous boundary law equation (37) can be written in
terms of the discrete convolution as

$u=(Q_{0}*u)^{d}.$

This in particular shows that the constant boundary law $u(i)=a$ for all
$i\in{\mathbb{Z}}_{q}$ solves the equation for the non-zero value of $a$ given
by $a=(a\|Q_{0}\|_{1})^{d}$.

The Markov chain transition operator (38) obtained for constant boundary law
is then the normalized transition operator itself, i.e.

$P^{u}(i,j)=\frac{Q_{0}(i-j)}{\|Q_{0}\|_{1}}$ (39)

which has as unique invariant distribution the equidistribution on
${\mathbb{Z}}_{q}$. The formula (35) for the Gibbs measure $\mu$ obtained for
constant boundary law then reduces to

$\mu(\omega_{\Lambda\cup\partial\Lambda})=(Z^{\omega}_{\Lambda})^{-1}\prod_{b\cap\Lambda\neq\emptyset}Q(\omega_{b}),$
(40)

from which we also see with (39) that our first definition of a homogeneous
tree-indexed Markov chain is satisfied. The r.h.s. of (40) on the other hand
is the obvious formula for the open boundary condition finite-volume Gibbs
measure in $\Lambda$. In this way we see that the free state in clock models
is rediscovered as the tree-indexed Markov chain which is obtained for
constant boundary law.

####  4.1.3 Central states of perturbed clock models

We now consider more generally models which are perturbations of clock models,
by which we mean that

$Q=Q_{0}+\tilde{Q}$

where $\tilde{Q}$ is small in (some) matrix norm. Such a situation arises for
example in the important special case of a clock model which is perturbed by
additional single-site terms coming from a non-trivial potential
$\Psi_{\\{v\\}}\equiv\Psi_{0}$. In this case the transfer operator does not
describe a clock model anymore but takes the general matrix form

$Q(i,j)=a(i)Q_{0}(i,j)a(j),$

where $a(i)=e^{-\beta\Psi(i)/(d+1)}$ is close to one for all $i$, and
$Q_{0}(i,j)$ only depends on $|i-j|$. Assume without loss of generality the
normalization $\|Q_{0}\|_{1}=1$, let $x:=u^{\frac{1}{d}}$ and write the
homogeneous equation (37) as $F(x,Q)=0$ with

$F_{i}(x,Q):=x(i)-\sum_{j}Q(i,j)x(j)^{d}\text{ for all }i\in{\mathbb{Z}}_{q}.$
(41)

Then we have

######  Lemma 4.1.

Suppose that all eigenvalues of $Q_{0}$ are different from $1/d$. Then there
is a neighborhood of $Q_{0}$ such that for all $Q$ in this neighborhood there
exists a continuously differentiable solution $Q\mapsto\bar{x}(Q)$ of the
boundary law equation (41) for $Q$, which has the property that
$\bar{x}(Q_{0})=1$.

###### Proof of Lemma 4.1.

It is a straightforward application of the implicit function theorem. We have
$F(1,Q_{0})=0$, so we can find a neighborhood around $Q_{0}$ in matrix norm
such that a perturbed solution exists, if $D_{x}F(1,Q_{0})=id_{q}-dQ_{0}$ is
invertible. ∎

Note that the eigenvalues of the matrix $Q_{0}$ are given by discrete Fourier
transform of the vector $(Q_{0}(j))_{j\in{\mathbb{Z}}_{q}}$ and hence directly
computable. We note in particular that for the $\beta$-dependent normalized
form

$Q_{0}(j)=e^{-\beta\Phi_{0}(j)}/\|e^{-\beta\Phi_{0}}\|_{1}$

with a potential $\Phi_{0}:{\mathbb{Z}}_{q}\mapsto[0,\infty)$ which has a
minimimum at $0$, and satisfies $0<u\leq\Phi_{0}(i)$ for $i\neq 0$, the
condition of Lemma 4.1 is valid for large enough $\beta$. More quantitatively,
this is ensured by

$\frac{d-1}{d+1}>(q-1)e^{-\beta u}$ (42)

which follows from the Lemma 4.2 below.

######  Definition 4.2.

Consider the continuously differentiable solution $\bar{x}$ of Lemma 4.1. We
call central state the Markov chain Gibbs state associated to the pair
$(\bar{x}(Q),Q)$ solving (41).

As a consequence of Lemma 4.1, the corresponding single-site marginal

$Q\mapsto\pi_{Q}=\frac{\bar{x}(Q)^{d+1}}{\|\bar{x}(Q)^{d+1}{\|_{1}}}$

and transition matrix

$Q\mapsto
P_{Q}(i,j)=\frac{Q(i,j)\bar{x}(Q)^{d}_{j}}{\sum_{k}Q(i,k)\bar{x}(Q)^{d}_{k}}$

are continuously differentiable perturbations of the values
$\pi_{Q_{0}}=\frac{1}{q}1$ and $P_{Q_{0}}=Q_{0}/{\|Q_{0}\|_{1}}$ of the free
state of the reference clock model $Q_{0}$ (where we denote by $1$ the vector
with all coefficients equal to 1).

Moreover, in this normalization, the perturbed solution is close to the free
state of the clock model as

$\bar{x}(Q)=1+(id_{q}-dQ_{0})^{-1}(Q-Q_{0})1+o(\|Q-Q_{0}\|)$ (43)

in any matrix norm $\|Q-Q_{0}\|$.

We now give lower bounds on the eigenvalues of the transition operator of a
clock model, which are larger than $1/d$ for $\beta$ large enough, ensuring
the condition in Lemma 4.1.

