# Extremal decomposition for random Gibbs measures: From general metastates to
metastates on extremal random Gibbs measures

Codina Cotar111 Department of Statistical Science, University College London,
1-19 Torrington Place, London WC1E 6BT, UK<EMAIL_ADDRESS>, Benedikt
Jahnel222 Weierstrass Institute for Applied Analysis and Stochastics,
Mohrenstrasse 39, 10117 Berlin, Germany<EMAIL_ADDRESS>and
Christof Külske333 Ruhr-University of Bochum, Fakultät für Mathematik,
Postfach 102148, 44721, Bochum, Germany<EMAIL_ADDRESS>
###### Abstract

The concept of metastate measures on the states of a random spin system was
introduced to be able to treat the large-volume asymptotics for complex
quenched random systems, like spin glasses, which may exhibit chaotic volume
dependence in the strong-coupling regime. We consider the general issue of the
extremal decomposition for Gibbsian specifications which depend measurably on
a parameter that may describe a whole random environment in the infinite
volume. Given a random Gibbs measure, as a measurable map from the environment
space, we prove measurability of its decomposition measure on pure states at
fixed environment, with respect to the environment. As a general corollary we
obtain that, for any metastate, there is an associated decomposition
metastate, which is supported on the extremes for almost all environments, and
which has the same barycenter.

AMS 2010 subject classification: 82B44, 60K35

Keywords: Gibbs measures, disordered systems, extremal decomposition,
metastates

##  1 Introduction

In this note, we consider quenched random spin models in the infinite volume
on lattices or more generally countably-infinite graphs. This includes
examples like the random-field Ising model and the Edwards-Anderson nearest-
neighbor spin glass, but also more generally continuous spin models. We prove
an extended version of the known extremal decomposition (see [Föl75] and
[Geo11, Theorem 7.26]) for the infinite-volume Gibbs measures in terms of pure
states, i.e., the extremal Gibbs measures, which is measurable w.r.t. the
random environment, see Theorem 2.1. We also present a connection between this
result and the theory of metastates via the notion of the decomposition
metastate, see Corollary 2.3.

What are the difficulties? Indeed, measurability w.r.t. random environments in
the infinite volume, which can be taken for granted in simple situations,
becomes nontrivial for general systems. The difficulty comes from the
following physical phenomenon that may happen for general random systems and
in particular those with frustrated interactions like spin glasses. A system
is frustrated, if it has competing interactions, in the sense that it is
impossible to minimize the total finite-volume Hamiltonian by minimizing all
individual local interactions terms in the same configuration. Hence a
groundstate has to be determined as a result of a non-trivial optimization
which may depend on the precise shape and size of the volume. The same holds
for low-temperature states. There may be chaotic volume-dependence, see [NS97,
NS98, NS13], of the finite-volume Gibbs measures along increasing volume
sequences with fixed nonrandom boundary conditions, in the phase-transition
regime. Hence, to construct a measurable random Gibbs measure, i.e., a
measurable map $\xi\mapsto\mu[\xi]$ from the environment to the Gibbs measures
via $\xi$-dependent volume sequences, is in general difficult. Conversely, for
fixed increasing volume sequence, pure states may only be found by
$\xi$-dependent boundary conditions. This phenomenon of possible chaotic
volume-dependence leads to the natural definition of a metastate
$\kappa[\xi](d\mu)$ as a probability measure on the (random) Gibbs measures of
the system, see [NS97, Bov06]. From a physical point of view, this is not just
an unwanted complication, but it is an important object for the description of
the large-volume asymptotics. Its intuitive meaning is that it describes the
limiting empirical measure for the occurrence of Gibbs states for fixed
environment $\xi$ in a sufficiently sparsely chosen increasing volume
sequence, see equation (4). Hence, it carries the additional information about
the weight, or relevance, of a particular Gibbs measure, compare [IK10]. To
show existence of a metastate, e.g., in situations with a compact local spin
space, is always possible, see [AW90, NS97, Bov06] for Ising systems, and as
we sketch in Section 2.2. Note that metastates were used to show uniqueness
results for short-range spin-glass models with independent couplings in two
dimensions, see [ADNS10], in higher dimensions, see [ANS18], and for other
applications, see also [Bov06]. Having constructed the metastate, we note that
there is always an associated measurable map from the environment to the Gibbs
measures, which appears as its barycenter of mean value over all the Gibbs
measures the metastate sees.

In the present note, we therefore start with the assumption of existence of
such a measurable Gibbs measure and investigate its extremal decomposition. In
general, there is no reason to assume that a metastate is supported only on
extremal Gibbs measures. For a (mean-field) example where this occurs see,
e.g., the Hopfield model in [Kül97]. Thus, we need to discuss the
decomposition into pure states also at the metastate level.

We keep the setup as general as possible and assume only measurability of the
Gibbsian specification for most of our results. We do not need to make the
requirements of quasilocality or non-nullness, which are otherwise very
important in the theory of infinite-volume Gibbs measures, see [Geo11].

Our results may be useful for random systems, but they may be applied also to
other classes of parametrized specifications, be it by finite-dimensional or
infinite-dimension parameters.

##  2 The extreme decomposition

###  2.1 Setup, examples, and first theorem

Let $S$ be a countable set of sites and $(E,{\cal E})$ the local state space
of each site. We assume that $(E,{\cal E})$ is a standard Borel space, see
[Geo11] for details. We consider models on $\Omega=E^{S}$ equipped with the
product sigma-algebra ${\cal F}=\mathcal{E}^{S}$, given by _specifications_
$\gamma[\xi]$ which depend on a random _parameter_ $\xi\in H$, where $(H,{\cal
H},{\mathbb{P}})$ is a probability space. A (non-random) specification
$\gamma=(\gamma_{\Lambda})_{\Lambda\Subset S}$ is a family of probability
kernels from $\Omega$ to the set of probability measures ${\cal
P}(\Omega)={\cal P}(\Omega,{\cal F})$ indexed by finite sets of sites
$\Lambda\Subset S$. We want to impose only minimal conditions on the
specification. First, we require consistency, i.e., for all finite volumes
$\Delta\subset\Lambda$, events $A\in{\cal F}$ and boundary conditions
$\omega\in\Omega$ we have that
$\int\gamma_{\Lambda}(d\sigma|\omega)\gamma_{\Delta}(A|\sigma)=\gamma_{\Lambda}(A|\omega)$.
Second, we require properness, i.e., for all $\Lambda\Subset S$ we have that
$\gamma_{\Lambda}(A|\omega)={\mathds{1}}_{A}(\omega)$ if $A$ is measurable
outside of $\Lambda$. We do not require for example non-nullness (lower bounds
on the specification) or quasilocality (continuity of the specification w.r.t.
the boundary condition), which play an important role for the existence theory
and variational characterization of Gibbs measures. In what follows, we will
assume that the random specification $\gamma[\cdot]$ is measurable in the
sense that for all $\Lambda\Subset S$ and $A\in{\cal F}$, the mapping

