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%\begin{frontmatter}
\title{Measures for lattice systems%
\thanks{Work partially supported by FAPESP (grant 95/0790-1,
Projeto Tem\'atico ``Fen\^omenos cr\'\i ticos e processos evolutivos
e sistemas em equil\'\i brio''), CNPq (grant 301625/95-6) and FINEP
(N\'ucleo de Excel\^encia ``Fen\'omenos cr\'\i ticos em
probabilidade e processos estoc\'asticos'', PRONEX-177/96).}}
\author{
\\
{\normalsize Roberto Fern\'andez\thanks{Researcher of the Consejo
Nacional de Investigaciones Cient\'{\i}ficas y T\'ecnicas
(CONICET), Argentina.}}\\[-1.5mm]
{\normalsize\it Instituto de Estudos Avan\c{c}ados, %}\\[-1.5mm]
Universidade de S\~ao Paulo}\\[-1.5mm]
{\normalsize\it Av.\ Prof.\ Luciano Gualberto, %}\\[-1.5mm]
Travessa J, 374 T\'erreo} \\[-1.5mm]
{\normalsize\it 05508-900 - S\~{a}o Paulo, Brazil}\\[-1mm]
{\normalsize\tt rf@ime.usp.br}\\[-2mm]
}
\date{July 1998}
\maketitle
\begin{abstract}
I review issues related with the presence or absence of
Gibbsianness in measures describing random fields in lattices.
After a brief exposition of the definition and properties of Gibbs
measures, I discuss the phenomenon of non-Gibbsianness. The
discussion involves three aspects: (1) Examples among renormalized
measures and invariant measures of stochastic transformations; (2)
probabilistic characterization of non-Gibbsian measures
---lack of quasilocality and ``wrong'' large-deviation properties---,
and (3) proposed clasification schemes and notions of generalized
Gibbsianness. The review closes with comments on directions for
further work.
\end{abstract}
\medskip
{\bf Keywords:} Non-Gibbsian measures, Gibbs measures, renormalization
transformations, lattice spin systems, cellular automata,
quasilocality, almost-Markovianness, generalized Gibbsian measures,
chains with complete connections.
\medskip
{\it AMS Subject Classification:} Primary 82B20, 82B28, 60K35
%\end{frontmatter}
\section{Introduction}
\emph{Gibbs measures} are the central objects of rigorous classical
statistical mechanics. In the established formalism, due to
Dobrushin, Lanford and Ruelle \cite{dob68b,lanrue69}, Gibbsianness is
a property encoded in the finite-volume conditional expectations. A
measure is Gibbsian if these expectations are determined by
Hamiltonians defined by sums of local terms or, more
precisely, of terms forming a summable interaction. The theory of
Gibbs measures is so well established in physics and probability
theory \cite{pre76,geo88} that in many instances a measure is assumed
to be Gibbsian almost by default. Gibbsianness brings a package of
useful properties: an efficient parametrization in terms of
interactions and inverse temperatures, an extremal principle and its
associated theory of large deviations, and a host of arguments and
techniques developed during one century of work in statistical
mechanics: contour arguments, cluster expansions, correlation
inequalities, uniqueness criteria, $\ldots$
Over the last few years, however, a number of studies of equilibrium
and dynamical classical spin systems showed the need to trascend the
framework of Gibbsian theory. Some unexpected features were detected
when rather simple and well known distributions ---like the
equilibrium measures of the Ising model--- were subjected to natural
transformations. The initial call to attention, due to Griffiths and
Pearce \cite{gripea78,gripea79,gri81}, came from the study of
renormalization transformations. It was soon understood \cite{isr79}
that these transformations \cite{fis83,gol92}, designed to study the
behavior of systems close to critical points, lead to probability
measures that can not be described by any summable interaction, thus
\emph{non-Gibbsian measures}. Other instances of non-Gibbsianness
were detected in the study of measures involving spin ``contractions''
\cite{lebmae87,dorvan89} and lattice projections \cite{sch89}, and,
not unexpectedly, among the stationary measures of stochastic time
evolutions \cite{lebsch88}. These works can be associated to an
initial stage of the study of non-Gibbsian measures, centered in the
\emph{symptomatology} of the phenomenon.
The second stage of this study ---the \emph{diagnosis} stage---
originated in the pioneer article by Israel \cite{isr79}, which was
formalized and exploited only a decade later
\cite{vEFS_LAT90,vEFS_PRL,vEFS_Prague,vEFS_LesHouches,vEFS_JSP,vEFS_Brazil,ent96}.
In this stage, the different known occurrences of non-Gibbsianness
were systematized and some key probabilistic aspects were emphasized.
The non-Gibbsianness of renormalized measures was traced to the lack
of continuity (in an appropriate sense, see below), with respect to
the external (or boundary) conditions, of some finite-volume
conditional expectations. This continuity, also known as
\emph{quasilocality} or \emph{almost-Markovianness}, is lost because
of the existence of ``hidden'' degrees of freedom that develop
long-range correlations. Changes in the exterior conditions occurring
arbitrarily far away can propagate, via these ``hidden'' correlations,
and alter the expectations around the origin, even in the absence of
(``nonhidden'') fluctuations in the intermediate regions. In these
examples non-Gibbsianness is thus a manifestation of first-order phase
transitions taking place in the system of ``hidden'' variables even
when the ``non-hidden'' (or \emph{block}) variables were fixed.
The initial studies of ``contractions'', ``projections''and measures
invariant under spin-flip or other types of dynamics
\cite{lebmae87,dorvan89,lebsch88,sch89,marsco91}, contained a
complementary type of diagnosis, based on the existence of
large-deviation probabilities that are either too large or too small
for the measure to be Gibbsian. For some of these examples the
diagnosis was later narrowed down to absence of quasilocality. A
detailed exposition of these arguments is presented in the long
monography \cite{vEFS_JSP}. As complementary references I mention
\cite{maevel94} which focuses on stationary measures for interacting
particle systems, and \cite{ferpfi96} which presents the phenomenon in
general probabilistic terms.
The last stage of the study of non-Gibbsian measures corresponds to
what could be called the \emph{treatment} of the phenomenon. A number
of classification schemes have been proposed aiming to establish
``hierarchies'' or ``degrees'' of non-Gibbsianness. One such a scheme
considers the behavior upon further decimation
\cite{maroli93,maroli94,lorvel94} to distinguish the so-called
\emph{robust} non-Gibbsianness. A second scheme is based on the size
of the set of external configurations where the discontinuities take
place \cite{ferpfi96} and leads to the notion of
\emph{almost-Gibbsian}, or, more properly, \emph{almost-quasilocal}
measures. A third scheme focuses on the existence of
almost-everywhere summable potentials to define the \emph{weakly
Gibbsian} measures
\cite{dob95,maevel97,dobshl97,brikuplef97,dobshl98}. Every
almost-quasilocal measure is weak Gibbsian and the converse is
false \cite{redmaemof98}. On the other hand there seems to
be no relation between these two categories and robust
non-Gibbsianness \cite{ent96,lor98,entshl98}.