######  Lemma 4.2.

Consider a clock model with Hamiltonian in the class (9) fulfilling the bound
(2.1). We have the lower bounds on the eigenvalues $\lambda_{j}$ of the
transition operator $Q_{0}=\frac{e^{-\beta\Phi}}{\|e^{-\beta\Phi}\|_{1}}$ of
the form

$\lambda_{j}\geq\frac{1-(q-1)e^{-\beta u}}{1+(q-1)e^{-\beta u}}.$

###### Proof.

Clearly $\|e^{-\beta\Phi}\|_{1}\leq 1+(q-1)e^{-\beta u}$. To estimate the
spectral radius of the symmetric matrix $e^{-\beta\Phi}-id_{q}$, observing
that for $v$ with $\|v\|_{2}=1$ we have

$\begin{split}&|\langle v,(e^{-\beta\Phi}-id_{q})v\rangle|\leq e^{-\beta
u}\sum_{i\neq j}|v_{i}||v_{j}|\cr&=e^{-\beta u}((\sum_{i}|v_{i}|)^{2}-1)\leq
e^{-\beta u}(q-1).\end{split}$ (44)

This implies that the eigenvalues of $e^{-\beta\Phi}$ are bounded below by
$1-e^{-\beta u}(q-1)$, which proves the lemma. ∎

####  4.1.4 Lazyness assumption

Recall the lazyness parameter $p_{1}(\mu)$ in Definition 2.2. For our main
Theorem 3.4 to be meaningful we need sufficient smallness of the quantity
$p_{1}(\mu)$.

From the above follows that the central states obtained by small perturbations
of clock models have an associated transition matrix which keep the small
$p_{1}$ property. Indeed, assume again the normalization of $Q_{0}$ such that
$\|Q_{0}\|_{1}=1$, and $Q_{0}(0)=1$. Suppose that all eigenvalues of $Q_{0}$
are different from $1/d$, and let $\mu_{Q}$ denotes the central state, defined
in a sufficiently small neighborhood of $Q_{0}$. Then $Q\mapsto
p_{1}(\mu_{Q})$ is a continuously differentiable function. In particular,
whenever the reference clock model $Q_{0}$ has small

$\displaystyle p_{1}(\mu_{Q_{0}})=\frac{\sum_{i\neq
0}Q_{0}(i)}{\|Q_{0}\|_{1}}\leq(q-1)e^{-\beta u},$ (45)

this smallness carries over to $p_{1}(Q)$ for $Q$ in a sufficiently small
neighborhood around $Q_{0}$. Specifically for the Potts model as a reference
clock model we have

$p_{1}(Q_{0})=\frac{q-1}{\mathrm{e}^{\beta}+q-1}$ (46)

which tends to 0 as $\beta$ tends to infinity.

###  4.2 Contours and Excess Energy Lemma

######  Definition 4.3 (Contour with respect to a fixed configuration
$\omega^{0}$).

__
Let $\omega^{0}\in\Omega_{0}^{V}$ be a fixed reference configuration. A
contour for the spin configuration $\omega\in\Omega_{0}^{V}$ relative to
$\omega^{0}$ is a pair

$\bar{\gamma}=(\gamma,\omega_{\gamma})$

where the support $\gamma\ {\subset}\ \\{v\in
V:\omega_{v}\neq\omega_{v}^{0}\\}$ is a connected component of the set of
incorrect points for $\omega$ (with respect to $\omega^{0}$), and
$\omega_{\gamma}=(\omega_{v})_{v\in\gamma}$.

Given a fixed reference configuration $\omega^{0}$, we will work with a
contour representation of the spin partition function in a finite volume
$\Lambda$ with boundary condition equal to $\omega^{0}$, which reads

$\begin{split}Z_{\Lambda}^{\omega^{0}}={\sum_{n\in{\mathbb{N}}}}\sum_{\bar{\gamma}_{1},\dots,\bar{\gamma}_{n}}\prod_{i=1}^{n}\rho[\omega^{0}](\bar{\gamma}_{i})\end{split}$
(47)

where the sum is over pairwise compatible contours $\bar{\gamma}$ with
activities

$\rho[\omega^{0}](\bar{\gamma})=\exp({-\beta(H_{\gamma\cup\partial\gamma}(\omega_{\gamma\cup\partial\gamma})-H_{\gamma\cup\partial\gamma}(\omega^{0}_{\gamma\cup\partial\gamma}))})$

given in terms of the excess energy of $\omega$ with respect to $\omega^{0}$.
We write $\bar{\gamma}_{1}\sim\bar{\gamma}_{2}$ whenever $\bar{\gamma}_{1}$
and $\bar{\gamma}_{2}$ are compatible. This representation together with the
tree structure of our graph (where contours have no interior) imply the
following Peierls bound:

######  Lemma 4.3 (Peierls bound).

Consider $\omega^{0}\in\Omega$ and a volume $\Lambda\Subset V$. Then

$\displaystyle\mu^{\omega^{0}}_{\Lambda}(\sigma_{v}\neq\omega^{0}_{v})$
$\displaystyle=\sum_{\bar{\gamma}:\gamma\ni
v}^{\Lambda}\rho[\omega^{0}](\bar{\gamma})\cdot\frac{{\sum_{n}}\sum^{\Lambda}_{\bar{\gamma}_{1},\dots,\bar{\gamma}_{n}\sim\bar{\gamma}}\prod_{i=1}^{n}\rho[\omega^{0}](\bar{\gamma}_{i})}{{\sum_{n}}\sum^{\Lambda}_{\bar{\gamma}_{1},\dots,\bar{\gamma}_{n}}\prod_{i=1}^{n}\rho[\omega^{0}](\bar{\gamma}_{i})}$
$\displaystyle\leq\sum_{\bar{\gamma}:\gamma\ni
v}^{\Lambda}\rho[\omega^{0}](\bar{\gamma}).$ (48)

where the sum is over contours $\bar{\gamma}$ in the finite volume $\Lambda$.