$(\xi,\omega)\mapsto\gamma_{\Lambda}[\xi](A|\omega)$

is $({\cal H}\otimes{\mathcal{F}})-\mathcal{B}[0,1]$ measurable.

We think of the parameter $\xi$ mostly, but not exclusively, as describing
disorder. Often we imagine a Gibbsian specification given in terms of an
absolutely summable potential $\Phi[\xi]=(\Phi_{A}[\xi])_{A\Subset S}$ where
each of the potential functions $\omega\mapsto\Phi_{A}[\xi](\omega)$ depends
on the spin variable $\omega$ only inside $A$, and depends measurably on the
parameter $\xi$ and the configuration $\omega$. Then, we may build the
associated quenched specification as the $\lambda$-modification, in which the
product measure $\lambda^{\Lambda}$ is modified by an exponential factor.
Keeping the parameter $\xi$ fixed, this means that we define $\xi$-dependent
specification kernels via

$\gamma_{\Lambda}^{\Phi}[\xi](d\omega_{\Lambda}|\eta)=\frac{\lambda^{\Lambda}(d\omega_{\Lambda})\exp(-\sum_{A\cap\Lambda\neq\emptyset}\Phi_{A}[\xi](\omega_{\Lambda}\eta_{\Lambda^{c}}))}{\int_{E^{\Lambda}}\lambda^{\Lambda}(d\tilde{\omega}_{\Lambda})\exp(-\sum_{A\cap\Lambda\neq\emptyset}\Phi_{A}[\xi](\tilde{\omega}_{\Lambda}\eta_{\Lambda^{c}}))},$
(1)

where $\lambda\in{\cal P}(E)$ is some a priori measure on the single site
space $E$ and $\lambda^{\Lambda}$ denotes the $|\Lambda|$-fold product
distribution on $E^{\Lambda}$. We write $\omega_{\Lambda}\eta_{\Lambda^{c}}$
for the concatenation of the configuration $\omega_{\Lambda}$ in $\Lambda$ and
$\eta_{\Lambda^{c}}$ in $\Lambda^{c}=S\setminus\Lambda$. Two examples which
fit this framework are the _random-field Ising model_ , and the _Edwards-
Anderson nearest-neighbor spin glass_. We may consider these models on a
general graph with countable vertex set $S$, with specification kernels
defined as modifications w.r.t. the uniform measure $\lambda$ on
$E=\\{-1,1\\}$.

For the random-field Ising model, the potential functions are given by the
pair interaction terms $\Phi_{A}[\xi](\omega)=-\beta\omega_{i}\omega_{j}$ for
$A=\\{i,j\\}$ for nearest neighbors $i,j\in S$, and single-site terms
$\Phi_{A}[\xi](\omega)=\xi_{i}\omega_{i}$ for $A=\\{i\\}$. For all other
$A\Subset S$ we set $\Phi_{A}[\xi]=0$. Note that the disorder configuration
$\xi=(\xi_{i})_{i\in S}$ enters only via the single-site terms, and are
usually assumed to be given by i.i.d. random variables w.r.t. ${\mathbb{P}}$
on $(H,{\cal H})$.

For the Edwards-Anderson spin glass (in zero external field), only the
nearest-neighbor pair-interaction term is non-vanishing with
$\Phi_{A}[\xi](\omega)=\beta\xi_{i,j}\omega_{i}\omega_{j}$ for $A=\\{i,j\\}$
and $i,j$ nearest neighbors, where $\xi$ is an $H$-valued configuration on the
edges $\\{i,j\\}$ of the underlying graph. One usually assumes that
$\xi_{i,j}$ are mean-zero random variables, which are i.i.d. over edges.

Both models are different in the following sense. For the random-field Ising
model, monotonicity arguments (based on the FKG inequality) guarantee
existence of a maximal and a minimal limiting measure, for any disorder
configuration, which simplifies the analysis substantially. Nevertheless, to
decide in which dimensions, and when they are equal, is far from trivial, see
[BK88, AW90, Bov06, AP18]. On the other hand, for free or periodic boundary
conditions, the random-field Ising model in three dimensions does show chaotic
size-dependence.

For the Edwards-Anderson spin glass, the aforementioned chaotic volume-
dependence is expected to occur for deterministic boundary conditions. On the
other hand, if there is only one pair of Gibbs states, related by spin-flip
symmetry (which should happen in two dimensions), any volume sequence with
symmetry-preserving boundary conditions cannot cause chaotic volume-
dependence. Examples of such boundary conditions are free and periodic
boundary conditions. There may be chaotic volume-dependence also for these
boundary conditions in higher dimensions, if there should be many of such
pairs of Gibbs measures. In that sense, both models display ordinary and size-
dependent convergence, at least in some regimes and for complementary sets of
boundary conditions. As another similarity, in the two examples above,
disorder enters via local terms in a Hamiltonian which is used to define the
Gibbsian specification, and the disorder takes values in a product space. This
is nice, but our setup is more general and assumes only measurability of the
specification depending on a parameter $\xi$.

A lack of monotonicity occurs also for gradient Gibbs measures. These appear
for unbounded spin systems where one is forced to restrict to gradient
observables (these are observables which depend only on field increments) in
order to have a well-defined infinite-volume measure. Viewed on the resulting
incremental field variables there is no monotonicity, even if the initial
model in the finite-volume Gibbsian formulation has monotonicity. A concrete
example where this is very explicit is the Gaussian free field. Gradient Gibbs
measures are beyond the framework of the present note, but we refer to [FS97,
BK07, vEK08, CK12, CK15].

