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\begin{document}
\begin{titlepage}
\vspace{5mm}
\begin{center}
{ \bf {\large PERCOLATION TECHNIQUES IN DISORDERED \\
SPIN SYSTEMS: THE UNIQUENESS REGIME \\
I. STATICS\\
II. DYNAMICS }}\\[3mm]
G. Gielis\footnote[1]{I.I.K.W Onderzoeker Belgium. e--mail:
Guy.Gielis@fys.kuleuven.ac.be}
and C. Maes\footnote[2]{Onderzoeksleider N.F.W.O. Belgium. e--mail:
Christian.Maes@fys.kuleuven.ac.be }\\[2mm]
Instituut voor Theoretische Fysica \\
K.U. Leuven \\
Celestijnenlaan 200D \\
B--3001 Leuven, Belgium\\[5mm]
{\bf Abstract}
\end{center}
We consider lattice spin systems with short range but random and unbounded
interactions.\newline
{\it Statics} : We give an elementary proof of uniqueness of Gibbs
measures at high temperature or strong magnetic fields, and of the
exponential decay of the corresponding quenched correlation functions.
The analysis is based on the study of disagreement percolation
(as initiated in van den Berg--Maes (1994)).\newline
{\it Dynamics} : We give criteria
for ergodicity of spin flip dynamics and estimate the
speed of convergence to the unique invariant measure. We find for
this convergence a stretched exponential in time
for a class of ``directed'' dynamics
(such as in the disordered Toom or
Stavskaya model). For the general case, we show
that the relaxation is faster than any power in time. No
assumptions of reversibility are made.\\
The methods are based on relating the problem to an oriented
percolation problem (contact process) and
(for the general case) using a slightly
modified version of the multiscale analysis of e.g. Klein (1993).
\noindent {\bf Keywords} : quenched disorder, spin glasses,
disagreement percolation, spin flip dynamics,
Gibbs measures, Griffiths'singularities,
multiscale analysis, ergodicity.
\end{titlepage}
\section{Introduction}
Adding disorder to a system of many interacting particles may in general
be a highly non--trivial perturbation. The study of spin glasses and
spin flip systems with quenched disorder is certainly not so well developed
as
their corresponding versions without disorder. In this paper we show how
percolation techniques can be useful for investigating that part of the
phase diagram in which the disordered system typically forgets about
initial data or
about boundary conditions. This behavior appears in the high noise, high
temperature or low density limit in a variety of interacting particle
systems.
Our analysis concerns two types of classical problems in the study of
disordered systems (topics (B) and (C) in the review of Fr\"ohlich (1986)):
one is in the context of
equilibrium and the other in the theory of stochastic spin flip dynamics.
Our analysis is however restricted to the uniqueness regime (high
temperature, strong magnetic field, high noise, strong bias,...) and the
main problem consists therefore in circumventing the static and dynamical
consequences of the so--called Griffiths' singularities, see Griffiths
(1969).\\
For the equilibrium part we give conditions assuring
the uniqueness of Gibbs fields with a random potential and the exponential
decay of the corresponding quenched correlation functions.\\
For the non--equilibrium part we investigate the relaxation properties
of spin flip dynamics with random transition rates.
Both questions have of course been considered before. For the case of
random Gibbs fields, there are the standard works of Olivieri {\it
et al} (1983), Berretti
(1985), Fr\"ohlich--Imbrie (1984) and Bassalygo--Dobrushin (1986).\\
Moreover, more recently there came out a series of papers containing
a simple and detailed
analysis of the uniqueness
regime of short range spin glasses in equilibrium, cf. Perez (1993), Klein
(1994) and von Dreifus {\it et al} (1994). In this
paper we continue along the lines of Bassalygo--Dobrushin (1986).
Their formulation is closely related to a bond percolation problem. We
use disagreement site percolation as in van den Berg--Maes (1994).\\
For Glauber dynamics, Zegarlinski (1994) recently studied the problem
in one
dimension. We study more general models, making the correspondence with
(temporal-) oriented percolation.
For an important subclass of models the problem is
further reduced to the study of space--time oriented percolation.
No assumptions of reversibility are made.
An important ingredient already appeared in the work of Campanino--Klein
(1991), Klein (1993) and Klein (1994).
Our results can be summarized as follows : the influence of the
boundary conditions on Gibbs states with a random potential can be
analyzed via a corresponding
percolation problem with random densities. The uniqueness
conditions
for Gibbs states are stated in terms of absence of percolation. Truncated
correlation functions are bounded by connectivity functions for
independent site percolation. The arguments are direct
applications of the van den Berg--Maes (1994) results.
For the non--equilibrium case we describe the influence of the initial data
on the asymptotic state via oriented percolation (contact process) in a
random environment. We give explicit criteria under which ergodic
behavior typically results and obtain bounds for the rate of convergence to
the unique stationary measures. For the subclass of dynamics in which the
spatial dependence of the transition rates is ``directed'', we get a
stretched exponential for the decay in time. For the more general case, we
only get a decay faster than any power of time.
The outline of this paper is as follows : Section 2 contains general
definitions. In Section 3 the equilibrium problem is investigated.
Section 4 is
devoted
to the anologous problem for both discrete time and continuous time spin
flip dynamics. In the Appendix we collect the more technical arguments and
modifications with respect to the multiscale analysis of Klein (1993).
\section{Definitions}
For convenience we consider spin systems defined on the regular
$d$-dimensional lattice ${\Zbar}^d$.
As will become clear, the arguments that follow are valid on a more
general periodic lattice (e.g. the triangular or the FCC lattice) .
${\Zbar}^d$ comes
equipped with the usual structure of nearest neighbor sites $x,
y$ connected
by bonds $\langle x,y \rangle$. If two sites $x$ and $y$ are nearest
neighbors (or
adjacent) we will write $x\sim y$.
A configuration $\sigma$ puts a spin value $\sigma(x)=1$ or $\sigma(x)=-1$
on every site $x\in {\Zbar}^d$. $\Omega =\{-1,+1\}^{{\Zbar}^d}$ is the set
of all
configurations. Our results can easily be extended to
other finite single site state spaces.
One typically considers two types of quenched disorder. One is
realized by the nearest neighbor couplings $\{J_{xy}\}_{x\sim y}$ and the
other by a set of single site parameters $\{h_x\}$ (also denoted below by
$\{\gamma_x\}$ if not referring to a random magnetic field).
We assume that the $\{J_{xy}\}$ are real (possibly
infinite) valued mutually independent
and identically distributed random variables. Similarly for the
$\{h_x\}(\{\gamma_x\})$.
Examples will follow where these parameters enter explicitly.