######  Definition 4.4 (Set of broken bonds).

__
We define the set of broken bonds of the configuration $\omega\in\Omega$ by

$D(\omega)=\\{(v,w)\in E:\omega_{v}\neq\omega_{w}\\}$

and denote the set of edges attached to $\gamma\subset V$ by

$\begin{split}E(\gamma)&=\\{\\{x,y\\}\in
E,\\{x,y\\}\cap\gamma\neq\emptyset\\}.\end{split}$ (49)

######  Lemma 4.4 (Entropy bound, [[16],Lemma 6]).

We have the following upper bound on the number of connected subsets of
vertices of $\mathcal{T}^{d}=(V,E)$:

$\\#\\{\gamma\subset V:\gamma\text{ connected ,}\gamma\ni
0,|\gamma|=\ell\\}\leq(d+1)^{2(\ell-1)}.$

######  Lemma 4.5 (Excess Energy Lemma [[10], Lemma 2]).

__
Let $\mu$ be a central state of any model with Hamiltonian in the class (10)
fulfilling the bounds (2.1) and (11). Let
$\bar{\gamma}=(\omega^{\prime}_{\gamma},\gamma)$ be a contour relative to the
fixed configuration $\omega$. Denote
$\omega^{\prime}=(\omega^{\prime}_{\gamma}\omega_{\gamma^{c}})$ the
corresponding excited spin configuration. Then the excess energy satisfies

$\begin{split}H(\omega^{\prime})&-H(\omega)\geq\\\
&(d-1)u|\gamma|-(U+u)|D(\omega)\cap
E(\gamma)|+\sum_{v\in\gamma}(\Psi(\omega^{\prime}_{v})-\Psi(\omega_{v})).\end{split}$
(50)

In particular, if there exist $\delta\geq 0$ and $\epsilon>0$ such that
$|D(\omega)\cap E(\gamma)|\leq\delta|\gamma|$ and
$\|\Psi\|_{\infty}\leq\epsilon/2$ then

$H(\omega^{\prime})-H(\omega)\geq|\gamma|\Big{(}(d-1)u-(U+u)\delta-\epsilon\Big{)}.$
(51)

Observe that in (51), if $\epsilon=0$ and $\delta<\frac{(d-1)u}{U+u}$ then the
energy cost of the contour $\bar{\gamma}$ is positive and proportional to its
volume $|\gamma|$. This fact was crucially used in our paper [10] to exhibit
low temperature local ground states which give rise to a wide family of
inhomogenous extremal states. We thus introduce the following definition of
bad contours.

######  Definition 4.5 (Bad events).

Consider any model in the class (10). Let

$\delta_{0}:=\frac{1}{2}\cdot\frac{(d-1)u}{(u+U)}$ (52)

where $u,U\in{\mathbb{R}}^{+}$ are the bounds on energy costs defined in (2.1)
and $d$ is the branching number of the tree.
Denote by $B_{v}$ the bad event that there exists a contour around $v$ with
respect to $\omega$ which does not have good enough excess energy in the sense
that

$\begin{split}B_{v}&:=\bigcup_{\gamma:\gamma\ni v}B(\gamma)\text{ where
}B(\gamma):=\\{\omega:|D(\omega)\cap
E(\gamma)|\geq\delta_{0}|\gamma|\\}.\end{split}$ (53)

######  Definition 4.6.

Suppose that $\mu$ is a homogeneous tree-indexed Markov chain for a
Hamiltonian in the class (10) fulfilling the bounds (2.1) and (11). Let
$p_{1}=p_{1}(\mu)$ be its lazyness parameter as in Defintion 2.2. Let
$\delta_{0}$ as in (52). Define $\lambda(p_{1})=\lambda(p_{1},u,U,d)$ as

$\begin{split}&e^{-\lambda(p_{1})}:=\inf_{t\geq
0}e^{-t\delta_{0}}(p_{1}e^{t}+1-p_{1})^{(d+1)}=\left(\frac{p_{1}}{\delta_{0}}\right)^{\delta_{0}}\Bigl{(}\frac{1-p_{1}}{d+1-\delta_{0}}\Bigr{)}^{1-\delta_{0}}.\end{split}$
(54)

Note that $\lambda(p_{1})\uparrow\infty$ or equivalently
$e^{-\lambda(p_{1})}\downarrow 0\text{ as }p_{1}\downarrow 0$. In particular,
if $\mu_{\beta}$ is the free state of the $q$-clock model at inverse
temperature $\beta$ (see (46)),

$\lambda(p_{1}(\mu_{\beta}))\uparrow\infty\text{ as }\beta\uparrow\infty.$

###  4.3 Key lemma

######  Lemma 4.6 (Key Lemma).

Let $\mu$ be the central state of any model with Hamiltonian in the class (10)
fulfilling the bounds (2.1) and (11). Then for $\beta$ large enough we have,
for $\mu$-almost every $\omega$

$\begin{split}\pi(\sigma_{v}\neq\omega_{v}|\omega)\leq
1_{B_{v}}(\omega)+\epsilon_{1}(\beta)\end{split}$ (55)

where the event $B_{v}$ is defined in (4.5), and
$\epsilon_{1}(\beta)\downarrow 0$ as $\beta\uparrow\infty$. Moreover,

$\mu(B(\gamma))\leq e^{-\lambda(p_{1})|\gamma|}$ (56)

where $\lambda(p_{1})$ is defined in (LABEL:def-lambda-p1) and for $p_{1}$
small enough, for any $v\in V$

$m(p_{1}):=\mu(B_{v})\leq\epsilon_{2}(p_{1})$ (57)

where $\epsilon_{2}(p_{1})\downarrow 0$ as $p_{1}\downarrow 0$.