These schemes have been used as the basis for a more ambitious, and
largely incomplete, program to develop a ``generalized Gibbsian
theory'' that includes some non-Gibbs distributions. Actually, this
effort was started rather early by people working in stochastic
evolutions \cite[and references therein]{lebsch88}. See
\cite{redmaemof98} for an updated analysis of the different attempts
to extend Gibbsianness and its properties.
In this review I shall start with a brief presentation of the
definition and main properties of Gibbsian measures, followed by an
overview of the most representative examples of non-Gibbsianness and a
discussion of the different ``treatments'' proposed for the
phenomenon. I will close with a personal view of further directions
of research. In the preparation of this article I have benefited from
the concise and complete account presented in \cite{ent96}.
\section{Gibbs measures}
In this section I review some key facts about Gibbs measures
emphasizing its role in the issues that follow. The default reference
for this section is the treatise by Georgii \cite{geo88}, though in
some cases I may provide more specific references.
\subsection{Basic definition}
Our systems will be formed by finite spins placed at the sites of
$\Zset^d$. The \emph{configuration space} is therefore a product
space $\Omega=\Omega_0^{\Zset^d}$ where $\Omega_0$ is a finite set.
Its elements ---the \emph{configurations}--- are families
$\omega=\{\omega_x\}_{x\in\Zset^d}$ where each $\omega_x$ takes values
in $\Omega_0$. Ising spins --- $\omega_x=-1,+1$ --- or Potts spins
\newline
--- $\omega_x=1,\ldots,q$ --- are typical examples.
The statistical mechanical description starts with ``finite windows'',
that is finite regions $\Lambda\subset\Zset^d$. The system there is
described via probability distributions on the corresponding space of
finite-volume configurations $\Omega_\Lambda\bydef\Omega_0^\Lambda$,
defined by the well known Boltzmann-Gibbs weights. These are
proportional to $\e^{-\beta H_\Lambda}$, where $\beta$ is the inverse
temperature and $H_\Lambda$ is the \emph{Hamiltonian} for the region
$\Lambda$. Two important requirements must be considered at this
point.
\begin{enumerate}
\item The Hamiltonians must be sums of \emph{local terms}, that is
of terms depending on spins at finite sets of sites. A
Hamiltonian for a larger region is obtained simply by adding new
local terms to the Hamiltonian for a smaller region.
\item The exterior of each window $\Lambda$ is taken to be frozen in
some configuration $\sigma_{\Lambda^\cc}$. The corresponding
Hamiltonian includes terms coupling spins inside and outside
$\Lambda$. Suitable summability conditions are required for such an
expression to be well-defined.
\end{enumerate}
The first requirement implies that the basic objects in the
construction of Boltzmann-Gibbs weights are not the Hamiltonians but
the \emph{interactions}, namely families
$\Phi=\{\Phi_B\}_{B\subset\subset\Zset^d}$, indexed by the finite
subsets $B$ of $\Zset^d$, where each $\Phi_B$ is a real- (or complex-)
valued function of the configurations, which depends only of the spins
in $B$. Given $\Phi$, the Hamiltonian on a finite region $\Lambda$
with external configuration $\sigma_{\Lambda^\cc}$ is the function on
$\Omega_\Lambda$ defined by the sum
\begin{equation}
\label{eq:1}
H(\omega_\Lambda|\sigma_{\Lambda^\cc}) \;\bydef\; \sum_{B\,:\,
B\cap\Lambda\neq\emptyset} \Phi_B(\omega_\Lambda\sigma_{\Lambda^\cc})\;.
\end{equation}
Here and in the sequel the notation
$\omega_\Lambda\sigma_{\Lambda^\cc}$ stands for the configuration
taking values $\omega_x$ for $x\in\Lambda$ and values $\sigma_x$ for
$x\not\in\Lambda$.
In order for \reff{eq:1} to be well defined it is natural to
demand the uniform and absolute summability condition:
\begin{equation}
\sup_{x\in\Zset^d}\,\sum_{B\ni x} \|\Phi_B\|_\infty \;<\; \infty\;.
\label{eq:5}
\end{equation}
Physically, this condition means that the overturning of a single spin
produces a finite change in the total energy.
Given an interaction and a external condition, the Boltzmann-Gibbs
prescription assigns to each configuration
$\omega_\Lambda\in\Omega_\Lambda$ the probability weight
\begin{equation}
%\bgl{\omega_\Lambda}{\sigma} \;\bydef\;
{\exp[-\beta\,H(\omega_\Lambda|\sigma_{\Lambda^\cc})] \over {\rm
Norm.}}\;.
\label{eq:10}
\end{equation}
These weights describe \emph{equilibrium} in finite volume. They
imply the averaging prescription
\begin{equation}
\bgl{f}{\sigma} \;=\;
\sum_{\omega_\Lambda\in \Omega_\Lambda}
f(\omega_\Lambda\sigma_{\Lambda^\cc})\,
{\exp[-\beta\,H(\omega_\Lambda|\sigma_{\Lambda^\cc})] \over {\rm
Norm.}}\;,
\label{eq:15}
\end{equation}
for all \emph{observables} $f$ (\ie measurable functions $f$).
To describe bulk properties it is necessary to pass to the limit
$\Lambda\to\Zset^d$ in some appropriate sense. In this limit the
notion of Hamiltonian loses its meaning; one must consider instead the
limit of the expectations \reff{eq:15}. An equivalent (in the present
setting) approach, introduced by Dobrushin, Lanford and Ruelle
\cite{dob68b,lanrue69}, transcribes the fact that the infinite-volume
analogue of the probability measures \reff{eq:15} should describe
equilibrium in the full space. This means that each finite volume
must be in equilibrium with the whole, that is, the finite-volume
prescriptions must be weighted by the full-volume prescription. This
leads to the definition that $\mu$ is a \emph{Gibbs measure} (for a
given interaction and inverse temperature) if
\begin{equation}
\int_\Omega f(\omega)\,\mu(d\omega) \;=\; \int_\Omega \bgl{f}{\sigma}\,
\mu(d\sigma)
\label{eq:20}
\end{equation}
for each observable $f$. These are the celebrated \emph{DLR
equations} (for Dobrushin, Lanford and Ruelle).
[Mathematical correctness requires, at this point, the definition of
the measure space where these measures are defined. That is, we must
decide which events or functions (=thought experiments or
measurements) are observable and which have to be left out of the
theory. As $\Omega$ is uncountable we can not declare \emph{all}
events to be part of our theory without running into contradictions.
