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\begin{document}
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\title{\vspace*{-2.4cm} \bf Robustness of the non-Gibbsian property:
some examples}
\author{
{\normalsize Aernout C. D. van Enter and J\'ozsef L\H{o}rinczi} \\[0.8cm]
{\normalsize\it Institute for Theoretical Physics, Rijksuniversiteit
Groningen} \\[0.1cm]
{\normalsize\it Nijenborgh 4, 9747 AG Groningen, the Netherlands} \\[0.3cm]
%{\normalsize\tt aenter\@th.rug.nl} \\[-1.5mm]
%{\normalsize\tt lorinczi\@th.rug.nl} \\[-2mm]
}
\vspace{1cm}
\date{\small\datetitle}
\maketitle
\vspace{4cm}
\begin{abstract}
We discuss some examples of measures on lattice systems, which lack the
property of being a Gibbs measure in a rather strong sense.
\end{abstract}
\thispagestyle{empty}
\clearpage
%
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\section{Introduction}
In recent years extensive research has been done on the occurrence of
states (probability measures) on lattice systems which are not of Gibbsian
type. Such measures occur for example in renormalization-group studies
\cite{GP1,GP2,Isr,DvE,vE,vEFK,vEFS0,vEFS1,vEFS2,Sal}, non-equilibrium
statistical mechanical models \cite{Spe,LSch,MVV2,MS}, image analysis
\cite{BP,GG,MVV3}, probabilistic cellular automata \cite{LMS,MVV2} and random
cluster models \cite{Gr,PVV}. The possibility of their occurrence and their
properties have been considered by various authors
\cite{BMO,Dob2,FP,Ken2,Ken1,LM,Lor1,Lor2,Lor3,LVV,MVV1,MO1,MO2,Sch,VV}. This
non-Gibbsian behaviour has often been considered `pathological'
---~ undesirable~---, and there have been various attempts to control the
non-Gibbsianness.
One approach, advocated by Martinelli
and Olivieri \cite{MO1,MO2}, is to study how the non-Gibbsian measures
behave under decimation transformations, that is, to consider the restriction
of the measure to some sufficiently sparse periodic sublattice. Various
examples where a once renormalized measure is non-Gibbsian have been shown
to result in Gibbs measures again after such mappings, mostly, but not
exclusively, in the regime where the original model has no phase transition
\cite{MO1,MO2,LVV}.
In another approach, developed by Fern\'andez and Pfister
\cite{FP}, one studies the
size of the set of `pathological' configurations and tries to show that
it is small, i.e. of measure zero. In this case one says that the
non-Gibbsianness is `weak' \cite{FP,MVV3,Lor1,Gr}.
An even stronger control
was recently obtained by Dobrushin in an example first studied in \cite{Sch}.
In this example one considers the restriction of the plus-phase of the
two-dimensional Ising model to a one-dimensional sublattice.
Here the non-Gibbsian measure can be described as the Gibbs measure for an
almost everywhere defined potential \cite{Dob2}.
In this paper we present some examples in which the non-Gibbsianness
is `robust', either in the sense of stable under decimations, or
in the sense of being due to a large-measure set. It is known \cite{MVV4}
that the two notions are not equivalent; indeed, there are examples of
measures which have a large set of pathological configurations but which
become Gibbsian after decimation.
\section{Notation and some standard results}
First we will introduce some notation and recall some known facts. For
details we refer to \cite{Geo,vEFS1}.
We consider spins placed at the vertices of the {\it lattice} $\mathbb{Z}^d$.