###### Proof.

We decompose

$\displaystyle\pi(\sigma_{v}\neq\omega_{v}|\omega)=\pi(\sigma_{v}\neq\omega_{v}|\omega)(1_{B_{v}}(\omega)+1_{B^{c}_{v}}(\omega))$
(58)

and we use the infinite-volume version of the Peierls bound of Lemma 4.3
(removing the restriction to finite volume always gives an upper bound)
applied to the complement of a bad event at $v$ to bound the probability of a
mismatch at $v$ by the sum over contours attached at $v$. This gives

$\begin{split}&\pi(\sigma_{v}\neq\omega_{v}|\omega)1_{B^{c}_{v}}(\omega)\leq\sum_{\bar{\gamma},\gamma\ni
v}\rho[\omega](\bar{\gamma})1_{B^{c}_{v}}(\omega)\end{split}$ (59)

where $\rho[\omega](\bar{\gamma})$ is the contour activity relative to
$\omega$. Note that $\omega$ is not assumed to be a local ground state (in the
sense of [10]), but we may still use the bound on the excess energy of a local
perturbation given in Lemma 4.5.
This delivers, using the assumptions on the potential in the class (10) and
the Definition 4.5 of $B_{v}$,

$\begin{split}&\rho[\omega](\bar{\gamma})1_{B^{c}_{v}}(\omega)\cr&\leq\exp({-\beta(d-1)u|\gamma|+\beta(u+U)|D(\omega)\cap
E(\gamma)|+{\|\Psi\|}_{\infty}|\gamma|})1_{B^{c}_{v}}(\omega)\cr&\leq\exp({-\beta|\gamma|((d-1)u-(u+U)\delta_{0}-{\|\Psi\|}_{\infty})})\cr&=\exp({-\beta|\gamma|{(d-1)u}/4})\end{split}$
(60)

Using the entropy bound of Lemma 4.4 we deduce that there exists
$C^{\prime},c^{\prime}>0$ such that

$\begin{split}&\pi(\sigma_{v}\neq\omega_{v}|\omega)1_{B^{c}_{v}}(\omega)\leq\sum_{\bar{\gamma},\gamma\ni
v}e^{-\beta c|\gamma|}{\leq\sum_{\ell\geq
1}(d+1)^{2(\ell-1)}(q-1)^{\ell}e^{-\beta\ell(d-1)u/4}}\cr&\leq
C^{\prime}e^{-c^{\prime}\beta}=:\epsilon_{1}(\beta)\end{split}$ (61)

for $\beta>4\log((q-1)(d+1)^{2})/(u(d-1))$, which proves (55).

Next we use the exponential Markov inequality to bound $\mu(B_{v}(\gamma))$
and sum over the contours afterwards to bound $\mu(B_{v})$, as will be
described now.

$\displaystyle\mu(B_{v}(\gamma))$ $\displaystyle=\mu\bigl{(}|D(\omega)\cap
E(\gamma)|\geq\delta_{0}|\gamma|\bigr{)}$ (62) $\displaystyle\leq\inf_{t\geq
0}e^{-t\delta_{0}|\gamma|}\mu(e^{t|D(\omega)\cap E(\gamma)|})$ (63)

It suffices to fix any $x\in\gamma$, look at the conditional measure
$\mu_{a}(\cdot):=\mu(\cdot|\sigma_{x}=a)$ for any fixed
$a\in{\mathbb{Z}}_{q}$, and find an upper bound which does not depend on $a$.

Using the Markov chain property of the measure $\mu_{a}$, successive
applications of the homogeneous transition matrix $P$ yield

$\begin{split}&\mu_{a}(e^{t|D(\omega)\cap
E(\gamma)|})\leq(p_{1}e^{t}+1-p_{1})^{(d+1)|\gamma|}\end{split}$ (64)

which follows from

$\max_{a\in{\mathbb{Z}}_{q}}\sum_{b\neq a}P(a,b)e^{t}+P(a,a)\leq
p_{1}e^{t}+1-p_{1}$

and the bound $|E(\gamma)|\leq(d+1)|\gamma|.$ Optimizing over $t$ leads to the
bound

$\begin{split}&\mu(B_{v}(\gamma)){\leq\Bigl{(}\inf_{t\geq
0}e^{-t\delta_{0}}(p_{1}e^{t}+1-p_{1})^{(d+1)}\Bigr{)}^{|\gamma|}}=e^{-\lambda(p_{1})|\gamma|}\end{split}$
(65)

which proves (56), since the definition (LABEL:def-lambda-p1) of
$\exp(-\lambda(p_{1}))$ is made to be the value of the above infimum. Finally
using the entropy bound of Lemma 4.4 on the number of contours of fixed length
which are attached to a given point, the estimate for $\mu(B_{v})$ follows.
Indeed, there exists $C>0$ such that for any $c\in(0,1)$ we have

$\displaystyle m(p_{1}):=\mu(B_{v})$ $\displaystyle\leq\sum_{\gamma:\gamma\ni
v}\mu(B_{v}(\gamma))\leq\sum_{\gamma:|\gamma|\geq
1}e^{-\lambda(p_{1})|\gamma|}$ $\displaystyle\leq\sum_{\ell\geq
1}(d+1)^{2(\ell-1)}(q-1)^{\ell}e^{-\lambda(p_{1})\ell}$ (66)
$\displaystyle\leq Ce^{-c\lambda(p_{1})}=:\epsilon_{2}(p_{1})$ (67)

for $p_{1}$ sufficiently small, which proves (57). ∎

###  4.4 Positivity of the Edwards-Anderson parameter

###### Proof of Theorem 3.2.

The proof is short and elementary, given the reconstruction bound of Theorem
3.1. Observe that

$\begin{split}&q_{\rm
EA}^{\mu}=\sum_{a\in{\mathbb{Z}}_{q}}\Bigl{(}\mu\Bigl{(}\pi({\sigma_{0}=a}|\cdot)^{2}\Bigr{)}-\mu(\sigma_{0}=a)^{2}\Bigr{)}\cr&=\sum_{a\in{\mathbb{Z}}_{q}}\Bigl{(}\sum_{b\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=b)\mu\Bigl{(}\pi({\sigma_{0}=a}|\cdot)^{2}|\sigma_{0}=b\Bigr{)}-\mu(\sigma_{0}=a)^{2}\Bigr{)}\cr&\geq\sum_{a\in{\mathbb{Z}}_{q}}\Bigl{(}\mu(\sigma_{0}=a)\mu\Bigl{(}\pi({\sigma_{0}=a}|\cdot)^{2}|\sigma_{0}=a\Bigr{)}-\mu(\sigma_{0}=a)^{2}\Bigr{)}\cr\end{split}$
(68)