In another (simpler) application, the variable $\xi$ may also encode a
parameter such as a random inverse temperature, modifying a given (random or
non-random) potential in a multiplicative way. More generally, we may consider
any parametrized probability distribution on Gibbsian potentials with the same
state space. Also in such a case it is clear that at different realizations,
the structure of Gibbs measures may be completely different for different
parameter values.

If there is a smooth dependence on this parameter, it may be expected that the
extremal states should depend smoothly on the parameter, at least in some
regions of the phase diagram. This is known to be true for example for ordered
states of ferromagnetic low-temperature models, in the regime where Pirogov-
Sinai is applicable. For more complicated disordered systems, the possibility
for such a ”state-following” may no longer be true. A phenomenon of
temperature chaos as it is called by the physicists may appear, see [vER07,
CH10, WMK15, CRL15, ASZ18]. We show that, while such chaos may happen, at
least measurability is not lost.

For a given non-random specification $\gamma$, we call a measure $\mu\in{\cal
P}(\Omega)$ an _infinite-volume Gibbs measure_ associated to $\gamma$, if it
satisfies the DLR equations, i.e., if for all $\Lambda\Subset S$ and
$A\in{\cal F}$ we have that
$\int\mu(d\omega)\gamma_{\Lambda}(A|\omega)=\mu(A)$. We denote by ${\cal
G}(\gamma)$ the set of infinite-volume Gibbs measures associated to $\gamma$.
In the random case, we will assume that ${{\cal G}(\gamma[\xi])}\neq\emptyset$
for ${\mathbb{P}}$-almost all $\xi$.

In particular, for ${\mathbb{P}}$-almost all $\xi$, ${{\cal G}(\gamma[\xi])}$
is a non-empty simplex of probability measures in ${\cal P}(\Omega)={\cal
P}(\Omega,{\cal F})$ (the set of probability measures on $(\Omega,{\cal F})$).
${{\cal G}(\gamma[\xi])}$ can be represented by its extreme elements, which we
denote by ${{\rm ex}{\cal G}(\gamma[\xi])}$. As a result of the non-random
Gibbsian theory, applied at fixed $\xi$, any $\mu\in{{\cal G}(\gamma[\xi])}$
is the barycenter of a unique probability measure $w[\xi]_{\mu}$ on ${{\rm
ex}{\cal G}(\gamma[\xi])}$, see [Geo11, Chapter 7]. The interpretation here is
that elements of the extreme boundary represent _pure_ macroscopic states of
the system, where the macroscopic variables are not random. Therefore, the
weight measure $w[\xi]_{\mu}$, which appears in this decomposition of $\mu$,
then represents the uncertainty encountered by experimentalists about which
pure states have been observed.

We will be concerned with _random infinite-volume Gibbs measures_ given by the
_measurable_ map $\xi\mapsto\mu[\xi]$. Measurability w.r.t. the parameter
$\xi$ is very important, here it is defined w.r.t. the _evaluation sigma
algebra_ ${\it e}({\cal P}(\Omega))$ given by the smallest sigma algebra such
that for all $A\in{\cal F}$, the mappings

$\xi\mapsto\mu[\xi](A)$

are ${\cal H}-\mathcal{B}[0,1]$ measurable. For fixed $\xi$, the weight
distribution $w[\xi]_{\mu[\xi]}$, associated to $\mu[\xi]$, is an element of
${\cal P}({\cal P}(\Omega))={\cal P}({\cal P}(\Omega),{\it e}({\cal
P}(\Omega)))$, the space of probability measures on the space of random fields
${\cal P}(\Omega)$ equipped with the evaluation sigma algebra on ${\cal
P}(\Omega)$. In order to consider measurability of $\xi\mapsto
w[\xi]_{\mu[\xi]}$, we need to equip also ${\cal P}({\cal P}(\Omega))$ with a
sigma algebra, which is the evaluation sigma algebra ${\it e}({\cal P}({\cal
P}(\Omega)))$ where ${\cal H}-\mathcal{B}[0,1]$ measurability of

$\xi\mapsto w[\xi]_{\mu[\xi]}(M)$

has to be checked for all $M\in{\it e}({\cal P}(\Omega))$.

Our first main result now states the existence and measurability of the
function $\xi\mapsto w[\xi]_{\mu[\xi]}$.

######  Theorem 2.1.

Assume $(E,{\cal E})$ to be standard Borel. Let $\gamma[\xi]$ be a random
specification on $\Omega=E^{S}$ such that ${\cal G}(\gamma[\xi])\neq\emptyset$
for ${\mathbb{P}}$-almost all $\xi$ and let $\mu[\xi]$ be a random infinite-
volume Gibbs measure in ${{\cal G}(\gamma[\xi])}$ which exists
${\mathbb{P}}$-almost surely. Then, also the mapping $\xi\mapsto
w[\xi]_{\mu[\xi]}$ exists and is ${\cal H}-{\it e}({\cal P}({\cal
P}(\Omega)))$ measurable. Moreover,

  1. 1.

the weight function is almost surely supported on the quenched extremal Gibbs
measures, that is $w[\xi]_{\mu[\xi]}\in{\cal P}({{\rm ex}{\cal
G}(\gamma[\xi])},{\it e}({{\rm ex}{\cal G}(\gamma[\xi])}))$ for
${\mathbb{P}}$-almost every $\xi\in H$, and

  2. 2.

the random Gibbs measure has the measurable extremal decomposition

$\mu[\xi]=\int_{{{\rm ex}{\cal G}(\gamma[\xi])}}w[\xi]_{\mu[\xi]}(d\nu)\nu.$
(2)

The map $\xi\mapsto w[\xi]_{\mu[\xi]}$ is defined by a $\xi$-measurable kernel
$\pi[\xi]$ via $w[\xi]_{\mu}(M)=\mu(\omega:\pi[\xi](\cdot|\omega)\in M)$ for
all $M\in{\it e}({\cal P}(\Omega))$ and $\mu\in{\cal P}(\Omega)$.