The {\it configuration space} is $\Omega = S^{\mathbb{Z}^d}$, where $S$ is
the {\it single spin space}. The notation $\omega_\Lambda$ for the finite
volume projection of $\omega$ to $S^\Lambda$ will be used. The configuration
space will be endowed with its product Borel $\sigma$-field $\cal F$. A
product measure $\chi$ will be chosen on $(\Omega,\cal F)$ as a {\it
reference measure}. An {\it interaction} is a
family of real valued functions $\Phi_\Lambda$ on $S^\Lambda$, indexed by
${\cal P}_f(\mathbb{Z}^d)$, the set of finite subsets of $\mathbb{Z}^d$, and
with the property $\Phi_\emptyset = 0$. We consider {\it translation
invariant} interactions, i.e. $\Phi_{\Lambda+k}(\omega_{\Lambda+k}) = \Phi_
\Lambda(\omega_\Lambda)$, for all $k\in\mathbb{Z}^d$. The interaction $\Phi$
is called {\it absolutely summable} whenever
\begin{equation}
\sum_{\Lambda \ni 0 \atop \Lambda\in {\cal P}_f(\mathbb{Z}^d)}||\Phi_\Lambda||_
\infty < \infty
\label{1}
\end{equation}
where $||\cdot||_\infty$ denotes the sup-norm. The energy content of a volume
$\Lambda$ is given by the {\it Hamiltonian}
\begin{equation}
{\cal H}_\Lambda^\Phi(\omega_\Lambda|\xi_{\Lambda^c}) = \sum_{X \cap \Lambda
\neq \emptyset} \Phi_X(\omega_X)
\label{2}
\end{equation}
where $\Lambda^c = \mathbb{Z}^d \smallsetminus \Lambda$.
Here $\xi_{\Lambda^c}$ is a particular configuration fixed outside the volume
$\Lambda$, and plays the role of the {\it boundary condition}. Whenever the
configuration space is compact, absolute summability of the interaction is a
natural condition since it guarantees the existence of finite volume
Hamiltonians. A
probability measure on $(\Omega,\cal F)$ is called a {\it Gibbs measure} for
the interaction $\Phi$ at inverse temperature $\beta$ if a version of its
conditional probabilities $\pi_\Lambda(\omega_\Lambda,\xi_{\Lambda^c})$
satisfies the {\it DLR-equation}:
\begin{equation}
\frac{\pi_\Lambda(\omega_\Lambda,\xi_{\Lambda^c})}
{\pi_\Lambda(\tau_\Lambda,\xi_{\Lambda^c})} =
e^{-\beta \{{\cal H}^\Phi(\omega_\Lambda|\xi_{\Lambda^c}) -
{\cal H}^\Phi(\tau_\Lambda|\xi_{\Lambda^c})\}}
\label{3}
\end{equation}
for every finite $\Lambda$. We denote the collection of these conditional
probabilities by $\Pi := \{\pi_\Lambda\}_{\Lambda\in{\cal P}_f(\mathbb{Z}^d)}$.
We will use the following notion of `locality' for conditional probabilities:
$\Pi$ is called {\it quasilocal} if
\begin{equation}
\lim_{\Lambda' \rightarrow \mathbb{Z}^d} \sup_{\xi, \eta \in \Omega \atop
\xi_{\Lambda'} = \eta_{\Lambda'}}
|\pi_\Lambda(\cdot,\xi) - \pi_\Lambda(\cdot,\eta)| = 0
\label{4}
\end{equation}
for all $\Lambda \subset \Lambda' \in {\cal P}_f(\mathbb{Z}^d)$.
$\Pi$ is called {\it quasilocal at the point} $\eta$ if
\begin{equation}
\lim_{\Lambda' \rightarrow \mathbb{Z}^d} \sup_{\xi \in \Omega \atop \xi_
{\Lambda'} = \eta_{\Lambda'}}
|\pi_\Lambda(\cdot,\xi) - \pi_\Lambda(\cdot,\eta)| = 0
\label{4'}
\end{equation}
for all $\Lambda \subset \Lambda' \in {\cal P}_f(\mathbb{Z}^d)$.
For the
models we will consider in the sequel, quasilocality coincides with
the continuity of conditional probabilities with respect to the boundary
conditions (in the product topology).
%$\Pi$ is said to be {\it nonnull} with respect to the reference measure $\chi$
%if for every $\xi_{\Lambda^c} \in S^{\Lambda^c}$, and $\omega \in \Omega$,
%$\Lambda \in {\cal P}_f(\mathbb{Z}^d)$, $\chi_\Lambda(\omega_\Lambda) > 0$
%implies $\pi_\Lambda(\omega_\Lambda,\xi_{\Lambda^c}) > 0$. Moreover, $\Pi$ is
%{\it uniformly nonnull} if it is nonnull and for every $\omega \in \Omega$
%there is an $\varepsilon > 0$ such that $\pi_\Lambda(\omega_\Lambda,\xi_
%{\Lambda^c}) \geq \varepsilon$, for all $\Lambda\in{\cal P}_f(\mathbb{Z}^d)$.