which, by the quantitative reconstruction Theorem 3.1 gives the lower bound

$\begin{split}q_{\rm
EA}^{\mu}&\geq\sum_{a\in{\mathbb{Z}}_{q}}\Bigl{(}\mu(\sigma_{0}=a)(1-\epsilon_{1}(\beta))^{2}(1-\epsilon_{2}(p_{1}))-\mu(\sigma_{0}=a)^{2}\Bigr{)}\\\
&\geq(1-\epsilon_{1}(\beta))^{2}(1-\epsilon_{2}(p_{1}))-\max_{a\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=a)\end{split}$
(69)

which proves the theorem. ∎

###  4.5 Overlap control for typical extremals

####  4.5.1 Exponential decorrelation of bad sites

Suppose that $\mu$ is a homogeneous tree-indexed Markov chain. We do not need
to assume any invariances of the measure $\mu$ under joint transformations of
the local spin-spaces (like a discrete clock-rotation or even permutation
symmetry). We will only assume that the jump probability $p_{1}$ (see
Definition 2.2), which gives a bound on the density of broken bonds in the
configuration drawn from $\mu$, is small enough. Using this assumption, we
derive decorrelation between two bad events occuring at the sites $u$ and $v$
(see Definition 4.5), with an upper bound that is exponential small in the
distance between $u$ and $v$.

######  Lemma 4.7.

Consider the central state $\mu$ of any model with Hamiltonian in the class
(10) fulfilling the bounds (2.1) and (11). Suppose that the transition matrix
$P$ of the state $\mu$ is irreducible and aperiodic. Recall the Definition 4.5
of a bad event $B_{v}$ at $v\in V$ and the notation for its expectation
$\mu(B_{v})=:m(p_{1})$ (which does not depend on $v$). Let

$\mbox{$\mathrm{C}ov$}(u,v):=\mu\Bigl{(}(1_{B_{v}}(\omega)-m(p_{1}))(1_{B_{u}}(\omega)-m(p_{1}))\Bigr{)}.$
(70)

Then for small enough $p_{1}$, there exist $C>0$ such that for any
$c_{1}\in(0,\frac{1}{3})$ and $c_{2}\in(0,1)$, for any $u,v\in V$,

$\begin{split}|\mbox{$\mathrm{C}ov$}(u,v)|&\leq Ce^{-c|v-u|}\end{split}$ (71)

where $|v-u|$ is the graph distance in the tree and

$\begin{split}&e^{-c}=\max\\{e^{-c_{1}\lambda(p_{1})},|\lambda_{2}(P)|^{c_{2}}\\}\end{split}$
(72)

where $\lambda(p_{1})$ is defined in (LABEL:def-lambda-p1) and
$\lambda_{2}(P)$ is the second largest eigenvalue (in modulus) of the
transition matrix $P$ of $\mu$.

The proof uses a combination of two facts. First, long bad contours attached
to a given site are exponentially improbable, see Lemma 4.6. This explains the
occurence of $\lambda(p_{1})$. Second there is exponential relaxation of the
one-dimensional Markov chain obtained by restriction of $\mu$ to a branch,
conditionally on a root, which explains the appearance of the second largest
eigenvalue $\lambda_{2}(P)$ of the transition matrix. Of course
$|\lambda_{2}(P)|<1$.

###### Proof.

We first define several objects and events which are depicted on Figure 1.

Figure 1: Objects appearing in the proof of Lemma 4.7 of exponential
decorrelation of bad events at $u$ and $v$. On the picture $d=2,|u-v|=11,r=3$.
Half-trees $H_{u}(3)$ and $H_{v}(3)$ are delimited by the two vertical lines.
One of the midpoints between $u$ and $v$ is denoted by $x$. The support of the
bad event $B_{u}^{1}$, depicted with a dashed line, stays inside $H_{u}(3)$,
while the support of the bad event $B_{v}^{2}$, depicted with a dotted line,
exits $H_{v}(3)$. The vertex $y$ is the closest to $u$ on the path from $u$ to
$x$ which does not touch the support of $B_{u}^{1}$.

Denote by the half-trees $H_{u}(r)$ and $H_{v}(r)$ the sets

$\begin{split}&H_{u}:=H_{u}(r):=\\{x\in
V,|x-u|\leq|x-v|-r\\}\cr&H_{v}:=H_{v}(r):=\\{x\in
V,|x-v|\leq|x-u|-r\\}\cr\end{split}$ (73)

We will choose $r=\lfloor|u-v|/3\rfloor$. Clearly $H_{u}$ and $H_{v}$ are
disjoint, with their distance growing proportionally to $|u-v|$. Define

$\begin{split}&B^{1}_{u}:=\bigcup_{\gamma:H_{u}\supset\gamma\ni
u}B_{u}(\gamma),\quad B^{2}_{u}:=B_{u}\backslash
B^{1}_{u}\cr&B^{1}_{v}:=\bigcup_{\gamma:H_{v}\supset\gamma\ni
u}B_{v}(\gamma),\quad B^{2}_{v}:=B_{v}\backslash B^{1}_{v}.\cr\end{split}$
(74)

The sets $B^{1}_{u}$ and $B^{1}_{v}$ describe bad events at $u$ (resp. $v$)
which are produced by contours included in $H_{u}$ (resp. $H_{v}$). The sets
$B^{2}_{u}$ and $B^{2}_{v}$ describe bad events which are not produced by the
previous contours and may connect $u$ and $v$.