Considering the proof of Theorem 2.1, in particular the input given by
Proposition 3.1, we actually establish measurability, w.r.t. $\xi$, of the
affine mapping $w[\xi]:\mu\mapsto w[\xi]_{\mu}$ from ${\cal P}(\Omega)$ to
${\cal P}({\cal P}(\Omega),{\it e}({\cal P}(\Omega)))$ equipped with the
evaluation sigma algebra $\sigma\big{(}{\it e}_{\mu,M}:(\mu,M)\mapsto
w[\xi]_{\mu}(M),\mu\in{\cal P}(\Omega),M\in{\it e}({\cal P}(\Omega))\big{)}$.
Additionally, under the assumptions of the theorem, for ${\mathbb{P}}$-almost
all $\xi$, $\mu\mapsto w[\xi]_{\mu}$ is a bijection between ${{\cal
G}(\gamma[\xi])}\subset{\cal P}(\Omega)$ and ${\cal P}\big{(}{{\rm ex}{\cal
G}(\gamma[\xi])},{\it e}({{\rm ex}{\cal G}(\gamma[\xi])})\big{)}$.

What have we gained? Ideally one would like to have a measurable extreme
decomposition $\xi\mapsto\mu^{\alpha}[\xi]$ taking values in the extremals for
given $\xi$, of the same type, indexed by some index $\alpha$. This can be
shown to be true for some systems, e.g., for the random-field Ising model,
where $\alpha\in\\{+,-\\}$. Indeed, by the FKG inequality, one may construct a
maximal measure $\mu^{+}[\xi]$ with a fixed $+$-boundary condition, for any
realization of $\xi$. However, in general this will not be possible, and so
the best result we can hope to obtain is to go one level higher and construct
a measurable map from $\xi$ to the probability measures on the _extremals_ for
the same $\xi$. This is precisely the result of Theorem 2.1.

###  2.2 Metastates

Let us start this section by giving some background on metastates. In [AW90],
Aizenman and Wehr constructed a metastate as a $\xi$-dependent probability
measure on the Gibbs states of a system that describes the large-volume
asymptotics. The main underlying ideas for this are the following. Consider
the probability distribution of the pair of the finite-volume Gibbs measure
and the disorder variable

$\displaystyle(\gamma_{\Lambda_{n}}[\xi](\cdot|\omega),\xi)$ (3)

under the governing measure of the disorder variable, ${\mathbb{P}}(d\xi)$,
for some fixed boundary condition $\omega$. Let us note that it is useful to
consider also more generally open or periodic boundary conditions, instead of
fixed configurations $\omega$. Suppose now that a limit exists for the random
pair (3) in the sense of weak convergence, along the cofinal sequence
$(\Lambda_{n})_{n\in{\mathbb{N}}}$. Call the resulting limiting distribution
$K(d\mu,d\xi)$. Of this limit, we may take a conditional distribution,
obtained by conditioning on the disorder variable $\xi$, and this provides us
with a measure on the first variable, which we call
$\kappa^{\text{AW}}[\xi](d\mu)$, the Aizenman-Wehr or conditional metastate.
The existence of the limit $K$ can be ensured for volume-subsequences
$(\Lambda_{k_{n}})_{n\in{\mathbb{N}}}$, for compact local state spaces $E$,
and nice specifications, although we are not aware that this result appears in
full generality. To carry this out in detail however is not a problem for
example for the aforementioned random Ising models using the finite local
state space and Markovianity of the interaction. The independence of the limit
on the choice of the subsequence has not been proved in general, but it is
very plausible in all examples that have been studied.

There is also a relation to the _Newman-Stein metastate_ , see [NS98], which
is constructed as a limit point of

$\begin{split}\kappa_{N}[\xi]=\frac{1}{N}\sum_{n=1}^{N}\delta_{\gamma_{\Lambda_{n}}[\xi](\cdot|\omega)}\end{split}$
(4)

for sufficiently sparse $\Lambda_{n}$. This provides the interesting and
intuitive reinterpretation of the Aizenman-Wehr-metastate as giving a
distribution on the Gibbs states at fixed $\xi$ by “drawing a large volume at
random, independently of the choice of $\xi$”. Covering both approaches, let
us define _metastates_ as follows.

######  Definition 2.2.

A measurable map $\kappa:H\rightarrow{\cal P}({\cal P}(\Omega))$ is called a
_metastate_ if for ${\mathbb{P}}$-almost all $\xi$, the Gibbs measures
corresponding to $\xi$ obtain full mass, i.e.,

$\kappa[\xi]\big{(}{\cal G}(\gamma[\xi])\big{)}=1.$

Coming back to our results, in Theorem 2.1, we assumed the existence of a
measurable Gibbs state $\mu[\xi]$. To see how this assumption can be
satisfied, note that for a metastate $\kappa[\xi](d\mu)$ one may take its
expected value (or barycenter) $\int\kappa[\xi](d\nu)\nu=\mu_{\kappa}[\xi]$ to
obtain a measurable map into the (not necessarily extreme) Gibbs measures for
$\xi$. Further, in the sense of Definition 2.2, the measurable weight function
$\xi\mapsto w[\xi]_{\mu[\xi]}$ is a metastate. The statement of Theorem 2.1
then has the following rephrasing: For any measurable Gibbs measure $\mu[\xi]$
there is a metastate $w[\xi]_{\mu[\xi]}$ supported on the quenched pure states
${{\rm ex}{\cal G}(\gamma[\xi])}$ such that measurable Gibbs measure can be
written as the barycenter of this metastate, in the form $\int_{{\cal
P}(\Omega)}w[\xi]_{\mu[\xi]}(d\nu)\nu$.

In particular, applying Theorem 2.1 and using the intermediate step to define
$\mu[\xi]=\mu_{\kappa}[\xi]$, we obtain the following result.

######  Corollary 2.3.

Suppose $\kappa:H\rightarrow{\cal P}({\cal P}(\Omega))$ is a metastate. Then
there is an associated decomposition metastate $\kappa^{\rm
ex}:H\rightarrow{\cal P}({\cal P}(\Omega))$ which has the following two
properties.

  1. 1.