Here, as usual in the theory of random fields, we
consider the $\sigma$-algebra generated by the cylinder sets, that is,
the sets determined by the spins at finitely many sites (microscopic
observables).]
>From a more probabilistic point of view, formula \reff{eq:20} means
that the \emph{conditional expectation} of $\mu$ on the finite region
$\Lambda$ given $\sigma$ outside coincides with the Boltzmann-Gibbs
average \reff{eq:15}:
\begin{equation}
\mu(f\,\vert\,\sigma_{\Lambda^\cc}) \;=\; \bgl{f}{\sigma}\;.
\label{eq:30}
\end{equation}
Thus, Gibbs measures are defined in terms of its conditional
distributions. The richness of statistical mechanics comes from the
fact that, unlike marginal distributions (Kolmogorov theorem),
conditional distributions do not necessarily determine a measure in a
unique way. It is known that there is always at least one such a
measure, but there may be more than one. The central problem in
classical statistical mechanics is, precisely, the determination of
all measures satisfying \reff{eq:20} for a given interaction.
Watching formula \reff{eq:20} [or \reff{eq:30}] one realizes that it
is not altered if $\bgl{f}{\sigma}$ is modified, or even undefined,
for a set of configurations $\sigma$ of $\mu$-measure zero. This
observation is the genesis of the generalization proposed by
Dobrushin to be discussed below.
\subsection{Quasilocality. The characterization theorem}
The best known spin systems ---Ising and Potts models--- have
\emph{finite range} interactions. That is, there exists an $r>0$
---the range---- such that $\Phi_B=0$ whenever $\mathrm{diam} B >r$.
In this case, the finite-volume expectations $\bgl{f}{\sigma}$,
defined in \reff{eq:15} have a Markovian property: They only depend on
the value of the external configuration $\sigma$ at sites at most a
distance $r$ from the set $\Lambda$. In the general case, where the
interaction has an infinite range, but satisfies the summability
condition \reff{eq:5}, the expectations have instead an
\emph{almost}-Markov, or \emph{quasi}-Markov, property: While
expectations do depend on the values taken by $\sigma$ at sites
arbitrarily far away, this dependence goes to zero at infinity.
More formally, we say that a function $f$ on $\Omega$ is
\emph{quasilocal} at a certain $\sigma$ if
\begin{equation}
\sup_{\xi_{\Gamma^c}\,,\,\eta_{\Gamma^c}} \left|
f(\sigma_{\Gamma}\xi_{\Gamma^\cc}) -
f(\sigma_{\Gamma}\eta_{\Gamma^\cc}) \right|
\;\longrightarrow\; 0
\label{eq:31}
\end{equation}
as $\Gamma \to \Zset^d$. Equivalently, $f$ is \emph{continuous} in
the product topology of $\Omega$. Let us call a measure $\mu$ on
$\Omega$ \emph{almost-Markovian} if one can find finite-volume
conditional expectations $\mu(f\,\vert\,\sigma_{\Lambda^\cc})$ that
are quasilocal at all $\sigma$ for all finite regions $\Lambda$ and
all quasilocal functions $f$. It is rather straightforward to verify
that \emph{every Gibbsian measure is almost-Markovian}. The
non-trivial part, due to Kozlov \cite{koz74} and Sullivan
\cite{sul73}, is the converse.
\begin{thm}\label{t.char}
A measure $\mu$ is Gibbsian if and only if all the conditional
probabilities $\mu(\omega_\Lambda\,\vert\,\sigma_{\Lambda^\cc})$ are
\begin{itemize}
\item quasilocal at all $\sigma$ for all $\omega$, and
\item strictly positive for all $\omega$ and $\sigma$.
\end{itemize}
\end{thm}
[We have indulged in a common abuse of notation and denoted
$\mu(\omega_\Lambda\,\vert\,\sigma_{\Lambda^\cc})$ instead of
the more pedantic
$\mu(\bone\{\eta\in\Omega:\eta_\Lambda=\omega_\Lambda\}\,\vert\,
\sigma_{\Lambda^\cc})$,
where $\bone\{\,\cdot\,\}$ is the characteristic function of the set
$\{\,\cdot\,\}$.]
The easy part of this theorem (necessity) is behind the proof of
non-Gibbsianness of renormalized measures, while the hard part
(sufficiency) justifies the notion of weak Gibbsianness.
We see that the definition of Gibbs measure is a combination of
probabilistic (conditional expectations) and topological
(continuity=quasilocality) notions. The generalized theories
discussed below can be interpreted as attempts to remove topological
constraints so as to leave the theory in a purely probabilistic
framework. Among the most immediate consequences of such attempts is
the loss of one-to-one-ness of the map ``measures $\to$ conditional
probabilities''. Indeed, being defined by integral equations, the
conditional probabilities of a measure can be freely changed in sets
of measure zero. A choice of conditional probabilities for each set
$\Lambda$ and configuration $\sigma$ constitutes a
\emph{realization}. The multiplicity of realizations disappears if
one adds the continuity requirement:
\begin{thm}\label{t.uniq}
A measure has at most one quasilocal realization of its finite-volume
conditional probabilities.
\end{thm}
In particular, this result, which is rather elementary from the point
of view of probability theory, shows that a measure can not be
simultaneously a Gibbs measure for different temperatures or
interactions producing different Boltzmann-Gibbs weights. For
translation-invariant interactions this implies that the pressure is a
strictly convex function of any linear parameter in the interaction,
if one modules-out interactions leading to the same Boltzmann-Gibbs
weights (``physically equivalent interactions'') \cite{grirue71}. The
loss of this physically very appealing property could be a potential
source of discomfort for generalized theories that dispose of topology
altogether.
The multiplicity of realizations must be taken into account when
trying to prove non-almost-Markovianness (and hence non-Gibbsianness).
One must show that the violation of quasilocality at a given external
configuration, happens for \emph{every} possible realization of a
particular conditional probability. A discontinuity of this type is
termed \emph{essential} in measure-theoretical jargon.