$\Pi$ is said to be {\it uniformly nonnull} with respect to the reference
measure $\chi$ if for every $\xi_{\Lambda^c} \in S^{\Lambda^c}$ and $\omega
\in \Omega$, there is an $\varepsilon > 0$ such that $\chi_\Lambda(\omega_
\Lambda) > 0$ implies $\pi_\Lambda(\omega_\Lambda,\xi_{\Lambda^c}) \geq
\varepsilon$, for all $\Lambda\in{\cal P}_f(\mathbb{Z}^d)$. (In percolation
theory uniform nonnullness is called `finite energy condition', a terminology
which is quite suggestive of a Gibbsian description of the probabilities
involved.)
%If the single spin space is finite, then nonnullness and uniform nonnullness
%are equivalent.
For Gibbs measures the following characterization theorem is known
\cite{Koz,Sul,vEFS1}:
\begin{theorem}
Let $\Pi$ be a consistent family of everywhere defined conditional
probabilities (a `specification'), and suppose a reference measure $\chi$
is given. The following two statements imply each other:
\begin{enumerate}
\item
There exists an absolutely summable interaction $\Phi$ such that $\Pi$ is a
family of conditional probabilities corresponding to a Gibbs measure for
$\Phi$.
\item
$\Pi$ is quasilocal, and uniformly nonnull with respect to the reference
measure $\chi$.
\end{enumerate}
\label{t1}
\end{theorem}
Another useful notion, relating different Gibbs measures, is the relative
entropy density. This is defined as follows. Suppose two
different probability measures $\varrho_1,\varrho_2$ are given on $(\Omega,
\cal F)$. Denote by $h_{\varrho_1;\varrho_2}$ the Radon-Nikodym derivative
of $\varrho_1$ with respect to $\varrho_2$, whenever it exists. Suppose
moreover that $\log h_{\varrho_1;\varrho_2} \in L^1(\varrho_1)$. The quantity
\begin{eqnarray}
I(\varrho_1 | \varrho_2) = \begin{cases} \int_\Omega h_{\varrho_1;\varrho_2}
(\omega) \log h_{\varrho_1;\varrho_2}(\omega) \varrho_2 (d\omega)
& \text{if} \, \varrho_1 \ll \varrho_2 \\
\infty & \text{otherwise} \end{cases}
\end{eqnarray}
is called the {\it relative entropy of $\varrho_1$ with respect to
$\varrho_2$}.
%$I(\varrho_1|\varrho_2) = \sup_{f \in {\cal C}(\Omega)}
%(\mathbb{E}_{\varrho_1}(f) - \ln \mathbb{E}_{\varrho_2}(e^f))$
%For all $a > 0$, \, $\varrho_1 \{ \omega \in \Omega: h_{\varrho_1;\varrho_2}
%(\omega) \geq e^a \} \,\leq \, \frac{1}{a} (I(\varrho_1|\varrho_2) + \ln 2)$
Denote by $\varrho^{{\cal F}_\Lambda}$ the restriction of $\varrho$ to
${\cal F}_\Lambda$, the product Borel $\sigma$-field for $S^\Lambda$. The limit
\begin{equation}
i(\varrho_1 | \varrho_2) = \lim_{\Lambda_n \in {\cal P}_f(\mathbb{Z}^d)}
\frac{1}
{|\Lambda_n|} I(\varrho_1^{{\cal F}_{\Lambda_n}} | \varrho_2^{{\cal F}_
{\Lambda_n}})
\label{5}
\end{equation}
defined in van Hove sense, is called the {\it relative entropy density} for
$\varrho_1$ with respect to $\varrho_2$. The relative entropy density actually
is the rate function describing the (level-3) large deviation behaviour of
$\varrho_1$ with respect to $\varrho_2$. However, the limit above need
not exist.