$\kappa^{\rm ex}$ has the same barycenter as $\kappa$, i.e., for
${\mathbb{P}}$-almost all $\xi$,

$\int_{{\cal P}(\Omega)}\kappa^{\rm ex}[\xi](d\nu)\nu=\int_{{\cal
P}(\Omega)}\kappa[\xi](d\nu)\nu\qquad\text{ and}$

  2. 2.

for ${\mathbb{P}}$-almost all $\xi$, $\kappa^{\rm ex}$ is supported on the
corresponding extremal Gibbs measures, i.e.,

$\kappa^{\rm ex}[\xi]\big{(}{{\rm ex}{\cal G}(\gamma[\xi])}\big{)}=1.$

###  2.3 Symmetries

So far we have not used any specific structure for the space $(H,{\cal H})$.
For the next result we want to introduce translations on $H$ as well as on
$\Omega$. For this, let us assume that $S={\mathbb{Z}}^{d}$ and equip $H$ with
a _shift group_ $\theta=(\theta_{i})_{i\in S}$, i.e., a family of measurable
bijections $\theta_{i}:H\to H,\,\xi\mapsto\theta_{i}\xi$. More specifically,
for $\omega\in\Omega$, we define $(\theta_{j}\omega)_{i}=\omega_{i-j}$. For
example in the random-field Ising model, where $H=\\{-1,1\\}^{S}$, the shift
group is defined exactly as for $\Omega$. We call a random specification
$\gamma$ _translation covariant_ , if for all $j\in S,\omega\in\Omega$,
$A\in{\cal F}$ and $\Lambda\Subset S$ we have that

$\gamma_{\theta_{j}\Lambda}[\theta_{j}\xi](\theta_{j}A|\theta_{j}\omega)=\gamma_{\Lambda}[\xi](A|\omega)\qquad\text{for
}{\mathbb{P}}\text{-almost all }\xi$

where $\theta_{j}A=\\{\theta_{j}\omega:\omega\in A\\}$ and
$\theta_{j}\Lambda=\\{i+j\in S:i\in\Lambda\\}$. Similarly we call a random
Gibbs measure $\xi\mapsto\mu[\xi]$ _translation covariant_ , if for all
$A\in{\cal F}$ and $j\in S$ we have that

$\mu[\theta_{j}\xi](A)=\mu[\xi](\theta_{-j}A)\qquad\text{for
}{\mathbb{P}}\text{-almost all }\xi.$

The following theorem expresses that the extremal decomposition of Theorem 2.1
for translation invariant models is also translation covariant.

######  Theorem 2.4.

Assume that $\gamma[\xi]$ and $\mu[\xi]$ satisfy the conditions of Theorem 2.1
and are both translation covariant. Then, the measurable map $\xi\mapsto
w[\xi]_{\mu[\xi]}$ from Theorem 2.1 is also translation invariant, i.e., for
all $j\in S$ and $M\in\it{e}({\cal P}(\Omega))$ we have that

$w[\theta_{j}\xi]_{\mu[\theta_{j}\xi]}(\theta_{j}M)=w[\xi]_{\mu[\xi]}(M)\qquad\text{for
}{\mathbb{P}}\text{-almost all }\xi,$ (5)

where $\theta_{j}M=\\{\theta_{j}\nu\in{\cal P}(\Omega):\nu\in M\\}$ with
$\theta_{j}\nu(A)=\nu(\theta_{-j}A)$ for all $A\in{\cal F}$.

Under suitable conditions, Theorem 2.4 can be extended also to cover other
group actions $\tau$ instead of $\theta$, such as spin flips or spin
rotations, generalizing [Geo11, Corollary 7.28] to the $\xi$-dependent
setting.

##  3 Proofs

The proof of Theorem 2.1 is based on the existence of the $({\cal
G}(\gamma[\xi]),{\cal T})$-kernel $\pi[\xi](\cdot|\omega)$ with the desired
measurability and support properties. Once this is established, the proof of
the main theorem follows by an application of [Geo11, Proposition 7.22] and
measure-theoretic arguments.

Let us introduce some more notation. For $\Lambda\Subset S$ we denote by
${\cal F}_{\Lambda}$, the sub-sigma algebra of sets in ${\cal F}$ which are
measurable in $\Lambda$. Further, the _tail sigma algebra_ ${\cal T}$ is
defined via ${\cal T}=\bigcap_{\Lambda\Subset S}{\cal F}_{\Lambda^{c}}$ where
$\Lambda^{c}=\Omega\setminus\Lambda$. As a particular case of [Geo11,
Definition 7.21], a probability kernel $\pi$ is called a _$({\cal
G}(\gamma),{\cal T})$ -kernel_, for $\gamma$ a specification, if the following
holds. For all measures $\mu\in{\cal G}(\gamma)$,

  1. 1.

$\pi$ is a version of the conditional probability given ${\cal T}$, that is,
for all $A\in{\cal F}$, $\mu(A|{\cal T})=\pi(A|\cdot)$ $\mu$-almost surely,
and

  2. 2.

for $\mu$-almost every $\omega$, the measure $\pi(\cdot|\omega)$ is an element
in ${\cal G}(\gamma)$ and $\\{\omega:\pi(\cdot|\omega)\in{\cal
G}(\gamma)\\}\in{\cal T}$.

Let us stress that $\pi$, if it exists, does not depend on individual measures
$\mu$, but rather on $\gamma$.

The following result establishes existence of a random version of a $({\cal
G}(\gamma),{\cal T})$-kernel and therefore extends [Geo11, Proposition 7.25]
to the random setting.

######  Proposition 3.1.

Assume $(E,{\cal E})$ to be standard Borel. Let $\gamma[\xi]$ be a random
specification on $\Omega=E^{S}$ such that ${{\cal
G}(\gamma[\xi])}\neq\emptyset$ for ${\mathbb{P}}$-almost all $\xi$. Then there
exists a map $\pi:H\times\Omega\times{\cal F}\mapsto[0,1]$,
$(\xi,\omega,A)\mapsto\pi[\xi](A|\omega)$, such that

  1. 1.

at any fixed $A\in{\cal F}$, we have $\bigcap_{\Lambda}({\cal H}\otimes{\cal
F}_{\Lambda^{c}})-\mathcal{B}[0,1]$ measurability of

$(\xi,\omega)\mapsto\pi[\xi](A|\omega),\qquad\text{ and}$

  2. 2.

for ${\mathbb{P}}$-almost all $\xi$, $\pi[\xi]$ is a $({\cal
G}(\gamma[\xi]),{\cal T})$-kernel.