\subsection{Large-deviation properties}
Given two measures $\mu$ and $\nu$ on $\Omega$, the \emph{information
gain} of $\mu$ relative to $\nu$ in a finite region $\Lambda$ is
\begin{equation}
\label{eq:40}
I_\Lambda(\mu|\nu) \;\bydef\; \sum_{\omega_\Lambda\in\Omega_\Lambda}
\mu(\omega_\Lambda) \,\log {\mu(\omega_\Lambda)\over
\nu(\omega_\Lambda)}\;,
\end{equation}
with the convention $0\,\log 0 \equiv 0$ and allowing the value
$+\infty$. When $\nu$ gives equal weight to each configuration, this
differs in a sign (plus the log of a normalization factor) from what
physics textbooks call the \emph{entropy} of $\mu_\Lambda$ (=$\mu$
restricted to $\Omega_\Lambda$). Roughly speaking, the number
$I_\Lambda(\mu|\nu)$ gauges how different the two measures are when
restricted to the window $\Lambda$. Indeed, on the one hand it is a
positive number with the distance-like property of being
zero if and only if the two measures coincide in $\Lambda$. On the
other hand, large-deviation theory shows that, again roughly speaking,
the $\nu$-probability of generating a sample that looks, in $\Lambda$,
as ``typical'' for $\mu$, decreases exponentially with the size of the
sample, the rate being precisely $I_\Lambda(\mu|\nu)$.
For statistical-mechanical measures, the thermodynamic limit of
\reff{eq:40} is of little use, because it is usually infinite.
Nevertheless, this divergence occurs at a rate not exceeding
$|\Lambda|\bydef$cardinality of $\Lambda$. Hence, in this limit the
object of interest is the \emph{density of information-gain} of $\mu$
relative to $\nu$:
\begin{equation}
\label{eq:41}
i(\mu|\nu) \;\bydef\; \lim_{\Lambda\to\Zset^d} {1\over
|\Lambda|}\, I_\Lambda(\mu|\nu)\;.
\end{equation}
Part of the problem is, of course, to show that such a limit exists.
This is indeed the case if both $\mu$ and $\nu$ are translation
invariant and the latter is a Gibbs measure. The heuristic
interpretation of the ensuing theory of large deviations is that
\begin{equation}
\label{eq:45}
\mathrm{Prob}_\nu(\omega_\Lambda \hbox{ is ``typical'' for } \mu)
\;\sim\; \e^{-|\Lambda|\,i(\mu|\nu)}\;.
\end{equation}
In passing to the densities \reff{eq:41} one loses the distance-like
property of being nonzero if $\mu$ and $\nu$ are different, as the
following theorem shows.
%
\begin{thm}\label{t.zero}
Assume $\nu$ is a translation-invariant Gibbs measure. Then
\begin{equation}
\label{eq:50}
i(\mu|\nu)=0 \quad \Longleftrightarrow \quad
\mu(\,\cdot\,\vert\,\sigma_{\Lambda^\cc}) \;=\;
\nu(\,\cdot\,\vert\,\sigma_{\Lambda^\cc})
\end{equation}
for all regions $\Lambda$ and all configurations $\sigma$.
\end{thm}
[In the right-hand side of \reff{eq:50}, ``='' actually means ``can be
chosen equal to''.]
In words, $\mu$ has zero density of information-gain relative to a
Gibbsian $\nu$ (both measures being translation invariant) if and only
if $\mu$ is also Gibbsian for the \emph{same} temperature and (class
of physically equivalent) interaction. Physically, this corresponds
to the fact that an untypical ``island'' should have a probabilistic
cost of the order of its boundary if it involves configurations
typical of a different Gibbs state for the \emph{same} interaction
(think of an island of ``$-$'' in the ``$+$''-state of the Ising model
at zero field and low temperature), while otherwise its cost is of the
order of the volume of the island (eg.\ an island of ``$-$'' for the
Ising model with positive field).
Theorem \ref{t.zero} has played an important role in the detection of
non-Gibbsianness. It has been applied in the following two
complementary ways.
\begin{enumerate}
\item Suppose $\mu$ is a well-known \emph{non}-Gibbsian measure ---for
instance a \emph{frozen-state}, that is, a (Dirac) delta
concentrated in a single configuration. Then every measure $\nu$
with $i(\mu|\nu)=0$ is non-Gibbsian. In view of \reff{eq:45} one
can say that some large deviations probabilities of $\nu$ are
``too-large'' for $\nu$ to be Gibbsian. Measures of this type were
obtained as a result of spin contractions \cite{lebmae87,dorvan89}
and as invariant measures of stochastic transformations
\cite{lebsch88,marsco91}.
\item Suppose $\mu$ and $\nu$ are such that \emph{if} $\nu$ admits
quasilocal conditional probabilities, then (i) those are also
conditional probabilities for $\mu$
---$\mu(\,\cdot\,\vert\,\sigma_{\Lambda^\cc}) \;=\;
\nu(\,\cdot\,\vert\,\sigma_{\Lambda^\cc})$ for all $\sigma$---, and
(ii) $i(\mu|\nu)> 0$. Then neither $\mu$ nor $\nu$ are Gibbsian.
Given \reff{eq:45} one can say that these measures have large
deviations probabilities that are ``too small'' for Gibbsianness.
This situation has been found in measures associated to lattice
projections \cite{sch89}.
\end{enumerate}
A third way to apply Theorem \ref{t.zero} is contained in the
following Corollary.
\begin{cor}[Dichotomy corollary]\label{c.dic}
If two translation-invariant measures $\mu$ and $\nu$ are such that
$i(\nu|\mu)=i(\mu|\nu)=0$, then either (1) both are Gibbsian and
yield the same finite-volume Boltzmann-Gibbs averages, or (2) both
are non-Gibbsian.
\end{cor}
This dichotomy corollary has been used to prove that cell (or local)
renormalization transformations at the level of interactions are never
many-valued \cite[Section 3.2]{vEFS_JSP}.
In view of Theorem \ref{t.zero}, it is tempting to use the density of
information-gain to estimate somehow the ``distance to Gibbsianness''
of a measure. For some remarks in this direction, see \cite[Section
5.1.2]{vEFS_JSP}.
\section{Transformations of measures}
We shall consider transformations sending measures on a configuration
space $\Omega=\Omega_0^{\Zset^d}$ ---the space of \emph{original} or
\emph{object} configurations--- to measures on a target space
$\Omega'=(\Omega'_0)^{\Zset^{d'}}$ ---the space of \emph{image} or
\emph{block} configurations. The latter can coincide with the former.
Two types of questions are usually posed regarding the action of these
transformations:
\begin{enumerate}
\item What happens after a \emph{single} application of the
transformation to a Gibbsian measure. This is the point of view of
renormalization transformations. In general the transformed measure
has a coarser $\sigma$-algebra (fewer observables) and the
transformation is interpreted as some sort of ``noise'' or
``blur-out''of the original measure. In renormalization-group
transformations this noise is introduced on purpose, to extract only
the ``blurred-out'' information characterizing critical points. In
image processing or sound recognition the noise is an unwanted
feature and the objective is to reconstruct the information
contained in the original measure. In both cases it is important to
determine whether the transformed measure is Gibbsian. The
Gibbsianness hypothesis is built into renormalization-group theory,
and it is the basis of important sampling and restoration
procedures.
\item What happens with the invariant measures of the transformation.