It is known to exist when $\varrho_2$ is chosen to be a
Gibbs measure, and hence in particular when it is a product measure.
\begin{theorem}
The relative entropy density has the following properties:
\begin{enumerate}
\item
$i(\varrho_1 | \varrho_2) \geq 0$
\item
Suppose $\varrho_1$ and $\varrho_2$ are two Gibbs measures for translation
invariant interactions. Then
\begin{enumerate}
\item
$i(\varrho_1|\varrho_2) > 0$ iff $\varrho_1$ and $\varrho_2$ are Gibbs measures
for different interactions
\item
$i(\varrho_1|\varrho_2) = 0$ iff $\varrho_1$ and $\varrho_2$ are Gibbs
measures for the same interaction
\end{enumerate}
\end{enumerate}
\label{t2}
\end{theorem}
Now we turn to considering transformations of Gibbs states. Take a positive
integer $b$, and consider the sublattice $b\mathbb{Z}^d$ having spacing $b$.
This will be the renormalized lattice. In our notation we will not use
rescaled distances.
A {\it renormalization transformation} is a probability kernel $T$ defined by
\begin{equation}
%&& T:(S^{\cal L},{\cal F})\rightarrow{\cal M}^+_1({S'}^{{\cal L}'},{\cal F}')
\varrho'(d\tau) = \int_{S^{\mathbb{Z}^d
% \smallsetminus b\mathbb{Z}^d
}}
T(\omega,d\tau) \varrho(d\omega)
\label{6}
\end{equation}
acting from the original to the image system.
\iffalse
satisfying the following properties.
\begin{enumerate}
\item
The image measure is invariant under a subgroup of the translation group
leaving ${\cal L}'$ invariant.
\item
It is strictly local in the sense that
\begin{enumerate}
\item
there exist two van Hove sequences $\{\Lambda_n\} \subset {\cal P}_f(\cal L)$
and $\{\Lambda'_n\} \subset {\cal P}_f({\cal L}')$ such that for each $E \in
{\cal F}_{\Lambda_n}$ the function $T(\cdot, E)$ is
${\cal F}'_{\Lambda'_n}$-measurable.
\item
there exists a finite $K > 0$, called {\it compression factor} such that
\[ \limsup_{n \rightarrow \infty}\frac{|\Lambda_n|}{|\Lambda'_n|} \leq K \]
\end{enumerate}
\end{enumerate}
\fi
In the most studied cases the renormalization transformation is a product of
kernels defined on blocks of internal spins:
\begin{equation}
T(\omega, d\tau) = \prod_{x \in b\mathbb{Z}^d} \hat{T}(\omega_{B(x)},d\tau_x)
\label{7}
\end{equation}
where $B(x)$ is a block attached to the site $x$, and $\hat{T}$ is blockwise
defined. We will use Ising spin variables $S=\{-1,+1\}$, and take a box $B(x)
\subset \mathbb{Z}^d$, a translate of a $d$-cube such that its first vertex
is $x$. The particular examples of renormalization transformations in which
we will be interested in the sequel are:
\begin{itemize}
\item
{\it Decimation transformation}
\begin{equation}
\hat{T}(\omega_{B(x)},d\tau_x) = \delta(\omega_{x} - \tau_x) d\tau_x
\label{8}
\end{equation}
\item
{\it Kadanoff transformation with parameter $p > 0$}
\begin{equation}
\hat{T}(\omega_{B(x)},d\tau_x) = \frac{\exp{(p \tau_x \sum_{y \in B(x)}
\omega_y)}}{2 \cosh (p \sum_{y \in B(x)} \omega_y)} \,
\frac{\delta(\tau_x - 1) + \delta (\tau_x + 1)}{2} d\tau_x
\label{9}
\end{equation}
\end{itemize}
The decimation transformation is an example of a {\it deterministic}
renormalization transformation while the Kadanoff transformation is an
example of a {\it stochastic} renormalization transformation. Kadanoff
transformations with trivial scaling have important applications in image
reconstruction problems \cite{GP1,GG,BP}. For further discussion on
renormalization transformations we refer to \cite{vEFS1} and references quoted
there.