Bad event caused by long contours are exponentially improbable in the minimal
length of such contours. Namely, by the Key Lemma 4.6 and the entropy bound
given in Lemma 4.4, there exist $C_{1}>0$ such that for any
$c_{1}\in(0,\frac{1}{3})$ we have

$\begin{split}\mu(B^{2}_{u})=\mu(B^{2}_{v})&\leq\sum_{\gamma:\gamma\ni
u,\gamma\cap
H^{c}_{u}\neq\emptyset}\mu(B_{u}(\bar{\gamma}))\leq\sum_{\gamma:\gamma\ni
u,|\gamma|\geq|u-v|/3}e^{-\lambda(p_{1})|\gamma|}\\\
&\leq\sum_{\ell\geq|u-v|/3}(d+1)^{2(\ell-1)}(q-1)^{\ell}e^{-\lambda(p_{1})\ell}\\\
&\leq C_{1}\exp({-c_{1}\lambda(p_{1})|u-v|}).\end{split}$ (75)

Spelling out the correlation function between the bad events in terms of the
decomposition (LABEL:bad-decomposition), we therefore have

$\begin{split}&|\mbox{$\mathrm{C}ov$}(u,v)|=|\mu(B^{1}_{u};B^{1}_{v})+\mu(B^{1}_{u};B^{2}_{v})+\mu(B^{2}_{u};B^{1}_{v})+\mu(B^{2}_{u};B^{2}_{v})|\cr&\leq|\mu(B^{1}_{u};B^{1}_{v})|+3\min\\{\mu(B^{2}_{u}),\mu(B^{2}_{v})\\}\\\
&\leq|\mu(B^{1}_{u};B^{1}_{v})|+3C_{1}\exp({-c_{1}\lambda(p_{1})|u-v|}).\cr\end{split}$
(76)

It remains to control the decay of $|\mu(B^{1}_{u};B^{1}_{v})|$. We would have
the exact decorrelation $\mu(B^{1}_{u};B^{1}_{v})=0$ in the free state of a
clock model, but for an non-symmetric model there is no reason to assume that
such decorrelation holds. Nevertheless, we may use the Markov chain property
of $\mu$ along the path from $u$ to $v$ and the exponential convergence of the
corresponding 1d-Markov chain, to arrive at the decorrelation which is
exponential in the distance $|u-v|$ (with our irreducibility and aperiodicity
assumptions).

Indeed, choose $x=x(u,v)$ to be the (or one of the at most two) middle sites
between $u$ and $v$ (which means that the distances $|u-x|$ and $|v-x|$ differ
at most by one). From the tree-indexed Markov chain property of $\mu$, using
conditional independence, we have

$\begin{split}&\mu(B^{1}_{u}\cap
B^{1}_{v})=\sum_{a\in{\mathbb{Z}}_{q}}\mu(B^{1}_{u}|\sigma_{x}=a)\mu(B^{1}_{v}|\sigma_{x}=a)\mu(\sigma_{x}=a).\end{split}$
(77)

Let $\Delta_{u,x}(a):=\mu(B^{1}_{u}|\sigma_{x}=a)-\mu(B^{1}_{u})$ and
similarly for $v$. We have

$\displaystyle|\mu(B^{1}_{u};B^{1}_{v})|$
$\displaystyle=\big{|}\sum_{a\in{\mathbb{Z}}_{q}}\Bigl{(}\mu(B^{1}_{u})\Delta_{v,x}(a)+\mu(B^{1}_{v})\Delta_{u,x}(a)+\Delta_{u,x}(a)\Delta_{v,x}(a)\Bigr{)}\mu(\sigma_{x}=a)\big{|}$
$\displaystyle\leq
2\max_{a\in{\mathbb{Z}}_{q}}|\Delta_{u,x}(a)|+\max_{a\in{\mathbb{Z}}_{q}}|\Delta_{v,x}(a)|.$
(78)

By symmetry it suffices to bound $|\Delta_{u,x}(a)|$. Choose $y$ on the path
$[x,u]$, between $x$ and $u$, such that it is the closest to $u$ with the
property that all the contours appearing in the definition of $B^{1}_{u}$ do
not touch $(x,y]$. To control the difference between
$\mu(B^{1}_{u}|\sigma_{x}=a)$ and $\mu(B^{1}_{u})$ we use the exponential
relaxation of the homogenous 1d Markov chain which arises as restriction of
$\mu$, on the path from $x$ to $y$. There exists $C_{2}>0$ such that for any
$c_{2}\in(0,1)$,

$\begin{split}&|\Delta_{u,x}(a)|=|\sum_{b\in{\mathbb{Z}}_{q}}\mu(B^{1}_{u}|\sigma_{y}=b)(\mu(\sigma_{y}=b|\sigma_{x}=a)-\mu(\sigma_{y}=b))|\cr&\leq\sum_{b\in{\mathbb{Z}}_{q}}|\mu(\sigma_{y}=b|\sigma_{x}=a)-\mu(\sigma_{y}=b))|\cr&=\sum_{b\in{\mathbb{Z}}_{q}}|P^{|x-y|}(a,b)-\mu(\sigma_{y}=b))|\cr&\leq
C_{2}|\lambda_{2}(P)|^{c_{2}|x-y|}\end{split}$ (79)

where the last line follows e.g. from [6, Example 4.3.9: Rates of Convergence
via the Perron-Frobenius Theorem]. As by construction
$|y-x|\sim\frac{1}{6}|u-v|$, as $|u-v|\uparrow\infty$, this implies the
desired exponential decorrelation of $|\mu(B^{1}_{u};B^{1}_{v})|$, and
concludes the proof. ∎

####  4.5.2 Almost sure convergence of empirical means of bad sites

As a consequence of the exponential decorrelation of bad events we harvest the
following lemma.

######  Lemma 4.8.

Consider the central state $\mu$ of any model with Hamiltonian of the form
(10) fulfilling the bounds (2.1) and (11). For any sequence of subsets
$\Lambda_{n}=\Lambda_{n}^{r}$ of a branch of the tree such that
$\sum_{n=1}^{\infty}{|\Lambda_{n}|}^{-1}<\infty$ and for $\mu$-almost every
$\omega$, the following limit holds

$\begin{split}&\lim_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{B_{v}}(\omega)=m(p_{1})\end{split}$
(80)

where $m(p_{1})=\mu(B_{0})$.

While the limit in question, if it exists is necessarily tail-trivial, the
statement is non-trivial as the measure $\mu$ is not extremal at low
temperature. So it must be proved manually.