We will use the following measure-theoretic generalities for the proofs. Since
$(E,{\cal E})$ is assumed to be standard Borel, also $(\Omega,{\cal F})$ is a
standard Borel space. Moreover, if the configuration space is standard Borel,
i.e., measure-theoretic isomorphic to a complete separable metric space with
Borel sigma-algebra, then, ${\cal F}$ can be generated by a countable core. We
recall that a core ${\cal C}$ is a finite-intersection stable generator of the
sigma algebra ${\cal F}$ with the following property. Convergence of
$(\mu_{n}(A))_{n\in{\mathbb{N}}}$ for all events $A\in{\cal C}$, where
$(\mu_{n})_{n\in{\mathbb{N}}}\subset{\cal P}(\Omega)$, uniquely determines a
limiting measure $\mu\in{\cal P}(\Omega)$ with
$\mu(A)=\lim_{n\uparrow\infty}\mu_{n}(A)$ for all $A\in{\cal C}$, see [Geo11,
Definition 4.A9].

Before we prove Proposition 3.1, let us show how it can be used to prove
Theorem 2.1.

###### Proof of Theorem 2.1.

By hypothesis, for ${\mathbb{P}}$-almost all $\xi$, we have that ${{\cal
G}(\gamma[\xi])}$ is non-empty. By Proposition 3.1, the measurable kernel
$\pi$ exists and, in particular, by part 2 of Proposition 3.1 is a $({\cal
G}(\gamma[\xi]),{\cal T})$-kernel. Thus, for ${\mathbb{P}}$-almost all $\xi$,
using [Geo11, Proposition 7.22], we have the following. For every
$\mu\in{{\cal G}(\gamma[\xi])}$, there exists a unique $w[\xi]_{\mu}$,
supported on ${{\rm ex}{\cal G}(\gamma[\xi])}$, such that the extremal
decomposition (2) holds. In particular, the mapping $\xi\mapsto
w[\xi]_{\mu[\xi]}$ exists and
$w[\xi]_{\mu[\xi]}(M)=\int\mu[\xi](d\omega){\mathds{1}}\\{\pi[\xi](\cdot|\omega)\in
M\\}$ for all $M\in{\it e}({\cal P}(\Omega))$. Hence, all that remains to be
shown is that the mapping $\xi\mapsto w[\xi]_{\mu[\xi]}$ is ${\cal H}-{\it
e}({\cal P}({\cal P}(\Omega)))$ measurable.

By the definition of the evaluation sigma algebra, it suffices to show ${\cal
H}-\mathcal{B}[0,1]$ measurability of

$\xi\mapsto(\xi,\mu[\xi])\mapsto w[\xi]_{\mu[\xi]}(M)$

for all $M\in{\it e}({\cal P}(\Omega))$. For this, note that the first mapping
is measurable by assumption since we supposed that $\xi\mapsto\mu[\xi]$ is
${\cal H}-{\it e}({\cal P}(\Omega))$ measurable. Thus, it suffices to
establish ${\cal H}\otimes{\it e}({\cal P}(\Omega))-\mathcal{B}[0,1]$
measurability of

$(\xi,\mu)\mapsto\int\mu(d\omega){\mathds{1}}\\{\pi[\xi](\cdot|\omega)\in
M\\}.$ (6)

By the definition of ${\it e}({\cal P}(\Omega))$ we can limit ourself to check
sets $M$ of the form $M(A,c)=\\{\nu\in{\cal P}(\Omega):\nu(A)\leq c\\}$ with
$A\in{\cal F}$ and $c\in[0,1]$ in which case (6) becomes

$(\xi,\mu)\mapsto\int\mu(d\omega){\mathds{1}}\\{\pi[\xi](A|\omega)\leq c\\}.$

Here, by Proposition 3.1 part 1 we have that
$\\{(\xi,\omega):\pi[\xi](A|\omega)\leq c\\}\in\bigcap_{\Lambda}({\cal
H}\otimes{\cal F}_{\Lambda^{c}})\subset({\cal H}\otimes{\cal F})$ for all
$A\in{\cal F}$ and $c\in[0,1]$. In fact, as is a standard statement in the
context of Fubini’s theorem, for any ${\cal H}\otimes{\cal
F}-\mathcal{B}[0,1]$ measurable bounded $g$ we have that

$(\xi,\mu)\mapsto\int\mu(d\omega)g(\xi,\omega)$

is ${\cal H}\otimes\text{{\it e}}({\cal P}(\Omega))-\mathcal{B}[0,1]$
measurable. Indeed, for any $C\in{\cal H}\otimes{\cal F}$ we have that

$(\xi,\mu)\mapsto\int\mu(d\omega){\mathds{1}}\\{(\xi,\omega)\in C\\}$

is ${\cal H}\otimes\text{{\it e}}({\cal P}(\Omega))-\mathcal{B}[0,1]$
measurable, since

${\cal D}=\\{C\in{\cal H}\otimes{\cal
F}:(\xi,\mu)\mapsto\int\mu(d\omega){\mathds{1}}\\{(\xi,\omega)\in C\\}\text{
is }{\cal H}\otimes\text{{\it e}}({\cal P}(\Omega))-\mathcal{B}[0,1]\text{
measurable}\\}$

contains the system of squares $Q=\\{A\times A^{\prime}:A\in{\cal
H},A^{\prime}\in{\mathcal{F}}\\}$ and is a Dynkin system. The statement that
$Q\subset{\cal D}$ follows, since

$(\xi,\mu)\mapsto\int\mu(d\omega){\mathds{1}}\\{(\xi,\omega)\in A\times
A^{\prime}\\}=\mu(A){\mathds{1}}\\{\xi\in A^{\prime}\\}$

is measurable as a product of measurable functions. The fact that it is a
Dynkin system is obvious since for $C_{1}\supset C_{2}$