\section{Examples of non-Gibbsianness which are stable under decimation }
Consider the Gaussian model on $\mathbb{Z}^d$. The configuration space is
$\mathbb{R}^{\mathbb{Z}^d}$ and the interaction is defined by
\begin{eqnarray}
\Phi_\Lambda = \begin{cases} \frac{1}{2} V_{jk} (\omega_j - \omega_k) & \text
{if $\Lambda = \{j,k\}$} \\
0 & \text{otherwise}
\end{cases}
\end{eqnarray}
where $\omega_j,\omega_k \in \mathbb{R}$. The functions $V_{jk}$ are even
functions of the differences $\omega_j-\omega_k$, and we assume them to be
translation invariant, i.e. $V_{j+l,k+l} = V_{jk}$, for all $j,k,l \in
\mathbb{Z}^d$. By particular choices of the potential one can describe in
general an {\it anharmonic crystal}. When all the functions $V_{jk}$ are
quadratic, the corresponding system is called a {\it harmonic crystal}. For
harmonic or anharmonic crystals one can ask the question whether Gibbs
measures can be constructed for the given potential (where as reference
measure the Lebesgue measure is chosen). It can easily be seen that such a
Gibbs measure for a harmonic crystal is actually an example of a {\it massless
Gaussian }, i.e. a probability measure defined by the covariance matrix
\begin{equation}
C_{jk} = \mbox{cov}(\omega_j,\omega_k) = \frac{1}{(2\pi)^d} \int_{-\pi}^{\pi}
... \int_{-\pi}^{\pi} \hat{c}(q) e^{iq(j-k)} dq
\label{10}
\end{equation}
with $\hat{c} \in L^1([-\pi,\pi]^d)$, positive and even, and the inverse of the
covariance matrix $B_{jk} = C_{jk}^{-1}$ satisfying the massless condition
\begin{equation}
\sum_{k \in \mathbb{Z}^d} B_{jk} = 0
\label{11}
\end{equation}
The mean of this Gaussian measure we will take to be zero. The link between
the harmonic crystal interaction $V$ and the massless Gaussian covariance is
given by the relation
\begin{equation}
V_{jk}(\omega) = \frac{1}{2}B_{jk}(\omega_j - \omega_k)^2
\label{12}
\end{equation}
For $d < 3$, such $B_{jk}$ define a long-range interaction, for $d \geq 3$
also nearest neighbour interactions can be obtained. For further details and
properties of massless Gaussians we refer to \cite{BD,BDZ,BLM,Dob1,LM,DvE,
vEFS1}.
Now we consider the {\it projected massless Gaussian model} obtained under the
map $\omega_j \mapsto \mbox{sign} \, \omega_j, \forall j$. (Since the set of
those configurations for which the sign is 0 is negligible, one can choose
for this case any value of the projected Gaussian spin variable.) The
projected system is thus a system of Ising spins with a probability measure
induced by the sign map.
Let us fix a particular Gaussian model which is defined by its covariance
matrix. We denote by $\mu$ the translation invariant (Gaussian) Gibbs measure
with mean zero, and denote the induced measure by $\varrho$. This measure is
known to be a non-Gibbsian measure in any dimension \cite{LM,DvE,vEFS1}.
It is known to remain non-Gibbsian under a general class of deterministic
transformations \cite{MO2}. Our new result is that this remains true for
stochastic maps like the Kadanoff transformations. Moreover, we can show
that the quasilocality property breaks down for stochastically transformed
measures, something which is as yet unknown in the deterministic case.
\begin{theorem}
Consider the Kadanoff transformation $K_p$ with parameter $p$, and a measure
$\varrho$ as defined above. For every $p > 0$, $K_p\varrho$ is non-Gibbsian.
In fact, $K_p\varrho$ is not quasilocal.
\label{t3}
\end{theorem}
First we need a lemma \cite{Voi,vEFS1}:
\begin{lemma}
Suppose $\varrho_1$ and $\varrho_2$ are two probability measures on a
measurable space $(X,\cal X)$, such that $i(\varrho_1|\varrho_2)$ exists.