###### Proof.

The almost sure limit holds, if we can show that for any fixed $\delta>0$ the
number of indices $n$ for which

$\begin{split}&\frac{1}{|\Lambda_{n}|}\Bigl{|}\sum_{v\in\Lambda_{n}}(1_{B_{v}}(\omega)-m(p_{1}))\Bigr{|}\geq\delta\end{split}$
(81)

is finite, for $\mu$-almost every $\omega$. By Borel-Cantelli it suffices to
show that

$\begin{split}&\sum_{n=1}^{\infty}\mu\Bigl{(}\frac{1}{|\Lambda_{n}|}\Bigl{|}\sum_{v\in\Lambda_{n}}(1_{B_{v}}(\omega)-m(p_{1}))\Bigr{|}>\delta\Bigr{)}<\infty.\end{split}$
(82)

For this we use the quadratic Chebychev inequality to bound the probability in
the above sum by

$\begin{split}&\frac{1}{|\Lambda_{n}|^{2}\delta^{2}}\sum_{v,u\in\Lambda_{n}}\mu\Bigl{(}(1_{B_{v}}(\omega)-m(p_{1}))(1_{B_{u}}(\omega)-m(p_{1}))\Bigr{)}.\end{split}$
(83)

Recalling the definition (70) of site covariances $\mbox{$\mathrm{C}ov$}(u,v)$
and using Lemma 4.7, we deduce that there exists $C>0$ such that

$\begin{split}&\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|^{2}}\sum_{v,u\in\Lambda_{n}}\mbox{$\mathrm{C}ov$}(u,v)\cr&\leq\sum_{n=1}^{\infty}\Biggl{(}\frac{1}{|\Lambda_{n}|}\mbox{$\mathrm{C}ov$}(0,0)+\frac{2}{|\Lambda_{n}|^{2}}\sum_{v\in\Lambda_{n}}\sum_{u\in\Lambda_{n},u>v}\mbox{$\mathrm{C}ov$}(u,v)\Biggr{)}\cr&\leq
C\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|}<\infty\end{split}$ (84)

since for any fixed vertex $v\in V$, the sum
$\sum_{u>v}\mbox{$\mathrm{C}ov$}(u,v)\leq C^{\prime}<\infty$ uniformly in $v$
by Lemma 4.7. This proves the existence of the almost sure limit of empirical
sums of indicators of bad points, along the quadratic volume sequences defined
in (LABEL:lambda-n-r). ∎

####  4.5.3 Concentration of spin-overlaps under typical extremals

######  Lemma 4.9.

Consider the central state $\mu$ of any model with Hamiltonian in the class
(10) fulfilling the bounds (2.1) and (11).
There exist a sequence of spacings $r=(r_{i})_{i\in{\mathbb{N}}}$, in general
depending on model parameters, such that for any sequence of subsets
$\Lambda_{n}=\Lambda^{r}_{n}$ of a branch of the tree such that
$\sum_{n=1}^{\infty}{|\Lambda_{n}|}^{-1}<\infty$, the following holds.
For $\mu$-almost every $\omega$, for $\pi(\cdot|\omega)$-almost every
realizations of $\sigma$ we have

$\begin{split}\limsup_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\Bigl{|}\sum_{v\in\Lambda_{n}}(1_{\sigma_{v}\neq\omega_{v}}-\pi(\sigma_{v}\neq\omega_{v}|\omega))\Bigr{|}=0.\end{split}$
(85)

###### Proof.

It suffices to prove that, for any $\delta>0$ we have, for $\mu$-a.e.
$\omega$,

$\begin{split}\pi\left(\frac{1}{|\Lambda_{n}|}\Bigl{|}\sum_{v\in\Lambda_{n}}(1_{\sigma_{v}\neq\omega_{v}}-\pi(\sigma_{v}\neq\omega_{v}|\omega))\Bigr{|}>\delta\text{
for infinitely many }n\,\middle|\,\omega\right)=0\end{split}$ (86)

Using the Borel-Cantelli lemma it suffices to show that, for $\mu$-a.e.
$\omega$,

$\begin{split}\sum_{n=1}^{\infty}\pi\left(\frac{1}{|\Lambda_{n}|}\Bigl{|}\sum_{v\in\Lambda_{n}}(1_{\sigma_{v}\neq\omega_{v}}-\pi(\sigma_{v}\neq\omega_{v}|\omega))\Bigr{|}>\delta\,\middle|\,\omega\right)<\infty.\end{split}$
(87)

Now by the quadratic Chebychev inequality, it is enough to show that

$\begin{split}\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|^{2}}\sum_{v,u\in\Lambda_{n}}|\pi(\sigma_{v}\neq\omega_{v};\sigma_{u}\neq\omega_{u}|\omega)|<\infty\end{split}$
(88)

holds for $\mu$-a.e. $\omega$. We would like to use the abstract fact that
$\pi(\cdot|\omega)$ is an extremal Gibbs measure, and hence decorrelates in
the sense of (12). Indeed, by extremality of $\pi(\cdot|\omega)$ at fixed
$\omega$, for any fixed $u$ and any sequence $v\uparrow\infty$ (meaning that
$v$ leaves any finite ball) the following non-quantative result holds: For
$\mu$-a.e. $\omega$,

$\begin{split}\lim_{v\uparrow\infty}|\pi(\sigma_{v}\neq\omega_{v};\sigma_{u}\neq\omega_{u}|\omega)|=0.\end{split}$
(89)

The problem is that the rate of convergence may depend on $\omega$, while we
want a speed which is uniform in all the extremals (in order for the limit
(85) to hold along the same sparse sequence of volumes for $\mu$-a.e.
$\omega$). An explicit analysis is difficult, due to the lack of spatial
homogeneity. To bypass this difficulty, note that to ensure (88) it is
sufficient to ensure that