This is the issue of interest in the study of cellular automata or
stochastic dynamics, where one investigates the result of infinitely
many iterations of a transformation starting from an arbitrary
initial state. These are models of systems out of equilibrium,
hence there is no reason to expect Gibbsianness of their stationary
measures. Nevertheless Gibbsiannes has been proven in certain
regimes, and at any rate, if this is not the case, it is meaningful
to wonder which properties of Gibbsianness are still present.
\end{enumerate}
\subsection{Deterministic transformations}
In a \emph{deterministic} transformation, the image
configuration is fully determined by the original one. It is
defined by a map
\begin{equation}
\label{eq:500}
\begin{array}{rcl}
t:\Omega &\to& \Omega'\\
\omega & \mapsto & \omega'=t(\omega)\;,
\end{array}
\end{equation}
which in turns defines a map that to a measure $\mu$ on $\Omega$
associates a measure $\mu'$ on $\Omega'$ with averages
\begin{equation}
\label{eq:51}
\int_{\Omega'} f'(\omega') \,\mu'(d\omega') \;=\;
\int_\Omega f'\bigl(t(\omega)\bigr) \,\mu(d\omega)\;.
\end{equation}
Let us call the transformation \emph{local}, or a \emph{block-} or
\emph{cell-transformation}, if there exists a \emph{finite} set
$B_{0'}\subset\subset\Zset^{d'}$ and some number $b$ such that the
sets $B_{x'}\bydef B_{0'}+ bx'$ ---the \emph{blocks} or
\emph{cells}--- satisfy: (i) Their union is all of $\Zset^{d'}$, and
(ii) each $\omega'_{x'}$ depends only of the original spins in the
corresponding block, that is, of $\omega_{B_{x'}}$.
Conspicuous examples of this type of transformations are:
\emph{1) Projection transformations.} The transformation $t$ is just
the restriction to a subset $S$ of $\Zset^d$. The transformed measure
applies only to functions depending on spins in this subset $S$ and
averages out all the other spins. The following two cases have been
studied in some detail:
\begin{itemize}
\item[\emph{1.1)}] \emph{Decimation of spacing $b$.} The subset $S$
is formed by lattice points all whose coordinates are multiples of
$b$. This subset is, in fact, isomorphic to the original $\Zset^d$,
hence $\Omega'=\Omega$ and $\omega_{x'} = \omega_{bx'}$.
\item[\emph{1.2)}] \emph{Projection on a hyperplane} \cite{sch89}. The
subset $S$ is (isomorphic to) $\Zset^{d-1}$ and it is identified
with the hyperplane $\{(x_1,x_2,\ldots,x_{d-1},0)\in \Zset^d\}$.
Formally, $\Omega'=\Omega_0^{\Zset^{d-1}}$ and
$\omega'_{x'}=\omega_{(x',0)}$. This is \emph{not} a cell
transformation.
\end{itemize}
\emph{2) Block-average transformations.} These are cell transformations
defined by
\begin{equation}
\label{eq:55}
\omega'_{x'} \;=\; {1\over |B_{x'}|}\, \sum_{y\in B_{x'}} \omega_y\;,
\end{equation}
where we are assuming that $\Omega_0$ is formed by consecutive
integers. This is an example where $\Omega'_0\neq\Omega_0$.
\emph{3) Majority-rule transformation.} For Ising spins
$\Omega_0=\{-1,1\}$, let
\begin{equation}
\label{eq:60}
\omega'_{x'}\;=\;\mathrm{sign}\Bigl(\sum_{y\in B_{x'}}\omega_y\Bigr)\;.
\end{equation}
This is a local transformation. If the block-size is even, a rule is
needed to decide ties. Often this rule is stochastic ($+1$ or $-1$
with equal probability). These would be the simplest example of
stochastic transformation (see below).
\emph{4) Spin contractions.} These are single-site transformations
($B_{x'}=\{x'\}$), where $d'=d$ but $\Omega_0'$ is strictly a subset
of $\Omega_0$. I mention two well studied cases:
\begin{itemize}
\item[\emph{4.1)}] \emph{Sign fields.} In these examples $\Omega_0$
is a symmetric subset of the real numbers while $\Omega'_0=\{-1,1\}$
or $\Omega'_0=\{-1,0,1\}$. The original and image lattices
coincide, $d=d'$. The map is
\begin{equation}
\label{eq:65}
\omega'_{x'}\;=\;\mathrm{sign}\,\omega_{x'}\;.
\end{equation}
Two particular cases are:
\begin{itemize}
\item[(i)] The sign-field of
(an)harmonic crystals. This corresponds to $\Omega_0=\Rset$. This
field was studied in \cite{lebmae87} in relation with the
phenomenon of entropic repulsion, and in \cite{dorvan89} in
reference to the renormalization-group theory of the Ising model in
dimensions larger than four.
\item[(ii)] The sign-field of the SOS model \cite{lor98,entshl98}. Here
$\Omega_0=\Zset$.
\end{itemize}
[Note that in both cases the original model has an infinite
single-spin space and hence it exceeds, rigorously speaking, the
framework adopted here.]
\item[\emph{4.2)}] \emph{Fuzzy Potts model}\cite{maevel95}. The original
spins,
with values in $\Omega_0=\{1, 2, \ldots, q\}$, are contracted into a
smaller number $n$ of values, where $n$ divides $p$: $\omega'_{x'}$
takes the value $i$ if $(i-1)q/n \le \omega_{x'}\le iq/n$.
\end{itemize}
\emph{5) Momentum transformations.} They are ``almost-local''
transformations. The image spins depend of all the initial spins, but
this dependence tends to zero for far-away spins. More precisely,
the transformation is defined by a law
\begin{equation}
\label{eq:650}
\omega'_{x'} \;=\; \sum_y F(b x' - y)\, \omega_y\;,
\end{equation}
where $F$ is the Fourier transform of a smooth function $\widehat
F(k)$ (representing a ``soft cutoff'').
Decimation, average, majority-rule and momentum transformations have
been intensively used in the renormalization-group analysis of various
systems. For references see \cite[Section 3.1.2]{vEFS_JSP} or
\cite[Section 1]{gripea79}.
\subsection{Stochastic transformations}
For these transformations, the procedure to obtain the image spins
involves some randomness. Formally (let me consider only the case of
cell transformations with ``parallel updating''), there is a
collection of weights $\{T(\omega'_{x'}|\omega_{B_{x'}})\}$ such that
\begin{equation}
\label{eq:70}
\sum_{\omega_{B_{x'}}} T(\omega'_{x'}|\omega_{B_{x'}}) \;=\; 1\;.