Sometimes it is however more convenient to speak about ``realizations'' in
general without specifying exactly how the disorder is frozen in in the
interactions or transition rates. Indeed, the relevant objects for our
analysis are the (random) specifications and/or transition kernels (see
below) and we do not need to refer to specific forms of the interaction.
We therefore write $\pi$ to denote such a general (random) realization (of
the disorder).\\
$\Pi$ is the set of all these realizations. {\bf Q} is the probability
law on the realizations. {\bf E} the
expectation value with
respect to the distribution {\bf Q}.
Independent percolation plays a crucial role in our construction.
We consider site percolation on
${\Zbar}^d$. One independently assigns to every site a
value $0$ or $1$ with a certain density.
A site with value one is called open. An open path is a
sequence of neighboring open sites.
We say that site percolation occurs if there is a positive
probability for the event that there exists an infinite open path.\\
In (time) oriented percolation, only paths with decreasing time coordinate
on the space--time lattice are considered. For more details and
definitions we refer to e.g. Grimmett
(1989).\\
In all cases, the densities at each site may (and frequently will)
differ, depending on the (random) realization.
\section{Equilibrium States with Random Potentials}
A probability measure $\nu$ on $\Omega$ is a Markov field if for every
finite $A\subset {\Zbar}^d, \eta \in \{-1,1\}^A $,
\be
\nu[\sigma = \eta \mbox{ on\ } A|\sigma(i), i \in A^c]
=\nu [\sigma=\eta \mbox{ on\ } A|\sigma(i), i\in \partial A],
\end{equation}
where $A^c={\Zbar}^d\setminus A$ and $\partial A$ is the set of all sites
in $A^c$ that are adjacent to $A$.
A major problem in statistical mechanics is to determine the
Markov fields $\nu$
which satisfy for all finite $A\subset {\Zbar}^d $, all $\eta \in \{-1,1\}^A$,
$\eta' \in \{-1,1\}^{\partial A}$
\be
\nu[\sigma=\eta|\sigma=\eta' \mbox{on\ } \partial
A]=Y_A(\eta,\eta'),
\end{equation}
with
\be
\{ Y_A(\cdot,\eta'), A \subset {\Zbar}^d\mbox{,\ finite}, \eta'
\in \{-1,1\}^{\partial A}\}
\end{equation}
a given set of self--consistent conditional probabilities (a specification)
possibly parametrized (among other things)
by the inverse temperature $\beta \geq 0$, external fields, etc. (see
Georgii (1988)). In that case we say that $\nu$
is a Gibbs measure with respect to the specification $\{Y_A\}$. We look for
conditions on the set $\{ Y_A(\cdot,\eta)\}$ such that there
exists just one associated Gibbs measure. \\
What is specific to our study
here is that the specification is {\it random}, i.e. the $Y_A=Y_A^\pi $
depend not only on the values of certain fixed parameters but also on the
realization $\pi$ of the randomness.
We do not wish to formalize
here the notion of
specification--valued random field. The reader who is bothered by
this is referred to the examples.
An important example is the following random field short range spin glass
with formal Hamiltonian:
\be
H=-\sum\limits_{\langle x,y\rangle} J_{xy}\sigma(x)\sigma(y)- b \sum
\limits_x h_x \sigma(x) - h \sum\limits_x \sigma(x),
\label{ham}
\end{equation}
determined by a realization of one--($h_x$) and two--point interactions
$(J_{xy})$.
The specification
$\{Y_A\}$ is obtained by taking the finite volume Gibbs measures
(fixed boundary conditions outside $A$) with respect to the Hamiltonian $H$
at inverse temperature $\beta$. For $A=\{x\}$ we then have, (with some
abuse of notations)
\be
Y_x(\sigma(x),\sigma)=\frac{\exp[\beta
\sum\limits_{y\sim x} J_{xy}\sigma(x)\sigma(y) + (\beta b
h_x + \beta h) \sigma(x)]}{Z_x^\beta (\{J_{xy}, \sigma(y)\}_{y\sim
x},h,h_x,b)}.
\label{spe}
\end{equation}
The construction of a Gibbs measure $\nu_\pi$ with respect to the
specification $\{Y_A =Y_A^\pi\}$ will obviously depend on the realization
$\pi$. The uniqueness of the Gibbs measure should be understood in the
sense that with {\bf Q}-probability one there is just one such Gibbs
measure. For these equilibrium measures, we define the truncated
correlation function of the local
functions
$f$ and $g$ on $\Omega$ as
\be
{\langle f;g \rangle}_\pi = \nu_\pi(fg) -
\nu_\pi(f)\nu_\pi(g).
\end{equation}
Bassalygo--Dobrushin (1986)
give a ``simple" proof for the uniqueness
of the Gibbs state for the specification (\ref{spe}) with random unbounded
interactions at sufficiently high
temperatures. A key idea is to reduce the problem to that of
the uniqueness of a Gibbs state with uniformly small pair interactions on
an associated ``arbitrary" locally finite graph. We do not make this
correspondence here but investigate how far we can take quite similar
stochastic--geometric arguments
to include random interactions. \\
von Dreifus {\it et al} (1994) carry out an
analysis which is also inspired by Bassalygo--Dobrushin (1986) and combine
it with expansions on duplicated systems (similar to the objects considered
in
Fr\"ohlich-Imbrie (1984)). These expansions very much resemble Hammersley
sums for independent percolation but the relation between the two problems
stops there because the ``densities'' in these sums can take values
larger than one.\\
In what follows we get a direct and explicit relation
with an independent percolation process, called disagreement percolation in
van den Berg--Maes (1994), from which all the results follow almost for
free.\\
Let
\be
q_x=\max\limits_{\eta,\eta'}{\rm var}
(Y_x(\cdot,\eta),Y_x(\cdot,\eta')),
\label{zes}
\end{equation}
where ${\rm var}(\cdot,\cdot)$ ($\in [0,1]$) is the variational distance.\\
Everything that follows is expressed in terms of the distribution of
the field $\{q_x\}$. Remember that the $q_x$ depend on the realization $\pi$
and on extra
parameters (such as the temperature and external fields) as inherited
from the specification. So instead of
referring to the high temperature or strong external field
regimes separately, the single phase regime of our disordered system will
be obtained if ``typically" the $q_x$ are ``small'' for
all $x$ in a sufficiently ``big'' set. Using Definition (\ref{zes}) one can
then
for every specific model get explicit conditions on the realizations and
the external parameters.\\
Important to observe
is that $q_x$ and $q_y$ may be correlated
for $x\neq y$. However, in all relevant examples the randomness in the
specification enters locally and has a high degree of independence. While
the arguments that follow essentially go through
unchanged under the assumption that there is a finite ``distance'' $R$
for which $q_x$ and $q_y$ are independent whenever the
``distance'' between $x$ and $y$ exceeds $R$, for simplicitly we
require that this already happens for $R=1$, i.e. $q_x $ and
$q_y $ may be correlated for $x\neq y$ only if $x\sim y$, otherwise
they are independent; $\{q_x\}$ is a one--dependent random field.