$\begin{split}\int\mu(d\omega)\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|^{2}}\sum_{v,u\in\Lambda_{n}}|\pi(\sigma_{v}\neq\omega_{v};\sigma_{u}\neq\omega_{u}|\omega)|<\infty.\end{split}$
(90)

Using monotone convergence for non-negative functions to interchange the
$\mu$-integral and the infinite sum, we can deduce this from

$\begin{split}\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|^{2}}\sum_{v,u\in\Lambda_{n}}\underbrace{\int\mu(d\omega)|\pi(\sigma_{v}\neq\omega_{v};\sigma_{u}\neq\omega_{u}|\omega)|}_{=:{c(u,v)}}<\infty\end{split}$
(91)

By dominated convergence we have, for any $u\in V$,

$\begin{split}{\lim_{v\uparrow\infty}c(u,v)=0}\end{split}$ (92)

We look at the off-diagonal terms in the double-sum (91) as in (LABEL:mll).
Now we can achieve the desired summability of

$\begin{split}\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|^{2}}\sum_{i=1}^{|\Lambda_{n}|}\sum_{j=1}^{i-1}c(v_{j},v_{i})<\infty\end{split}$
(93)

by iteratively choosing $v_{i}$ given $v_{1},\dots,v_{i-1}$ on the same branch
so large that

$\sum_{j=1}^{i-1}c(v_{j},v_{i})<i^{-2}.$

The above choice provides the following upper bound of (93) as

$\sum_{n=1}^{\infty}\frac{1}{|\Lambda_{n}|}\sum_{i=1}^{\infty}i^{-2}$

which is finite. ∎

####  4.5.4 Proof of Theorem 3.4

###### Proof.

Applying Lemma 4.9, the Key Lemma 4.6, and Lemma 4.8, we deduce that for
sparse enough sets $\Lambda_{n}=\Lambda_{n}^{\rm r}$ the following holds. For
$\mu$-almost every $\omega$, $\pi(\cdot|\omega)$-almost surely,

$\displaystyle\liminf_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{\sigma_{v}=\omega_{v}}$
$\displaystyle\geq
1-\limsup_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}\pi(\sigma_{v}\neq\omega_{v}|\omega)$
$\displaystyle\geq
1-\liminf_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{B_{v}}(\omega)-\epsilon_{1}(\beta)$
$\displaystyle=1-\epsilon_{1}(\beta)-m(p_{1})$ $\displaystyle\geq
1-\epsilon_{1}(\beta)-\epsilon_{2}(p_{1})$

with $\epsilon_{1}(\beta)=C^{\prime}e^{-c^{\prime}\beta}$, see (LABEL:def-
epsilon1-beta), and $\epsilon_{2}(p_{1})=Ce^{-c\lambda(p_{1})}$, see (4.3) and
Definition 2.2. This proves the theorem. ∎

###  4.6 From overlap control to almost sure singularity of extremals

###### Proof of Corollary 3.2.

On the one hand we have the following lower bound from Theorem 3.4 and
monotone convergence. For sparse enough sets $\Lambda_{n}=\Lambda_{n}^{\rm
r}$, for $\mu$ almost every $\omega$,

$\begin{split}&\pi(\underline{\phi}^{\omega}|\omega)=\liminf_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}\pi(\sigma_{v}=\omega_{v}|\omega)\geq
1-\epsilon_{1}(\beta)-\epsilon_{2}(p_{1}).\end{split}$ (94)

On the other hand, we use

$\begin{split}1_{\sigma_{v}=\omega_{v}}\leq
1_{\omega_{v}=\omega^{\prime}_{v}}+1_{\sigma_{v}\neq\omega^{\prime}_{v}}\end{split}$
(95)

to write the upper bound

$\begin{split}\pi(\underline{\phi}^{\omega}|\omega^{\prime})\leq\limsup_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{\omega_{v}=\omega^{\prime}_{v}}+\pi\left(\limsup_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{\sigma_{v}\neq\omega^{\prime}_{v}}\middle|\omega^{\prime}\right).\end{split}$
(96)

The first term in the r.h.s. is equal to ${1}/{q}$ for $\mu\otimes\mu$-a.e.
pair $(\omega,\omega^{\prime})$, in the case of clock models. Indeed, the
Borel-Cantelli argument, together with the Chebychev inequality as in the
proof of Lemma 4.8 shows that the desired strong law of large numbers follows
(since exponential decorrelation of the local events
$(1_{\omega_{v}=a})_{v\in\Lambda_{n}^{r}}$ follows from exponential
decorrelation of the 1d Markov chains as in (LABEL:exp-decorrelation-1dMC)).
In the general case of central states this term equals
$\sum_{a\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=a)^{2}$; it is close to $1/q$ for
perturbed free states of clock models, see Section 4.1.4.

The second term in the r.h.s. is $\pi(\cdot|\omega^{\prime})$-a.s. controlled
via Theorem 3.4 by

$\begin{split}\limsup_{n\uparrow\infty}\frac{1}{|\Lambda_{n}|}\sum_{v\in\Lambda_{n}}1_{\sigma_{v}\neq\omega^{\prime}_{v}}\leq\epsilon_{1}(\beta)+\epsilon_{2}(p_{1}).\end{split}$
(97)

So the statement of the theorem follows once

$1-\epsilon_{1}(\beta)-\epsilon_{2}(p_{1})>\sum_{a\in{\mathbb{Z}}_{q}}\mu(\sigma_{0}=a)^{2}+\epsilon_{1}(\beta)+\epsilon_{2}(p_{1})$

which concludes the proof. ∎

##### Acknowledgements.

C.K. and A.L.N. thank Labex Bézout (ANR-10-LABX-58) and Laboratory LAMA (UMR
CNRS 8050) at Université Paris Est Créteil (UPEC) for various supports.
Research of A.L.N. and L.C. have also been supported by the CNRS IRP
(International Research Project) EURANDOM “Random Graph, Statistical Mechanics
and Networks” and by the LabEx PERSY-VAL-Lab (ANR-11-LABX-0025-01) funded by
the French program Investissement d’avenir.

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