\end{equation}
These weights describe the probability of obtaining a spin
$\omega'_{x'}$ from the original configuration $\omega_{B_{x'}}$ of
the block $B_{x'}$. Correspondingly, the transformed $\mu'$ of a
measure $\mu$ is the measure that for each function $f'$ depending on
finitely many image spins yields an average
\begin{equation}
\label{eq:71}
\int_{\Omega'} f'(\omega') \,\mu'(d\omega') \;=\;
\int_\Omega f'(\omega')\,\prod_{x'} T(\omega'_{x'}|\omega_{B_{x'}})
\,\mu(d\omega)\;.
\end{equation}
[Of course, this transformations include the deterministic
transformations defined above as a particular case.]
As examples I mention:
\emph{1) Kadanoff transformations.} Defined, for Ising spins,
$\Omega_0=\{-1,+1\}$, by the weights
\begin{equation}
\label{eq:80}
T(\omega'_{x'}|\omega_{B_{x'}}) \; =\; {\exp\Bigl(p\,\omega'_{x'}\,
\sum_{y\in B_{x'}} \omega_y\Bigr) \over \mathrm{Norm.}}\;,
\end{equation}
where $p>0$ is a parameter. These transformations have been used to
study the critical properties of the Ising model. They admit several
generalizations and interesting limit cases, see \cite[Section
3.1.2]{vEFS_JSP}.
\emph{2) Stochastic smooth sign-fields.} Used in \cite{kue98} to
study continuous-spin systems in the presence of a random field.
These are single-site spin contractions with $\Omega_0=\Rset$ and
$\Omega'_0=\{-1,1\}$, defined by the probabilities
\begin{equation}
\label{eq:80.z}
T(\omega'_{x'}|\omega_{x'}) \; =\; {1\over 2} \Bigl(1+\omega'_{x'}
\tanh(a\, \omega_{x'})\Bigr)\;,
\end{equation}
parametrized by the constant $a$. When $a\to\infty$ these
transformations tend to the deterministic sign-field transformation
defined by \reff{eq:65}.
\emph{3) Transformations defining stochastic cellular automata.} In
this case the blocks $B_{x'}$ are usually overlapping, the
image space is identical to the original one and it is interpreted as
the latter at a later ``time'', and the numbers $T(\omega'|\omega)$
are thought as transition probabilities. A
large number of such automata has been proposed and studied. In
particular, I shall refer below to work done for the voter model
\cite{lebsch88}, a Swendsen-Wang-like dynamics studied in
\cite{marsco91} and a numerical study of the Toom model \cite{mak97}.
\section{Non-Gibbsian measures: The symptoms}
\subsection{The ``peculiarities''}
Griffiths and Pearce \cite{gripea78,gripea79,gri81} were the first to
point out problems with the assumption of Gibbsianness of measures
subjected to renormalization transformations. While their arguments
were not fully rigorous, many of their ideas and observations have
been later put on a rigorous footing. As an illustration let us
consider their discussion of what they call ``model I''. Take the
Kadanoff transformation with blocks of size one (\ie where the
original and image spin coincide), for the (nearest-neighbor
ferromagnetic) Ising model. If the map \reff{eq:71}--\reff{eq:80} is
applied to a finite region $\Lambda$, the distribution of
\emph{original} spins $\omega$ when the image spins $\omega'$ are all
set equal to $-1$ corresponds to an Ising model with field $h-p$.
Consider now the energy cost of flipping $\omega'_0$ from $-1$ to
$+1$:
\begin{eqnarray}
\label{eq:85} \exp{W'_{\{0\}}(+1|-1)} & \bydef & {\mu'_{\Lambda}(+'_0
-'_{\Lambda\setminus\{0\}}|-'_{\Lambda^\cc}) \over \mu'_{\Lambda}(-'_0
-'_{\Lambda\setminus\{0\}}|-'_{\Lambda^\cc})} \nonumber\\ & = &
\langle \e^{2p\sigma_0} \rangle_{\Lambda}^{h-p} \nonumber\\ & = &
\cosh 2p + \langle \sigma_0 \rangle_{\Lambda}^{h-p} \, \sinh 2p\;,
\end{eqnarray}
where $\langle \,\cdot\, \rangle_\Lambda^{h-p}$ stands for the Ising
Boltzmann-Gibbs factor for the region $\Lambda$ with field $h-p$. At
low temperature, the right-hand side has a multivalued thermodynamic
limit for $h=p$. Griffiths and Pearce conclude that this indicates
that there is not a well defined interaction behind the measure
$\mu'$.
While this does not constitute a mathematically complete
non-Gibbsianness argument, it already shows that the ``peculiarities''
---as Griffiths and Pearce call them--- are due to the existence of
phase transitions of the system of \emph{original} spins
\emph{constrained} by well-chosen image-spin configurations (they call
this a \emph{modified object system}). Therefore they need not happen
at transition regions of the original system.
Griffiths and Pearce proposed a second scenario for these
``peculiarities'', in which the renormalized interaction would be
well-defined, but would not be a smooth function of the parameters of
the original model. Soon after a number of numerical studies
appeared, suggesting the presence of multivaluedness and
discontinuities in the transformations at the level of Hamiltonians
(see references in \cite[Section 1.1]{vEFS_JSP}. Within the framework
of standard Gibbs theory, this scenario was, however, ruled out by
later studies \cite[Section 3]{vEFS_JSP}. Nevertheless, the
multivaluedness can occur if the transformations are non-cell, for
instance if they include projections to lower-dimensional manifolds
\cite{lorvel94}, or if one relaxes the theoretical framework by
allowing weakly Gibbsian measures \cite{dobshl97,dobshl98} (Section
\ref{ss.weak} below).
\subsection{Entropic repulsion and contracted Gaussians}
\label{z.entro}
Almost ten years later than Griffiths and Pearce, Lebowitz and Maes
\cite{lebmae87} produced an example of a different nature. They
considered harmonic crystals, that is, systems with spins
$\varphi_x\in\Rset$ and with formal Hamiltonian of the form
\begin{equation}
\label{eq:90}
H(\varphi) \;=\; \sum_{\langle x\, y\rangle} V_{xy}(\varphi_x-\varphi_y)
\end{equation}
where the functions $V_{xy}$ are even and convex. They showed that,
due to the shift-symmetry $\varphi_x\to\varphi_x+k$ of the system, the
probabilistic cost of shifting the spins within a region is
subexponential in the volume of this region (is like inserting a
bubble configured in a different Gibbs state). Physically, these
systems can be used to model the height of an interface. The result
implies that any linear perturbation of the interaction (``soft
wall'') sends the interface to infinity (entropic repulsion).
Mathematically, the measure obtained by taking the sign of the spins
has zero relative entropy with respect to the delta-measure
concentrated in the all-''$+$'' configuration. The resulting measure
is therefore non-Gibbsian by Theorem \reff{t.zero} (``too large''
large deviations).