This is verified in all examples discussed here.\\
Another feature present in all
our examples of interest is that, with {\bf Q}-probability one, there are
finite regions of all sizes
on which $q_x$ is large. These regions
are responsible for
the so called Griffiths' singularities, Griffiths (1969). \\
\vspace{5mm}
\noindent {\it Example 1} (spin glass)\\
In a spin glass, the realization $\pi$ consists of a collection of
real independent and
identically distributed variables $J_{xy}$ associated to the bonds $\langle
x,y \rangle $.
\be
Y_x(\sigma(x),\sigma)=\frac{\exp [\beta \sum\limits_{y\sim
x}J_{xy}\sigma(x)\sigma(y)+\beta h\sigma(x)]}
{Z_\pi^\beta(\{J_{xy}\sigma(y)\}_{y\sim x},h)},
\end{equation}
for $h>0$ an externally applied magnetic field.
Then
\be
q_x = 1/2 [\mbox{tanh}(\beta \sum\limits_{y\sim x}|J_{xy}| + \beta h)
+ \mbox{tanh}(\beta \sum\limits_{y\sim x}|J_{xy}| - \beta h)].
\end{equation}
Note that
$q_x$ can be made ``small'' both by taking
$\beta > 0$ small or by taking $h$ large.
Note also that
a realization where $J_{xy}=\infty $ for some bond $\langle x,
y\rangle$
causes $q_x=q_y=1$ for all values of $\beta,h > 0$.
Moreover,
with {\bf Q}-probability one, $\sup\limits_{x}
q_x=1$ because the interaction couplings are unbounded.
\vspace{5mm}
\noindent {\it Example 2} (random field Ising model)\\
For the Hamiltonian (\ref{ham}) with $0 \leq J_{xy} = J < \infty $
fixed and $h=0$ we have
\be
q_x=\frac{1}{2}[\mbox{tanh}\beta(2d J +b h_x)+ \mbox{tanh}\beta(2dJ - b
h_x)].
\end{equation}
$q_x$ can now also be made small by letting $b$ grow,
except when the realization puts $h_x=0$ because then
$q_x= \tanh (2d \beta J)$, independent of $b$.
\vspace{5mm}
\noindent {\it Example 3} (hard core lattice gas)\\
Here a realization is a random choice of activities $a_x=e^{\lambda
\gamma_x} -1$ with $\gamma_x\geq 0$ random and $\lambda \geq 0$ an extra
parameter.
\be
Y_x(1,\eta)= \left\{ \begin{array}{ll}
1-e^{-\lambda \gamma_x} & \mbox{if \ } \mbox{ for all } y\sim x \
\eta(y)=-1\\ 0 & \mbox{otherwise}.
\end{array}
\right.
\end{equation}
Then,
\be
q_x= 1-e^{-\lambda \gamma_x}.
\end{equation}
will be ``small'' for $\lambda $ large except in a
realization where $\gamma _x=\infty $.
\vspace{1cm}
The idea is now to use the $\{q_x\}$ as densities for independent site
percolation on ${\Zbar}^d $ (independently a site $x$ is open with
probability
$q_x$, and is closed with probability $1-q_x$). Denote by $G_\pi(x,y)$ the
probability to find an open path from $x$ to $y$ if the realization is $\pi$.
Next Proposition~\ref{peen} is a direct consequence of van den
Berg--Maes (1994) and it
is the first key-ingredient for the main result of this section.\\
Following notations are used.
The distance dist$(f,g)$ between the local functions $f$ and $g$ on
$\Omega$ is
\be
\mbox{dist}(f,g)=\min\limits_{\stackrel{x \in \mbox{{\scriptsize supp}}f}
{x \in \mbox{{\scriptsize supp}}g}}|x-y|,
\end{equation}
with $\mbox{supp}f$ the support of $f$ and
\be
|x-y|=\inf \{|\omega|,\omega \mbox{ a path connecting $ x$ and $y$} \}.
\label{dist}
\end{equation}
Definition (\ref{dist}) has a somewhat more general flavor than needed on
${\Zbar}^d$, but is useful when considering other lattices.
Further we denote by
$||f||$ the usual supremum norm of $f$ and
\be
\delta_x =\sup\limits_{\eta}|f(\eta^x)-f(\eta)|
\end{equation}
the oscillation of $f$ at $x$. The
total oscillation is then
\be
|||f|||=\sum\limits_{x \in {\Zbar}^d} \delta_x f.
\end{equation}
\begin{pro}\label{peen}
Absense of independent site percolation with densities
$\{q_x\}_{x \in {\Zbar}^d}$ implies
the uniqueness of the
Gibbs state for the specifications used in the definition of the
$\{q_x\}_{x\in {\Zbar}^d}$ .\\
Moreover, the truncated correlation function in that Gibbs state obey the
bound
\be
|{\langle f,g \rangle}_\pi| \leq ||f|| \ |||g|||
\max\limits_{x\in \mbox{{\scriptsize {\rm supp}}}g}\sum\limits_{y \in
\mbox{{\scriptsize {\rm supp}}}f}
G_\pi(x,y)
\end{equation}
for $f$ and $g$ local functions.
\end{pro}
\vspace{5mm}
\noindent {\it Proof}
The uniqueness of the Gibbs state follows directly from Corollary 2 in van
den Berg--Maes (1994). Let $\nu_\pi$ be the unique Gibbs measure.
The truncated
correlation function can be estimated as follows.
\be
\begin{array}{rl}
|\langle f;g\rangle_\pi |
&=|\int \nu_\pi(d\sigma) (\nu_\pi(fg|\sigma \mbox{ on supp}f))-\nu_\pi(f)
\nu_\pi(g)|\\
&=|\int \nu_\pi(d\sigma)f(\sigma)(\nu_\pi(g|\sigma
\mbox{ on supp}f)-\nu_\pi(g)) | \\
&\leq ||f|| \max\limits_{\eta,\eta'}|\nu_\pi(g|\sigma=\eta \mbox{ on supp}f)
\\
&\hspace*{20mm}-\nu_\pi(g|\sigma=\eta'\mbox{ on supp}f) |.