Consider a renormalization transformation $T$ given on this measure space.
Then the relative entropy density $i(T\varrho_1|T\varrho_2)$ exists and
\begin{equation}
i(T\varrho_1|T\varrho_2) \leq \mbox{const} \cdot i(\varrho_1|\varrho_2)
\label{13}
\end{equation}
\label{l1}
\end{lemma}
\begin{pft}
It is known that \cite{LM,DvE,vEFS1,BDZ}
\begin{equation}
i(\delta^+|\varrho) = 0 = i(\delta^-|\varrho)
\label{14}
\end{equation}
therefore by the Lemma above we have
\begin{eqnarray}
&& i(K_p \delta^+ | K_p \varrho) = 0 \\
&& i(K_p \delta^- | K_p \varrho) = 0
\end{eqnarray}
It can be seen by the definition of the Kadanoff transformation that it
transforms $\delta$-measures into product measures, thus there exist product
measures $\lambda_p^+ \neq \lambda_p^-$ such that
\begin{eqnarray}
&& K_p \delta^+ = \lambda^+_p \hspace{0.3cm} \forall p\\
&& K_p \delta^- = \lambda^-_p \hspace{0.3cm} \forall p
\end{eqnarray}
Since $\lambda^+_p$ and
$\lambda^-_p$ are trivially two Gibbs measures for two non-equivalent one-site
interactions, and $K_p\varrho$ cannot be a Gibbs measure simultaneously for
both of these one-site interactions, by Th. \ref{t2} we infer that there is
no absolutely summable interaction such that $K_p\varrho$ would be a Gibbs
measure for it.
Furthermore, it is known that the family of conditional probabilities
corresponding to the measure $\varrho$ is not uniformly nonnull \cite{LM,
vEFS1}, although the measure is strictly positive, that is, every cylinder
set has positive measure. Strict positivity is a weaker property than uniform
nonnullness, because for uniform nonnullness to hold one needs that each
cylinder set has positive measure which remains strictly bounded away from zero
under {\it arbitrary} conditioning.
However, it is easy to see that under the Kadanoff map the
family of conditional probabilities becomes uniformly nonnull, therefore by
Th. \ref{t1} we can conclude that $K_p \varrho$ is not quasilocal.
\end{pft}
\begin{corollary}
Consider an arbitrary decimation transformation $T$. Then neither the measure
$(T \circ K_p)\varrho$, nor the measure $(K_p \circ T)\varrho$ is Gibbsian.
This also holds when $T$ is replaced by any finite iterate of $T$.
\label{c1}
\end{corollary}
\begin{pf}
This follows by a similar argument applied to either of the measures by taking
note of the fact that a decimation transformation maps a product measure into
another product measure, and it maps a Dirac measure into another Dirac
measure. (Actually, this applies to a wider class of deterministic
transformations.)
\end{pf}
As was shown in \cite{DvE,vEFS1}, some of these projected Gaussians are
scaling limits for majority rule transformations, in particular of relevance
in high dimensions. Applying a different renormalization-group map to it
corresponds in renormalization-group language to making a move in a
`redundant' direction \cite{Weg}. Such a `redundant' direction corresponds
to taking a coordinate transformation in the (here not existing) space of
Hamiltonians.
\begin{remark}
A version of Th. \ref{t3} remains true for other examples of measures which are
strictly positive but not uniformly nonnull, in particular
for the invariant measures of both the
voter model and the Martinelli-Scoppola model.