$\int\mu(d\omega){\mathds{1}}\\{(\xi,\omega)\in
C_{1}{\backslash}C_{2}\\}=\int\mu(d\omega){\mathds{1}}\\{(\xi,\omega)\in
C_{1}\\}-\int\mu(d\omega){\mathds{1}}\\{(\xi,\omega)\in C_{2}\\}$

where the l.h.s. is measurable if the two terms on the r.h.s. are, and a
similar argument applies for countable disjoint unions. Thus, by Dynkin’s
theorem [Kle13, Theorem 1.19], the sigma-algebra generated by an intersection-
stable system of sets equals the Dynkin system generated by it. We have
$Q\subset{\cal D}$ and hence ${\cal H}\otimes{\cal F}=\sigma(Q)={\cal
D}(Q)\subset{\cal D}$, and so ${\cal H}\otimes{\cal F}={\cal D}$ follows. This
finishes the proof. ∎

###### Proof of Proposition 3.1.

Since $(E,{\cal E})$ is standard Borel, as outlined before the proof on
Theorem 2.1, there exists a countable core of ${\cal F}$ which we denote
${\cal C}$, for details see [Geo11, Theorem 4.A11]. Only at this moment we
will make use of this notion, namely in the definition of the following set.
Extending the construction in [Geo11, Proposition 7.25] we define

$\bar{\Omega}_{o}=\\{(\xi,\omega)\in
H\times\Omega:\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega){\text{
exists for all }}A\in{\cal C}\\},$

the set of disorder parameters and configurations such that the specification
has an infinite-volume limit on the core. Here
$(\Lambda_{n})_{n\in{\mathbb{N}}}$, denotes a cofinal sequence of finite
subsets of $S$. While for some $(\xi,\omega)$ the convergence property might
depend on the choice of the sequence, this will not matter in the
construction, as we will see. In particular,
$\bar{\Omega}_{o}=\bigcap_{A\in{\cal C}}\bar{\Omega}_{o}(A)$ where
$\bar{\Omega}_{o}(A)=\\{(\xi,\omega)\in
H\times\Omega:\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega){\text{
exists}}\\}$ since ${\cal C}$ is assumed to be countable. Further,
$\bar{\Omega}_{o}(A)\in\bigcap_{\Lambda}({\cal H}\otimes{\cal
F}_{\Lambda^{c}})$ for all $A\in{\cal C}$ and thus
$\bar{\Omega}_{o}\in\bigcap_{\Lambda}({\cal H}\otimes{\cal F}_{\Lambda^{c}})$.
Define $\Omega_{o}(\xi)=\\{\omega\in\Omega:(\xi,\omega)\in\bar{\Omega}_{o}\\}$
and note that $\Omega_{o}(\xi)\in{\cal T}$ lies in the tail sigma algebra
corresponding to ${\cal F}$.

Fixing a disorder parameter $\xi$, we can repeat the arguments from [Geo11,
Proposition 7.25] to define a kernel $\pi[\xi]$ and prove part 2 of the
Proposition. For this also fix $\omega\in\Omega_{o}(\xi)$. Then, by the
definition of the core ${\cal C}$, there is a unique probability measure
$\pi[\xi](\cdot|\omega)\in{\cal P}(\Omega)$ such that
$\pi[\xi](A|\omega)=\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega)$
for all $A\in{\cal C}$. We extend $\pi[\xi](\cdot|\omega)=\nu_{o}$ for an
arbitrary measure $\nu_{o}\in{\cal P}(\Omega)$ whenever $(\xi,\omega)$ are not
in the set $\bar{\Omega}_{o}$. Since, for ${\mathbb{P}}$-almost all $\xi$ we
have that ${{\cal G}(\gamma[\xi])}\neq\emptyset$, repeating the exact same
arguments as in [Geo11, Proposition 7.25], we see that for
${\mathbb{P}}$-almost all $\xi$, $\pi[\xi]$ is indeed a $({\cal
G}(\gamma[\xi]),{\cal T})$-kernel.

The preceding arguments established existence of the map $\pi$. What remains
to be shown is part 1. We must show that
$(\xi,\omega)\mapsto\pi[\xi](A|\omega)$ is $\bigcap_{\Lambda}({\cal
H}\otimes{\cal F}_{\Lambda^{c}})-\mathcal{B}[0,1]$ measurable at any
$A\in{\cal F}$. For this define

${\cal D}=\\{A\in{\cal F}:(\xi,\omega)\mapsto\pi[\xi](A|\omega)\text{ is
}\bigcap_{\Lambda}({\cal H}\otimes{\cal
F}_{\Lambda^{c}})-\mathcal{B}[0,1]\text{ measurable}\\},$

and recall that for any $A\in{\cal C}$,

$\pi[\xi](A|\omega)={\mathds{1}}\\{(\xi,\omega)\in\bar{\Omega}_{o}\\}\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega)+{\mathds{1}}\\{(\xi,\omega)\not\in\bar{\Omega}_{o}\\}\nu_{o}(A).$
(7)

Thus, ${\cal C}$ is contained in ${\cal D}$, since for all $A\in{\cal C}$ the
limit $\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega)$ as well as
the indicators ${\mathds{1}}\\{\bar{\Omega}_{o}\\}$ and
$\nu_{o}(A){\mathds{1}}\\{\bar{\Omega}_{o}\\}$ are $\bigcap_{\Lambda}({\cal
H}\otimes{\cal F}_{\Lambda^{c}})-\mathcal{B}[0,1]$ measurable function and
hence $(\xi,\omega)\mapsto\pi[\xi](A|\omega)$ is a $\bigcap_{\Lambda}({\cal
H}\otimes{\cal F}_{\Lambda^{c}})-\mathcal{B}[0,1]$ measurable function.