\end{array}
\end{equation}
Using Corollary 1 in van den Berg--Maes (1994) we get that
\be
\begin{array}{rl}
|\langle f;g\rangle _\pi |&\leq ||f||
\sum\limits_{\stackrel{x}{y \in \mbox{{\scriptsize {\rm supp}}}f}}
\delta_x g \ G_\pi(x,y)\\
&\leq||f||\ |||g||| \max\limits_{x \in \mbox{{\scriptsize {\rm
supp}}}g}\sum\limits_{y \in \mbox{{\scriptsize {\rm supp}}}f} G_\pi(x,y).
\end{array}
\end{equation}
This proves the Proposition.
\QED
>From Proposition~\ref{peen} we learn that the structure of the unique Gibbs
measure will inherit all the nice proporties of the connectivity function
of the associated percolation process. We must therefore study the latter
more closely.
\vspace{1cm}
\begin{theo}\label{een}
If $\{q_x\}_{x\in {\Zbar}^d}$, as defined by (\ref{zes}),
is a stationary one--dependent random field verifying
\be
{\bf E}(q_x) <\frac{1}{(2d-1)^2}
\label{C2}
\ee
then---with {\bf Q}-probability one---there is a unique Gibbs measure
$\nu_\pi$.\\
Moreover,
\be
{\bf E}(|\langle f;g \rangle_{\pi}|)\leq C(f,g) \mbox{{\rm e}}^{-m
\mbox{\ \scriptsize{\rm dist}}(f,g)}
\label{21}
\ee
for all local functions $f$ and $g$, with $m>0$ and $C(f,g)=C\|f\|\
|||g|||<\infty$.
\end{theo}
\vspace {5mm}
\noindent {\it Proof}
For any path $\omega$, we denote with $\omega_i$
the $i$-th site in the path.
\noindent The Theorem follows from Proposition~\ref{peen} and the
bound
\be
G_\pi(x,y)\leq \sum\limits_{n\geq
|x-y|}\sum\limits_{|\omega|=n}
\prod\limits_{i\ \mbox{\scriptsize even}}q_{\omega_{i}},
\label{G}
\ee
with the second sum running over all the self--avoiding walks $\omega,
|\omega|=n$ from $x$ to $y$. Taking the expectation of both sides of
(\ref{G}) and using the independence of $\{q_x\}_{\{x \mbox{ \scriptsize
even}\} }$ completes the proof.
\QED
\vspace{5mm}
\noindent{\it Remarks} \\
\noindent {\it 1.} From the examples it is clear that it is not
always possible to tune an
external parameter to make $q_x$ (pointwise) arbitrarily small on any site
$x$. For instance in Example~1~ (3) always $q_x=1$ if $J_{xy}=\infty
(\gamma_x=\infty)$ or in Example~2 $q_x=\mbox{tanh}2d\beta J$ when $h_x=0$,
independent of $b$. When this happens we call a site $x$ ``bad". Therefore,
a necessary condition for the assumption of Theorem~\ref{een} to be
verified is that the {\bf Q}--probability of a site to be ``bad" is itself
small enough. In a way this condition is also sufficient: see
e.g. Example~1, if ${\bf Q}\{J_{xy}=\infty\}<1/(2d-1)^2$, then for $\beta>0$
sufficiently small condition (\ref{C2}) is satisfied.
\vspace{5mm}
\noindent {\it 2.} Theorem~\ref{een} does not give the best possible bound
on the smallness of ${\bf E}(q_x)$. Depending on specific models and using
the main underlying idea (as from Proposition~\ref{peen}) we can improve
substantially on this bound. E.g. for example 2 with $\{h_x\}$ an
independent identically distributed
field, when ${\bf E}(q_x) \exp -c(\log t)^\frac{d}{d-1}$. This does not agree with
the simulations of
Ogielski (1985). Palmer {\it et al}. (1984) introduced different degrees
of freedom to
elucidate the relaxation in strongly interacting glassy materials. They
propose a serial relaxation process, slower modes are constrained by the
faster ones : the former can't decay before the latter are finished. The
formula
they obtain for $q(t)$ contains two parameters that can be chosen so that
a stretched exponential decay appears.
Here we do not restrict ourselves to {\it reversible} continuous
time
spin glasses. Also other continuous and discrete time spin flip dynamics
are regarded. The common characteristic is a short
range interaction governed by random and possibly unbounded parameters.
A consequence of such disorder is that there will typically be large
regions on which the spins are strongly interacting. This is completely
analogous
to the situation in equilibrium.
What is far worse here however, is that these ``bad'' regions
should be thought of
as infinitely extending in the time direction, i.e., spins there will relax
only very slowly in the course of time. Depending on the size and the
``badness'' of these regions, the relaxation time may become arbitrarily
large. Therefore we cannot expect to see {\it typically} an exponential
decay to the invariant measure. One could view this as a dynamical
consequence of the Griffiths' singularities.
Our main
goal for the rest of the paper is to see what to do about this.
\subsection{Disordered Probabilistic Cellular Automata}
A Probabilistic Cellular Automaton (PCA) is a parallel updating discrete
time evolution $\sigma_n,n=0,1,\ldots.$ on
$\Omega$. $\sigma_n$ is a Markov process defined by transition
probabilities
$p_x(\pm 1,\eta)$ and for every $\Lambda \in {\Zbar}^d$
\be
\mbox{Prob}[\sigma_n(x)=\xi(x),\forall x \in \Lambda | \sigma_{n - 1}
= \eta] =\prod\limits_{x \in \Lambda} p_x (\xi(x)|\eta)
\end{equation}
for $\xi,\eta \in
\Omega$.\\
For simplicity we assume that the $p_x(\cdot|\eta)$ depend only on the
nearest neighbor spins $\eta(y), y\sim x$ and on $\eta(x)$.
While the dynamics is time homogeneous we add
quenched disorder by letting the transition probabilities $\{p_x\}$ depend
on the realization $\pi$ having the same nature as in the previous
Section. It may for example enter in the coupling between $x\sim y$ or in
the bias (see the examples below). As before there will typically be extra
parameters
(such as the noise level or temperature) available to modify the behavior
of our PCA for the same distribution of realizations.\\
We use $P_\pi$ to denote the transfer operator of the PCA and $P_\pi^N =
P_\pi^{N-1}P_\pi$.
A probability measure $\nu_\pi$ on $\Omega $ is invariant if $\nu_\pi P_\pi =
\nu_\pi$. The PCA is ergodic if there exists a unique invariant measure
$\nu_\pi$, such that for all probability measures $\mu$ on $\Omega$
\be
\mu P_\pi^N \longrightarrow \nu_\pi.