The {\it voter model} is an interacting particle system defined by the
flip rates
\begin{equation}
c(\omega,x) = \frac{1}{2d} \sum_{y: |x-y|=1} \1_{\{\omega_y \neq \omega_x\}}
\label{15}
\end{equation}
and the variables (the `voters') $\omega_x$ placed on $\mathbb{Z}^d$ can take
the values 0 and 1. It is well-known \cite{Lig} that for $d=1$ and $d=2$ the
only extremal stationary measures are $\delta_{\bf 0}$ and $\delta_{\bf 1}$,
where the notations ${\bf 0}$ resp. ${\bf 1}$ correspond to the configurations
$\omega_x=0$ resp. $\omega_x=1$, for all $x\in\mathbb{Z}^d$. For $d\geq3$,
however, there is a one-parameter family of extremal stationary translation
invariant measures $\{\nu_z\}_{0\leq z \leq 1}$, parametrized by the density of
$\omega_x= 1$ with respect to $\nu_z$. For the voter model, the fact that the
relation (\ref{14}) holds for the extremal translation invariant stationary
measures $\nu_z$, has been proven for all $d\geq 3$ in \cite{LSch}. It is not
known in this case, nonetheless it is believed, that the invariant measures are
strictly positive, but the family of conditional probabilities corresponding
to them is not uniformly nonnull.
The {\it Martinelli-Scoppola model} \cite{MS} is a model with a stochastic
cluster dynamics on the lattice $\mathbb{Z}^2$. The single spin space is
$S=\{0,1\}$, where
$\omega_x=0$ corresponds to an empty site, and $\omega_x=1$ corresponds to an
occupied site. A maximal connected set of occupied sites is called a cluster.
A set $X\subset \mathbb{Z}^2$ is called connected if for any two sites $x,y
\in X$ there exists a sequence $\{x_k\}_{k=1,...,n}\subset X$ of sites (a
path) such that $x_1=x$, $x_n=y$ and $|x_k-x_{k+1}|=1, \forall k=1,...,n-1$.
The dynamics is defined as follows: At each time $t$ a configuration
$\omega^t\in\{0,1\}^{\mathbb{Z}^2}$ is given. The configuration $\omega^{t+1}$
is defined by a process consisting of a simultaneous creation and
annihilation operation. The creation operation consists in changing the empty
sites at time $t$ into occupied sites at time $t+1$ with probability $p$,
at each site independently of other sites. The annihilation operation
consists in removing the clusters belonging to the configuration $\omega^t$,
independently of each other and with probability 1/2. For sufficiently small
probabilities $p$ there is but one invariant measure $\varrho$ for this
process. For the Martinelli-Scoppola model the relation (\ref{14}) was proved
in \cite{MS}. It is not known, but it is suspected, that the family of
conditional probabilities of the stationary measure for this model too fails
to be uniformly nonnull \cite{MO2}. Since in this model there is no $+/-$
symmetry, $K_p \varrho$ has to be distinguished from a product measure by
means of e.g. some correlation functions. Indeed, it is known that there
exist some fast decaying non-trivial correlation functions \cite{MS}.
\label{r1}
\end{remark}
\section{An example of non-quasilocal behaviour on large sets}
In this section we show that mixtures of Gibbs measures for different
interactions are non-Gibbsian in a rather strong sense. These measures
can simply be shown to be non-Gibbsian \cite{vEFS1}; here we show that the
situation is worse in the sense that every configuration is a point at which
quasilocality does not hold.
Let $(\Omega, {\cal F}, \chi)$ be a measure space, with $\Omega =
S^{\mathbb{Z}^d}$, for some $S$. Suppose on this measure space $\varrho_1$ and
$\varrho_2$ are two Gibbs measures for the same interaction at different
temperatures $\beta_1$ and $\beta_2$. It is well-known that these two Gibbs
measures are singular with respect to each other, or equivalently $||\varrho_1
-\varrho_2||_{\mbox{\tiny var}} = 2$. For notational simplicity we will assume
that the interaction is of finite range.
Consider the convex combination $\varrho = \frac{1}{2}(\varrho_1 + \varrho_2)$.