The same phenomenon was generalized by Dorlas and van Enter to the
sign field of self-similar Gaussians \cite{dorvan89}, and later to
anharmonic crystals ($V_{xy}$ not-necessarily quadratic) \cite[Section
4.4]{vEFS_JSP}. Assuming that, as believed, block-average
transformations of the critical Ising model in $d\ge5$ converge to a
Gaussian fixed point, the results of \cite{dorvan89} imply that, after
a sequence of majority-spin transformations with larger and larger
block-size, the critical Ising-model measure converges to a
non-Gibbsian distribution.
Lebowitz and Schonmann showed that the extremal invariant measures of
the voter model ($d\ge 3$) are non-Gibbsian because they have also
too-large probabilities of having bubbles of spins frozen in the
all-''$+$'' configuration \cite[formula (3.8)]{lebsch88}.
\subsection{Projections on hyperplanes}
Schonmann \cite{sch89} provided the first example of non-Gibbsiannes
manifested via ``too-small'' large deviations. He considered the
projection of the two-dimensional Ising model onto the $x$-axis and
showed that changing the spins far away along this axis one can pass
from the projection of the ``$+$''-measure to the projection of the
``$-$''-measure. Therefore, if the projection of the ``$+$''-state has
quasilocal conditional probabilities, these must be also conditional
probabilities for the projection of the ``$-$''-state. But, on the other
hand, there are large-deviation results showing that there can be at
most one Gibbs translation-invariant projected state. Hence neither
projection is Gibbsian. The result is valid all the way up to the
critical temperature of the two-dimensional Ising model. The example
was later generalized and studied in more detail in \cite{ferpfi96}.
In particular it was shown there that projections of a $d$-dimensional
Ising model, $d\ge 2$, onto a coordinate hyperplane are non-Gibbsian
for temperatures smaller than the critical temperature of the initial
$d$-dimensional model.
\section{Non-Gibbsian measures: The diagnosis}
\subsection{Non-quasilocality for cell-renormalization transformations}
Israel \cite{isr79} provided a (practically) rigorous argument that
proved the existence, and clarified the nature, of Griffiths' and
Pearce's ``peculiarities'' for the decimation of the two-dimensional
Ising model at low temperatures. He showed how a phase transition in
the constrained system of original spins causes the lack of
quasilocality of one-point conditional probabilities. By Theorem
\ref{t.char} this implies non-Gibbsianness. This confirmed Griffiths'
and Pearce's first scenario ---lack of summable renormalized
interaction.
The essence of Israel's argument is rather simple. Consider
decimation of $2\times2$-blocks and fix the image (=nondecimated)
spins in the alternating configuration $\omega'_x=(-1)^{|x|}$.
These constrained spins act as additional magnetic fields over the
remaining original spins, but these fields have alternating signs and
cancel out. Therefore, the constrained system is a decorated Ising
model (Ising model with additional sites at the middle of each bond)
which is equivalent to a standard Ising model at a higher temperature.
The model has, thus, a phase transition at low temperatures and it is
not hard to see that one can select one or the other phase by choosing
the image spins all ''$+$'' or all ''$-$'' in a ring of unit thickness
and diverging radius. This, in turns, changes the magnetization at
the origin: At low-enough temperature there exists a constant
$\varepsilon>0$ such that for all square sets $\Gamma$ sufficiently
large and all image configurations $\eta'$ and $\xi'$:
\begin{equation}
\label{eq:900}
\Bigl| \langle \omega'_0 \,|\, {\sigma'}^{\pm}_\Gamma\,
+'_{\partial\Gamma}\,\eta'_{(\Gamma\cup\partial\Gamma)^\cc}\rangle -
\langle \omega'_0 \,|\, {\sigma'}^{\pm}_\Gamma\,
-'_{\partial\Gamma}\,\eta'_{(\Gamma\cup\partial\Gamma)^\cc}\rangle
\Bigr|
\; >\; \varepsilon\;,
\end{equation}
where ${\sigma'}^{(\pm)}$ denotes the alternating configuration. An
important technical point: The fact that the inequality holds
uniformly in the configurations $\eta'$ and $\xi'$ implies that the
jump involves two sets of configurations that are open in the product
topology, and hence of non-zero measure. It follows that the
discontinuity at ${\sigma'}^{(\pm)}$ is essential.
The main ingredients of this argument were abstracted and exploited in
\cite{vEFS_JSP}. The proof of the violation of quasilocality requires:
\begin{itemize}
\item[I] To exhibit a special image configuration
$\sigma'_{\mathrm{spec}}$ such that the resulting
\emph{constrained} system of \emph{original} spins has more than one
phase.
\item[II] To show that two of these phases can be selected by fixing
the \emph{image} spins arbitrarily far away in a suitable manner.
\item[III] Furthermore, the selection of these phases must be made via
\emph{open} sets of image configurations, so the lack of
quasilocality becomes essential.
\end{itemize}
Once this was understood, it was relatively straightforward to obtain
a large catalogue of transformations for the Ising model in dimensions
$d\ge 2$ leading to non-Gibbsianness: decimation with arbitrary
spacing, Kadanoff transformations for arbitrary block size and value
of $p$, block-averaging for even block sizes, and some cases of
majority rule ($d=2$) \cite[Section 4]{vEFS_JSP}. One must discover
special configurations $\sigma'_{\mathrm{spec}}$ such that the
constrained system exhibits a phase transition that can be treated
rigorously, for instance via Pirogov-Sinai theory as explained in
\cite[Appendix B]{vEFS_JSP}.
All these examples are at temperatures strictly below the Ising
critical temperature. Some of them, though, involve non-zero
magnetic fields, required to be low-enough for decimation and Kadanoff
transformations in $d\ge 3$ but that could have arbitrary values for
block-averaging. Soon other examples showed that any region of the
phase diagram could be hit by the phenomenon. For instance,
decimation for the high-$q$ Potts model leads to non-Gibbsianness for
an interval of temperatures higher than the critical
\cite{entferkot95}. Furthermore, for each fixed temperature there is
a (perversely designed) transformation leading to non-Gibbsianness
\cite{ent97}. Griffiths' and Pearce's suspicions that ``peculiarities
might be a fairly general phenomenon'' \cite[page 64]{gri81}, were
fully confirmed.