\ee
See Lebowitz {\it et al} (1990) for more details.\\
The PCA evolution is of course not deterministic because $P_\pi$
applied to a configuration $\eta\in \Omega $ (the
delta measure concentrated at $\eta$) gives the product measure
with densities $\{p_x(.|\eta)\}$. Our object of study is therefore a
stochastic dynamics in which the degree and/or nature of the stochasticity
(noise,...) itself
is randomly determined (via the quenched disorder realized by $\pi$).
\vspace{5mm}
\noindent Define
\be
k_x\equiv
\max\limits_{\eta,\eta'}{\rm var}(p_x(.|\eta),p_x(.|\eta')).
\end{equation}
$k_x$
is a function
of the realization $\pi$ but we assume that $\pi$ only enters locally:
$\{k_x\}$ is a one--dependent random field. In particular, the
$k_x$, $x$ in the even (odd) sublattice of ${\Zbar}^d$, are jointly
independent.
At the same time there typically will be large
regions on which the $k_x$ are large (close to one). This is similar to
what happens to the $\{q_x\}$ in the equilibrium case (see (5)) but, as
remarked before, these regions are copied at every time step and
therefore give rise to infinite ``cylinders'' on the space--time lattice.
This is the reason why, in contrary to the equilibrium case, we cannot allow
a $k_x$ to be one with positive probability.
\vspace{5mm}
\noindent {\it Example 4} (Discrete time spin glass)\\
This is the discrete time (non--equilibrium) analogue
of Example 1 ($h=0$). With $J_{xy}=J_{yx}$ for $x\sim y$ an independent
family of random couplings, the transition probability is
\be
p_x(\sigma(x)|\eta)=\frac1{2}
[1 +\sigma(x) \mbox{tanh} (\beta\sum\limits_{y\sim x}J_{xy} \eta(y)].
\end{equation}
Here,
\be
k_x = \mbox{tanh} \beta\sum\limits_{y\sim x}|J_{xy}|.
\end{equation}
\vspace{5mm}
\noindent{\it Example 5} (A random version of Stavskaya's PCA)\\
The $\{\gamma_x\}$ are independent and identically
distributed non--negative random variables. The transition probability is
\be
p_x(+1|\eta)=
\left\{
\begin{array}{ll}
1&\mbox{ if\ } \eta(x)=\eta(x+1)=+1 \\
\mbox{e}^{-\lambda \gamma_x} &\mbox{otherwise}
\end{array}
\right.
\end{equation}
In the case $\gamma_x=\gamma$ with $\gamma $ large, the PCA has more than
one invariant measure. Here,
\be
k_x =1-\mbox{e}^{-\lambda \gamma_x}.
\end{equation}
\vspace{5mm}
The basic coupling of a PCA with itself is a new Markov process (a new
PCA)
$(\sigma_N,\sigma'_N)$ on the product space $\Omega \times
\Omega$, whose transition probabilities satisfy
\be
\mbox{Prob}_\pi[\sigma_N(x)\neq {\sigma'_N}(x)|
\sigma_{N-1}=\eta,\sigma'_{N-1}=\eta']=\mbox{var}(p_x(.|\eta),
p_x( . | \eta')) \leq k_x
\label{h1}
\end{equation}
The basic coupling has (as every other coupling) the property that
\be
\sup\limits_{\sigma,\sigma'}|P_\pi^N f(\sigma) - P_\pi^N f(\sigma')|
\leq \sum\limits_x \delta_xf \mbox{Prob}_\pi [\sigma_N(x) \neq
{\sigma'}_N(x)|\sigma,\sigma']
\label{h2}
\end{equation}
What is however typical about the basic coupling is that with probability
one, if $\sigma_N(x)\neq\sigma'_N(x)$ then there must be some $y\sim x$ or
$y=x$ for which $\sigma_{N-1}(y)\neq\sigma'_{N-1}(y)$ as well, etc.
Or, disagreement can only be inherited. Hence it is natural to associate
the behavior of forgetting about initial conditions with the absence of
oriented percolation of disagreements on the space--time graph.
Before we come to that we must
specify the percolation part more explicitly.
The space--time graph of a PCA is obtained as follows. The set of vertices
are the space--time points $(x,n), x\in {\Zbar}^d, n=0, 1,... $ in
the time--ordered stacking of the spatial lattice ${\Zbar}^d $ ; $n$ is the
time--coordinate. The graph is completed by drawing arrows (oriented
edges) from each space--time point $(x,n)$ to another point $(y,n-1)$
($y=x$
or $y\sim x$ in so far that there is a non--trivial dependence of
$p_x(1|\eta)$ on $\eta(y)$ for some realization $\pi$).\\
A path $\omega $ on the space--time graph (starting at $(x_0,N)$) is an
(oriented) sequence of space--time points $\omega :
(x_0,N),(x_1,N-1),\dots , (x_k,N-k),
x_l \in {\Zbar}^d, l = 0,\ldots,k < N$ in which at each step
$(x,n)$ to $(y,n-1)$ there is an arrow in the graph from the point
$(x,n)$ to the point $(y,n-1)$.\\
\vspace{5mm}
Consider now the independent oriented site percolation problem on the
space--time graph with (random but highly correlated) densities
$\{k_{(x,N)}\}=\{k_x\}$; each point $(x,N)$ is open (closed) with
probability
$k_x(1-k_x)$. Let $G_\pi^N(x,y)$ denote the probability
to have an open path from $(x,N)$ to $(y,0)$.
\be
G_\pi^N (x,y)\equiv \mbox{Prob}_\pi \left[\mbox{There is a path of
disagreement from $(x,N)$ to $(y,0)$} \right].
\ee
It is immediate from the previous discussion
and the inequality in (\ref{h1})
that
\be
\mbox{Prob}_\pi[\sigma_N(x)\neq {\sigma'}_N(x)|\sigma_0,
{\sigma'}_0] \leq
\sum\limits_{y\in {\Zbar}^d} G_\pi^N(x,y).
\end{equation}
Combining this with (\ref{h2}) and using that there is at least one
invariant measure for the dynamics we get Proposition \ref{ptwee} and
see that all depends on how well we are able
to control the connectivity function $G_\pi(\cdot,\cdot)$.
\begin{pro}\label{ptwee}
For every local function $f$
\be
\|P_\pi^N f- \nu_\pi( f) \| \leq|||f||| \sup\limits_{x\in
\mbox{{\scriptsize supp}} f} \sum\limits_{y \in {\Zbar}^d }
G_\pi^N(x,y),
\label{hlp1}
\end{equation}
with $\nu_\pi$ an invariant measure for the dynamics.