Denote by $\pi_\Lambda$, $\pi^{(1)}_\Lambda$ resp. $\pi^{(2)}_\Lambda$ the
conditional probabilities for respectively $\varrho$, $\varrho_1$ and
$\varrho_2$. Then
\begin{eqnarray}
&& \pi_\Lambda(\cdot,\omega) = \pi^{(1)}_\Lambda(\cdot,\omega)
\hspace{0.3cm} \text{for $\varrho_1$-almost all $\omega \in \Omega$}
\label{16} \\
&& \pi_\Lambda(\cdot,\omega) = \pi^{(2)}_\Lambda(\cdot,\omega)
\hspace{0.3cm} \text{for $\varrho_2$-almost all $\omega \in \Omega$}
\label{17}
\end{eqnarray}
holds for all finite subsets $\Lambda' \subset \Lambda$. We denote by
${\cal C}_1$ the set of configurations for which (\ref{16}) holds, and by
${\cal C}_2$ the set of configurations for which (\ref{17}) holds. Also,
we take the neighbourhood basis
\[ {\cal U}_{\omega, \Lambda}
= \{\omega' : {\omega'}_\Lambda = \omega_\Lambda \} \]
Since the two measures are singular with respect to each other, the above
considerations lead to the following conclusion:
\begin{theorem}
Consider the sets
\[ {\mathcal V}^{(1)}_{\omega, \Lambda} = {\cal C}_1 \; \cap \;
{\cal U}_{\omega, \Lambda} \]
\[ {\mathcal V}^{(2)}_{\omega, \Lambda} = {\cal C}_2 \; \cap \;
{\cal U}_{\omega, \Lambda} \]
For every $\omega\in \Omega$ there exists a volume $\Lambda'$ such that for
each two configurations $\xi \in {\mathcal V}^{(1)}_{\omega, \Lambda}$ and
$\eta\in {\mathcal V}^{(2)}_{\omega,\Lambda}$, whenever $\dist(\partial\Lambda,
\partial \Lambda')$ is larger than the range of the interaction, there is a
constant $\varepsilon> 0$, independent of $\Lambda$, such that
\[ \lim_{\Lambda\rightarrow\mathbb{Z}^d}
|\pi_{\Lambda'}(\cdot,\xi_{\Lambda^c}) -
\pi_{\Lambda'}(\cdot,\eta_{\Lambda^c})|
\geq \varepsilon \]
\label{t4}
\end{theorem}
The point here is that the conditional probabilities in $\Lambda'$ are
computed at an inverse temperature $\beta_1$ or $\beta_2$, according to what
happens outside the larger volume $\Lambda$, but not depending on the
configuration restricted to the annulus between the boundaries of $\Lambda$
and $\Lambda'$. Theorem \ref{t4} above says that the mixture of two Gibbs
measures at different temperatures is non-quasilocal {\it at every}
configuration. This is an example of a measure which fails everywhere to be
Gibbsian, thus a case where the `pathology' is extremely severe. Note that
Th. \ref{t4} can actually be generalized in a straightforward way to any
convex combination of two Gibbs states for two non-equivalent interactions.
As a side remark, we observe that if the two Gibbs measures both remain
Gibbsian under decimation, then the strong non-Gibbsianness of their convex
combination is preserved under this decimation.
A particular example of a non-Gibbsian measure for which every configuration
is a point of non-quasilocality is provided by the following example: Consider
the nearest neighbour ferromagnetic Ising interaction on the two-dimensional
square lattice in the subcritical regime. Denote by $\mu^+$ resp. $\mu^-$
the $+$ phase respectively the $-$ phase. In \cite{LVV} it has been shown that
the projection to the one-dimensional sublattice $b\mathbb{Z}$ with $b \geq 3$
of $\mu^+$ and $\mu^-$ are Gibbs measures for two different absolutely summable
interactions. Hence we have by Th. \ref{t4}:
\begin{corollary}
The conditional probabilities for a projection of any mixture $\mu = \lambda
\mu^+ + (1-\lambda)\mu^-$, $0 < \lambda < 1$, onto $b\mathbb{Z}$, with $b \geq
3$, are non-quasilocal at every configuration.
\label{c2}
\end{corollary}
\vspace{2cm}
{\bf Acknowledgments:} \hspace{0.2cm}
We thank R.L. Dobrushin, R. Fern\'andez, C. Maes, E. Olivieri, S.B. Shlosman,
K. Vande Velde and M. Winnink for helpful discussions. This work was supported
by EC
grant CHRX-CT93-0411, and jointly by the `Stich\-ting Fundamenteel Onderzoek
der Materie' and `Stich\-ting Mathematisch Centrum'.
\newpage
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\end{document}