Note also that ${\cal D}$ is a Dynkin system. For, let $A\subset B$ be events
in ${\cal D}$, then

$\begin{split}\pi&[\xi](B\backslash
A|\omega)={\mathds{1}}\\{(\xi,\omega)\in\bar{\Omega}_{o}\\}\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](B\backslash
A|\omega)+{\mathds{1}}\\{(\xi,\omega)\not\in\bar{\Omega}_{o}\\}\nu_{o}(B\backslash
A)\cr&={\mathds{1}}\\{(\xi,\omega)\in\bar{\Omega}_{o}\\}\big{(}\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](B|\omega)-\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega)\big{)}+{\mathds{1}}\\{(\xi,\omega)\not\in\bar{\Omega}_{o}\\}\big{(}\nu_{o}(B)-\nu_{o}(A)\big{)}\cr&=\pi[\xi](B|\omega)-\pi[\xi](A|\omega).\end{split}$
(8)

Hence measurability of the r.h.s. carries over to that of the l.h.s.. In the
same way it is seen that
$\pi[\xi](\bigcup_{n}A_{n}|\omega)=\sum_{n}\pi[\xi](A_{n}|\omega)$ is
measurable for disjoint $A_{n}$ when each of the $\pi[\xi](A_{n}|\omega)$ is.
Thus, by Dynkin’s theorem we have that ${\cal C}\subset{\cal D}$ and hence
${\cal F}=\sigma({\cal C})={\cal D}({\cal C})\subset{\cal D}$, and so ${\cal
F}={\cal D}$ follows, this completes the proof. ∎

###### Proof of Theorem 2.4.

We need to go back to the definition of the metastate as an image of the
$\xi$-measurable kernel under the random Gibbs measure, and check how these
objects behave under translation. Recall that

$w[\xi]_{\mu[\xi]}(M)=\int\mu[\xi](d\omega){\mathds{1}}\\{\pi[\xi](\cdot|\omega)\in
M\\},$

and thus for ${\mathbb{P}}$-almost all $\xi$, by a change of variables,

$\begin{split}w[\theta_{j}\xi]_{\mu[\theta_{j}\xi]}(\theta_{j}M)&=\int\mu[\theta_{j}\xi](d\omega){\mathds{1}}\\{\pi[\theta_{j}\xi](\cdot|\omega)\in\theta_{j}M\\}\\\
&=\int\mu[\xi](d(\theta_{-j}\omega)){\mathds{1}}\\{\pi[\theta_{j}\xi](\cdot|\omega)\in\theta_{j}M\\}\\\
&=\int\mu[\xi](d\omega){\mathds{1}}\\{\pi[\theta_{j}\xi](\cdot|\theta_{j}\omega)\in\theta_{j}M\\},\end{split}$

where the second step follows from the translation covariance of $\mu[\xi]$.
Now, in order to conclude equation (5), it suffices to show that
$\pi[\theta_{j}\xi](\cdot|\theta_{j}\omega)\in\theta_{j}M$ if and only if
$\pi[\xi](\cdot|\omega)\in M$. For this, first note that the statement that
there exists $\nu\in\theta_{j}M$ such that for all $A\in{\cal F}$ we have

$\pi[\theta_{j}\xi](\theta_{j}A|\theta_{j}\omega)=\nu(\theta_{j}A),$

since $\nu(\theta_{j}A)=\theta_{-j}\nu(A)$, is equivalent to the statement
that there exists $\nu\in M$ such that for all $A\in{\cal F}$ we have

$\pi[\theta_{j}\xi](\theta_{j}A|\theta_{j}\omega)=\nu(A),$

since $\theta_{-j}\nu\in M$. Thus, it suffices to show that for all $j\in S$,
$A\in{\cal F}$ and $\omega\in\Omega$ we have that

$\pi[\theta_{j}\xi](\theta_{j}A|\theta_{j}\omega)=\pi[\xi](A|\omega)\qquad\text{for
}{\mathbb{P}}\text{-almost all }\xi.$

We modify slightly previous arguments to reflect the translation-invariant
setting. First, note that for a product spin space of the form $\Omega=E^{S}$
with $S={\mathbb{Z}}^{d}$ and with a standard Borel set $E$, the countable
core ${\cal C}$ can be chosen in a translation-invariant way, meaning that for
each subset of ${\cal C}$ also its translate belongs to ${\cal C}$, see
[Geo11, Corollary 4.A.13], and the discussion preceding it. We quickly recall
the arguments: $E$ is measure-theoretic isomorphic either to
$\\{1,\dots,|E|\\}$ with the product set-sigma algebra, if $|E|<\infty$;
${\mathbb{N}}$ with the product set-sigma algebra, if $|E|$ is countably
infinite; or $\\{0,1\\}^{{\mathbb{N}}}$ with sigma-algebra generated by the
cylinder sets, if $E$ is uncountably infinite. For instance, in the
uncountably infinite case $\Omega$ is isomorphic to
$\\{0,1\\}^{{\mathbb{N}}\times{\mathbb{Z}}^{d}}$, and the countable core can
be chosen as the set of finite cylinders of this set. The set of cylinders is
translation invariant, in particular under translations in ${\mathbb{Z}}^{d}$.
Second, we use a slight modification of the kernel

$\pi[\xi](A|\omega)={\mathds{1}}\\{(\omega,\xi)\in\bar{\Omega}_{o}\\}\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega)+{\mathds{1}}\\{(\omega,\xi)\not\in\bar{\Omega}_{o}\\}\nu_{o}(A),$

defined now via the set

$\bar{\Omega}_{0}=\\{(\xi,\omega)\in
H\times\Omega:\lim_{n\uparrow\infty}\gamma_{\theta_{i}\Lambda_{n}}[\xi](A|\omega){\text{
exists for all }}A\in{\cal C}\text{ and all }i\in{\mathbb{Z}}^{d},\text{ and
is independent of }i\\},$

where convergence is now guaranteed in a translation invariant way. Using
this, we have that

$\begin{split}\pi[\theta_{j}\xi](\theta_{j}A|\theta_{j}\omega)&={\mathds{1}}\\{(\theta_{j}\omega,\theta_{j}\xi)\in\bar{\Omega}_{o}\\}\lim_{n\uparrow\infty}\gamma_{\theta_{j}\Lambda_{n}}[\theta_{j}\xi](\theta_{j}A|\theta_{j}\omega)+{\mathds{1}}\\{(\theta_{j}\omega,\theta_{j}\xi)\not\in\bar{\Omega}_{o}\\}\nu_{o}(\theta_{j}A)\cr&={\mathds{1}}\\{(\omega,\xi)\in\bar{\Omega}_{o}\\}\lim_{n\uparrow\infty}\gamma_{\Lambda_{n}}[\xi](A|\omega)+{\mathds{1}}\\{(\omega,\xi)\not\in\bar{\Omega}_{o}\\}\nu_{o}(A)=\pi[\xi](A|\omega),\end{split}$

when we choose $\nu_{o}$ in a translation-invariant way. This concludes the
proof. ∎

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