\end{pro}
\subsubsection{Ergodicity for a ``directed'' PCA}
We consider here PCA with transition probabilities
of the form
\be
p_x(\cdot|\eta) = p_x(\cdot|\eta_x,\eta(x+e_
\alpha), \alpha=1,\ldots,d)
\end{equation}
where the $\{e_\alpha\}$ are the unit vectors on
${\Zbar}^d$.\\
The nice thing about ``directed'' PCA is that its space--time graph is not
only
timelike oriented but it is also spatially directed : there is no path
connecting $(x,N)$ with $(x,M)$ passing by another space--time point
$(y,n)$ with $y\neq x$. In particular, when we project this graph on the
spatial lattice (by identifying all points $(x,n), n =0, 1, \ldots$ with
$(x,0)$) we get an oriented spatial lattice. Every path $\omega :
(x,N), (x_1,N-1),\ldots, (x_l, N-l),
\ldots, (y,0)
$ from
the point $(x,N)$ to
the point $(y,0)$ on the space--time graph gives rise to a directed
path on this spatial lattice. The opposite
relation can also be uniquely defined
if in addition we specify the times $n_l = 1, 2, \ldots$ the path stays on
the site $x_l$ visited by the spatial path.\\
Here we restrict ourselves to
PCA's for which $\{k_x\}$ is a set of independent identically distributed
random
variables. Stavskaya's PCA (Example 5) is a well--known example. Another
one (in two dimensions) is Toom's model.
\vspace{5mm}
\noindent{\it Example 6} (Toom's model)\\
The transition probability is derived from a majority rule. With
$\{\gamma_x\}$ as in Example 5:
\be
p(\sigma(x)|\eta)=\frac{1}{2} [1+(1-\mbox{e}^{-\lambda
\gamma_x})\mbox{sgn}(\eta(x)+\eta(x+e_1)+\eta(x+e_2))].
\end{equation}
Then,
\be
k_x=1-\mbox{e}^{-\lambda \gamma_x}.
\end{equation}
\vspace{5mm}
In Propositions \ref{pdir1} and \ref{pdir2} we estimate the connectivity
functions $G_\pi^N(x,y)$ of a directed PCA. First we derive an
upperbound uniform in $N$.
\begin{pro}\label{pdir1}
Suppose
\be
{\bf E}(\frac{k_x}{1-k_x})<\frac{1}{d},
\end{equation}
then
\be
{\bf E}[G_\pi^N(x,y)] \leq C_1 \mbox{e}^{-\lambda' |x-y|}
\ee
for constants $C_1<\infty$, $\lambda'>0$.
\end{pro}
\vspace{5mm}
\noindent{\it Proof}
Let $|x-y|=m$. Every path $\omega :(x,N) \leadsto (y,0)$ on the directed
space--time graph gives rise to a spatial path $\omega':x\leadsto y$ on
${\Zbar}^d$ of lenght $|\omega'|=m$. Following the constuction explained
above we may thus write that
\be
\begin{array}{rl}
G_\pi^N(x,y)&\leq \sum\limits_{\stackrel
{\omega':x\leadsto y }{|\omega'|=m}}
\sum\limits_{\stackrel{l_1,\ldots,l_m \geq 1}{l_1+\ldots+l_m=N+1}}
\prod\limits_{i=1}^{m} k_{{\omega'}_i}^{l_i}\\
&\leq \sum\limits_{\stackrel
{\omega':x\leadsto y }{|\omega'|=m}} \prod\limits_{i=1}^{m}
\frac{k_{{\omega'}_i}}{1-k_{{\omega'}_i}}
\end{array}
\label{ster}
\end{equation}
from which the conclusion readily follows.
\QED
\vspace{5mm}
\noindent{\it Remarks}\\
\noindent {\it 1.} From the Borel--Cantelli lemma we have from
Proposition~\ref{pdir1} that {\bf Q}-a.s. there is a
$N_{x,1}=N_{x,1}(\pi,{\bf E}(\frac{k_x}{1-k_x}),d)<\infty$ such
that if $|x-y|>N_{x,1}$, then
\be
G_\pi^N(x,y)\leq \mbox{e}^{-\lambda|x-y|},
\ee
with $\lambda<\lambda'$.
\vspace{5mm}
\noindent {\it 2.} The same conclusions remain valid in $d=1$ under the
assumption that
\be
{\bf E}(\log \frac{k_x}{1-k_x})<0.
\ee
This is clear from (\ref{ster}) (but now there is only one path $\omega':x
\leadsto y$) by applying the strong law of large numbers.
\vspace{5mm}
\noindent Define $l^\ast_x$ such that for all $l>l^\ast_x$, $k_x^l <
\exp(-\lambda'l^\delta)$, with $0\leq \delta< 1$, i.e.
\be
l^\ast_x =(\frac{-\lambda'}{\log k_x})^{(\frac{1}{1-\delta})}.
\end{equation}
\begin{lem}\label{ldir}
Take $0\leq\beta\leq1$ and $v>0$ such that $(1-\beta)(1-\delta)v>\beta d +1$.
If
\be
{\bf Q}(k_x\geq 1-\tau)<\tau^v,
\end{equation}
then ---with {\bf Q}-probability one---
there exists a $L_0=L_0(\pi)$ such that for $L>L_0$
\be
l^\ast_x \leq L^{1-\beta} \mbox{ for all\ }x\in [-L^{\beta},L^{\beta}]^d
\cap {\Zbar}^d.
\end{equation}
\end{lem}
\vspace{5mm}
\noindent {\it Proof}
A straightforward calculation shows that
\be
{\bf Q}\{l^\ast_x\leq L^{1-\beta}:x\in [L^\beta,L^\beta]^d\}\geq
\left(1-\left(1-\exp(-\lambda'L^{(\delta-1)(1-\beta)v})\right)\right)
^{(2L^\beta+1)^d}.
\end{equation}
Remember that $\mbox{e}^{-x}\geq 1-x$ for $0\leq x\leq 1$ and that $\lambda'
L^{(1-\beta)(\delta-1)}<1$ for L large enough. So,
\be
{\bf Q}\{\exists x \in [L^\beta,L^\beta]^d \cap
{\Zbar}^d:l^\ast_x>L^{1-\beta} \}\leq
1-(1-\lambda'L^{(1-\beta)(\delta-1)v})^{(2L^\beta+1)^d}
\end{equation}
This is summable if $(1-\beta)(1-\delta)v>\beta d +1$. Hence we can use
the Borel-Cantelli lemma to conclude.