\subsection{Non-quasilocality of projections to hyperplanes}
The original argument \cite{sch89} proving the non-Gibbsianness of the
projection of the $2d$-Ising to the line is somehow delicate. Its
first part, proving that if quasilocality were present then both the
``$+$'' and ``$-$'' projections would have the same conditional
probabilities, resorts to percolation results that are specifically
two dimensional. This casted some doubts on whether the example could
be generalized to higher dimension, and, if this generalization were
possible, on which would be the limit temperature for the existence of
non-Gibbsianness. A natural candidate is the critical temperature of
the initial Ising model, but the use of percolation arguments pointed
towards the Peierls temperature, that is the temperature above which
there is percolation of minority spins (in two dimensions the Peierls
and critical temperatures coincide). The second part of the argument
is, in my opinion, even more subtle. It states that \emph{if} one of
the projections were Gibbsian, then the relative density of
information-gain between both projections \emph{would} exist and be
positive. As the projections turned out to be non-Gibbsian, the
actual existence of this density of information-gain was not proven.
In fact it remains unproven to date. The argument consists, thus, in
exposing a \emph{potential} smallness of large-deviation
probabilities.
Alternative arguments show that in fact the non-Gibbsianness is due to
lack of quasilocality and that it happens for any dimension $d\ge 2$.
This was first proven in \cite[Section 4.5.2]{vEFS_JSP}, via a Peierls
argument, for an alternating special configuration. In
\cite{vEFS_JSP,maevel94,ferpfi96} this lack of quasilocality was
related to the existence of a \emph{wetting} phenomenon.
If the spins in a coordinate hyperplane are all ``$+$'', the state of
the system in a half space is unique and independent of the external
conditions used for the other hyperplanes. The all-``$+$''
configuration causes the formation of a droplet of the corresponding
state whose thickness diverges in the thermodynamic limit. This
corresponds to a situation of \emph{complete wetting}. Similarly, the
all-``$-$'' configuration produces complete wetting. On the other
hand, there are configurations that lead to \emph{partial wetting}:
the width of the associated layer remains finite and the bulk phase is
decided by the boundary conditions. It is at these configurations
that the quasilocality of the projections is lost. When one of these
configurations is surrounded by an arbitrary far layer of ``$+$''
spins, complete wetting leads to local averages that are different
from those obtained for a layer of ``$-$''.
Furthermore, an inequality presented in \cite{ferpfi96} shows that
the quasilocality of the projections fails whenever the surface
tension of the wetting droplet is positive. This is known to happen
for all dimensions $d\ge 2$ and for all temperatures lower than the
critical \cite{fropfi87}. Hence, the non-Gibbsianness of the
projections happens up to this temperature, rather than the Peierls
temperature.
\subsection{Non-Gibbsianness of invariant measures}
The work in \cite{marsco91,maevel94} brings additional insight into
the non-Gibbsianness of stationary states for cellular automata. The
first reference studies a non-local dynamics for a lattice gas (\ie
$\Omega_0=\{0,1\}$). For its invariant measure, the probability that
a region be all filled with particles decreases only subexponentially
in the volume of the region. The measure has, therefore,
``too-large'' large deviations with respect to the delta-measure
concentrated on the ``all-occupied'' configuration. Furthermore, the
study exhibits the \emph{mechanism} behind this fact: Once a ring of
particles has been established, the dynamics will proceed to fill the
interior of the ring with particles. The probabilistic cost of
establishing an occupied region is, therefore, dictated by the
formation of the boundary of the region.
Perhaps the most important result for invariant measures is a
\emph{dichotomy theorem} analogous to Corollary \ref{c.dic}. The
theorem requires two properties from the stochastic transformations:
\begin{thm}[Dichotomy theorem] \label{t.dic}
Assume that the transformation satisfies (i) the transition
probabilities $T(\omega'_x|\omega_{B_{x'}})$ are all strictly
positive, and (ii) there exists $R>0$ such that the
$\cup_{x'\in[-L,L]^d} B_{x'} \subset [-L-R,L+R]^d$ for all $L>0$.
Then the translation-invariant measures that are invariant for the
transformation are either all Gibbsian or none Gibbsian.
\end{thm}
The theorem remains valid for spin-flip processes with positive (and
local) rates. For the latter, the theorem was first obtained by
K\"unsch \cite{kun84}; the form stated here was proven in
\cite{maevel94}. Most renormalization transformations fail to satisfy
the second assumption. Indeed if blocks do not overlap the image
spins within a square of size $L$ come from internal spins in a square
of size $bL$.
An immediate consequence of this theorem is that all invariant
measures are Gibbsian for transformations satisfying detailed balance
with respect to Bolzmann-Gibbs weights. For non-reversible
probabilistic cellular automata, like the Toom model \cite{too95}, the
situation is less clear. From Theorem \ref{t.zero} and the previous
dichotomy theorem we see that the appearance of two invariant measures
with strictly positive relative density of information-gain would
automatically imply the non-Gibbsianness of all the invariant
measures. In \cite{maevel94} this observation is transcribed into the
following heuristic test, related to the mechanism described in
\cite{marsco91}: Take a typical configuration of one of the invariant
measures, introduce a boundary typical of the other measure and
observe whether the dynamics tends to fill the interior of the region
with the phase dictated by the boundary. If not, this would be an
indication that the probabilistic cost of a region of such
``mistakes'' grows exponentially with the volume, and hence that the
relative information-gain is not zero. Both (and all) invariant
measures would then be non-Gibbsian. This test has been recently
numerically performed for the Toom model \cite{mak97}. The results
are not totally conclusive, but they give some evidence that the
information-gain density between the plus and minus invariant Toom
measures is zero, in agreement with the nonrigorous but plausible
argument presented in \cite{vel95}.
\subsection{Other instances of non-quasilocality}
\subsubsection{The random-cluster model}\label{ss.ran}
The random-cluster model, introduced by Fortuin and Kasteleyn
\cite{forkas72} (see also the historical references listed in
\cite{gri95}), is a correlated bond-percolation model. To each bond
configuration $\nnn$ the model assigns a (finite-volume) probability
weight
\begin{equation}
\label{eq:95}
p^{{\cal N}_1(\nnn)}\, (1-p)^{{\cal N}_0(\nnn)}\, q^{{\cal C}(\nnn)}
/ {\rm Norm.}
\end{equation}
where ${{\cal N}_1(\nnn)}$ is the number of open bonds, ${{\cal
N}_0(\nnn)}$ the number of closed bonds, and ${{\cal C}(\nnn)}$
the number of connected clusters. For $q=1$ one recovers independent
percolation, while for $q\ge 2$ the model is related by identities to
(is a ``representation'' of) the $q$-state Potts model. In the latter
case, $p$ is a function of the inverse temperature.
>From \reff{eq:95} one can define, in the obvious way, conditional
probabilities for various boundary conditions. As the $q$-dependence
in \reff{eq:95} is highly nonlocal, it is not difficult to see that
for $q\neq 1$ these conditional probabilities are not quasilocal
\cite{aizetal88,vEFS_JSP} for any $0T_{\mathrm{c}}/1,36$ ---almost complementing the
interval $T