\QED
\vspace{5mm}
\noindent The second step in the study of $G_\pi^N(x,y)$ concerns
deriving an upperbound uniformly in $|x-y|$.
\begin{pro}\label{pdir2}
If
\be
{\bf E}(\frac{k_x}{1-k_x}) <\frac{1}{d},
\label{voorw}
\end{equation}
then for every $0<\epsilon<1/2$ we have that if
\be
{\bf Q}\{k_x>1-\tau\}<\tau^v,
\end{equation}
with $v=v(\epsilon)$ high enough,
there exist a time
$N_{x,2}=N_{x,2}(\pi,d,{\bf E}(\frac{k_x}{1-k_x})) < \infty$ and
a constant $\lambda$, independent of $x$ and $\pi$ such
that for $N>N_{x,2}$
\be
G_\pi^N(x,y)\leq \exp(-\lambda N^\epsilon).
\end{equation}
\end{pro}
\vspace{5mm}
\noindent{\it Proof }
For $0<\beta<1$ (will be specified later on) take ${N'}_{x,2}$ such that
$({N'}_{x,2})^\beta>N_{x,1}$ (see Proposition~\ref{pdir1}) and
$\left( ({N'}_{x,2})^{1-\beta}-1 \right)^\frac{1}{1-\beta}>L_0$
(See Lemma \ref{ldir}).
\\ Let $N>N'_{x,2}$.\\
First we assume that
$m=|x-y| N^{1-\beta}$. So,
\be
\begin{array}{rl}
(\ref{27})&\leq\sum\limits_{\omega':x\leadsto y}\sum\limits_{i=1}^{m}
k_{{\omega'}_i}^{N^{1-\beta}-1}\prod\limits_{i=1}^{m}
(\frac{k_{{\omega'}_i}}{1-k_{{\omega'}_i}}).
\end{array}
\label{label}
\end{equation}
Using Lemma \ref{ldir} with $\lambda<\lambda'$ and $\delta$ as specified
later on, and the summing over $y$ gives
\be
(\ref{label})\leq \exp-(\lambda(N^{1-\beta}-1)^\delta )
\sum\limits_{y\in {\Zbar}^d}
\sum\limits_{\omega':x\leadsto y}\sum\limits_{i=1}^{m}
\prod_{i=1}^{m}\frac{k_{\omega_i}}{1-k_{\omega_i}}.
\end{equation}
Condition (\ref{voorw}) assures that the sum is finite with {\bf
Q}-probability one. Hence for $\beta'>\beta$ and $N$ large enough.
\be
G_\pi^N(x,y)\leq \exp-(\lambda N^{(1-\beta')\delta} )
\ee
In the case $|y-x|\geq N^\beta$ we can use Proposition \ref{pdir1} to see
that
\be
G_\pi((x,0),(y,N)) \leq \exp-(\lambda N^\beta).
\end{equation}
Now for $0<\epsilon<1/2$ we can choose $\delta$ and $\beta'>\beta$ such
that
$\beta>\epsilon$ and $(1-\beta')\delta>\epsilon$. This proves the
proposition for $N_{2,x}>{N'}_{2,x}$ large enough.
\QED
\begin{theo}\label{dir1}
If
\be
{\bf E}(\frac{k_x}{1-k_x}) <\frac{1}{d},
\end{equation}
then
the PCA has a unique invariant measure $\nu_\pi$.\\
Moreover for every $0<\delta<1/2$ we have that if
\be
{\bf Q}\{k_x>1-\tau\}<\tau^v,
\end{equation}
with $v=v(\delta)$ high enough,
then for every local function $f$ there exist a time
$N_{0}=N_{0}(\pi,\mbox{{\rm supp}}f,d,{\bf E}(\frac{k_x}{1-k_x})) < \infty $
and a
constant $m>0$ such that for $N>N_{0}$
\be
\|P_\pi^N f-\nu_\pi(f) \|\leq \exp(-m N^\delta).
\end{equation}
\end{theo}
\vspace{5mm}
\noindent{\it Proof}
Combining Proposition~\ref{ptwee} with Proposition~\ref{pdir1}
we see that with {\bf Q}-probability one there is a unique invariant
measure. Furthermore,
\be
\begin{array}{rl}
\|P_\pi^N f- \nu_\pi (f) \|
\leq&|||f||| \sup\limits_{z \in \mbox{{\scriptsize supp}} f}
\sum\limits_{|y-z|\leq
N^\epsilon} G_\pi^N(z,y) \\ \vspace{3mm}
&+ \sup\limits_{z \in \mbox{{\scriptsize supp}} f} \sum\limits_{|y-z|>
N^\epsilon} G_\pi^N(z,y) \}
\label{34}
\end{array}
\end{equation}
\noindent Take
$ N_0 > \max\{\sup\limits_{x \in \mbox{{\scriptsize supp}} f} N_{x,1},
\sup\limits_{x \in \mbox{{\scriptsize supp}} f} N_{x,2}\}$,
then for $N^\epsilon>N_0$ and $\delta<\epsilon$,
\be
(\ref{34})\leq \exp-(m N^\delta)
\end{equation}
with $0 2d^2(1+\sqrt{1+\frac{1}{d}}+\frac{1}{2d}). $$
If
\be
\mbox{{\bf E}}[{\{\log(1-\log(1-k_x))\}}^K] < \infty,
\label{pca1}
\end{equation}
then --- for {\bf Q}-almost every realization ---
there exists a $v(K, d) >1$ such that
for every $10$ we can find constants
$0C_1\} N_{0,x}$.
\end{pro}
Campanino--Klein (1991) proved an analogous result for independent
bond percolation and Klein (1993) for (continuous time) percolation on
${\Zbar}^d \times \Rbar$.
In their problems the set corresponding to $\{k_x \}$ contains
{\it independent} random variables. Although we deal with directed site
percolation and
$k_x $ and $k_y $ are {\it dependent} if
$x\sim y$, we can almost copy their proof. The modifications
are given in Appendix A.
\begin{theo}\label{0}
Take
$$ K > 2d^2(1+\sqrt{1+\frac{1}{d}}+\frac{1}{2d}). $$
If
\be
\mbox{{\bf E}}[{\{\log(1-\log(1-k_x))\}}^K] < \infty,
\label{PCA1}
\end{equation}
then--- for {\bf Q}-almost every
realization --- there exists a $v(K,d) >1$ such
that for every $1 < v < v(K, d) $ and $c>0$
we can find constants
$0C_1\} N_c$
\be
\|P_\pi^N f-\nu_\pi(f)\|\leq B(d,f)
\exp(-c(\log (1+N))^v)
\end{equation}
\end{theo}
\vspace{5mm}
\noindent {\it Proof}
Write (\ref{hlp1}) as
\be
\begin{array}{rl}
\|P_\pi^N f- \nu_\pi(f) \|
\leq&|||f|||\{ \sup\limits_{z \in \mbox{{\scriptsize supp}} f}
\sum\limits_{|y-z|\leq
(\log(1+N))^v} G_\pi^N(z,y) \\
&+ \sup\limits_{z \in \mbox{{\scriptsize supp}} f} \sum\limits_{|y-z|>
{(\log(1+N))}^v} G_\pi^N(z,y)). \}
\end{array}
\end{equation}
Using (\ref{help2}) and take\ $ N_0 > \sup
\limits_{x \in \mbox{{\scriptsize supp}} f}N_{0,x}$,
then for $N>N_0$
\be
\|P_\pi^N f- \nu_\pi(f) \|\leq B(d,f){\exp-c(\log(1+N))^v}.
\end{equation}
with $00$ .
\end{pro}
\vspace{5mm}
\noindent{\it Proof}
Let $m=|x-y|$ and $\omega'$ the projection of the space--time path $\omega$
on ${\Zbar}^d$.
\be
\begin{array}{rl}
G_{a,\pi}^t(x,y)
&\leq \sum\limits_{\omega':x\leadsto y }
\sum\limits_{\stackrel{l_1,\ldots,l_{m} \geq
1}{l_1+\ldots+l_{m}=\frac{t}{a}+1}}
\prod\limits_{i=1}^{m} 1-\mbox{e}^{-a \lambda_{{\omega'}_i}}
\prod\limits_{j=1}^{m}\mbox{e}^{-a\delta_{{\omega'}_j} {l_{{\omega'}_j}}}\\
&\leq \sum\limits_ {\omega':x\leadsto y } \prod\limits_{i=1}^{m}
\frac{1-\mbox{e}^{-a\lambda_{{\omega'}_i}}}{1-\mbox{e}^
{-a\delta_{{\omega'}_i}}}
\prod\limits_{j=1}^{m} \mbox{e}^{-a\delta_{{\omega'}_j}}
\end{array}
\label{z-y}
\end{equation}
Letting $a\downarrow 0$ the Proposition easily follows.
\QED
\noindent{\it Remark}\\
>From Proposition~\ref{pdir3} we can deduce that there exists a
$T_{x,1}=T_{x,1}(\pi,{\bf E}(\frac{k_x}{1-k_x}),d)$ such that for
$|x-y|>T_{x,y}$
\be
G_\pi^t(x,y)\leq \exp{\lambda |x-y|}
\ee
with $\lambda<\lambda'$.
\begin{pro}\label{pdir4}
If
\be
{\bf E}(\frac{\lambda_x}{\delta_x}) <\frac{1}{d},
\label{vor}
\end{equation}
then for every $0<\epsilon<1/2$ we have that if
\be
{\bf Q}\{\mbox{e}^{-\delta_x}>1-\tau\}<\tau^v,
\end{equation}
with $v=v(\epsilon)$ high enough,
there exist a time
$T_{x,2}=T_{x,2}(\pi,d,{\bf E}(\frac{\lambda_x}{\delta_x})) < \infty$ and a
constant $\lambda$ , independent of $x$ and $\pi$ such
that for $t>T_{x,2}$
\be
G_\pi^t(x,y)\leq \exp(-\lambda t^\epsilon).
\end{equation}
\end{pro}
\vspace{5mm}
\noindent{\it Proof}
Take $m$ and $\omega'$ as before. In the same way as we did in the proof of
Proposition~\ref{pdir2} for a PCA, we can show that for $T$ high enough and
$|x-y|\beta$
\be
G_\pi^t(x,y)\leq \exp (-\lambda t^{(1-\beta')\delta}).
\ee
If $|x-y|\geq T^\beta$, Proposition~\ref{pdir3} is saying that for T large
enough
\be
G_\pi^T(x,y) \leq \exp (-\lambda T^\beta).
\ee
Hence, for a given $0<\epsilon<1/2$ choose $\delta$ and
$\beta'>\beta$ such that
$\beta>\epsilon$ and $(1-\beta')\delta>\epsilon$.
\QED
Proposition~\ref{pdir3} and \ref{pdir4} allow us to prove the main result
for a ``directed" IPS.
\begin{theo}\label{dir2}
If
\be
{\bf E}(\frac{\lambda_x}{\delta_x })<\frac{1}{d},
\end{equation}
then
the IPS has a unique invariant measure $\nu_\pi$.\\
Moreover for every $0<\delta<1/2$ we have that if
\be
{\bf Q}\{\mbox{e}^{-\delta_x}>1-\tau\}<\tau^v,
\end{equation}
with $v=v(\delta)$ high enough,
then for every local function $f$ there exist a time
$T_{0}=T_{0}(\pi,\mbox{{\rm supp}}f,d,{\bf
E}(\frac{\lambda_x}{\delta_x})) < \infty $ and
a constant $m>0$ such that for $N>N_{0}$
\be
\|P_\pi^N f-\nu_\pi(f) \|\leq \exp(-m N^\delta).
\end{equation}
\end{theo}
\noindent The proof of Theorem~\ref{pdir4} is
similar to the one of Theorem~\ref{dir1}.
\subsubsection{Ergodicity in a general IPS}
Define
\be
\lambda_{\langle x,y \rangle}=\lambda_x+\lambda_y.
\ee
Note that
$\lambda_{\langle xy \rangle}$ and $\delta_x$ can be {\it
correlated}
if $x$ and $y$ are (next--) nearest neighbors.
\begin{pro}\label{4}
Let
$$
K > 2d^2(1+\sqrt{1+\frac{1}{d}}+\frac{1}{2d}).
$$
Suppose
\be
\Gamma\equiv\max\{ \mbox{ {\bf E} }
[{\{\log(1+\lambda_{\langle x,y \rangle})\}}^K],
\mbox{{\bf E }}[{\{\log(1+1/\delta_x)\}}^K]\}<\infty,
\label{ips2}
\end{equation}
Then there exists a constant $C_1$ such that if
\be
{\bf E}\{(\log(1+\frac{\lambda_{\langle x,y \rangle}}
{\delta_x}))^K\}1$ so that
for every $10$ we can find
constants $0 C_2\} T_{c,x}$.
\end{pro}
Proposition~\ref{4} is formally the same as Theorem 3.2 in Klein (1993).
However, we must take care of the consequences of the non-trivial
correlations between
$\delta_x$ and $\lambda_{\langle x,y \rangle}$. In Appendix B we
briefly show the necessary modifications to the argument of Klein. (1993)
\begin{theo}\label{t0}
Let
$$
K > 2d^2(1+\sqrt{1+\frac{1}{d}}+\frac{1}{2d}).
$$
Suppose
\be
\Gamma\equiv\max\{ \mbox{ {\bf E} }[{\{log(1+\lambda_{\langle
x,y\rangle})\}}^K],
\mbox{{\bf E }}[{\{\log(1+1/\delta_x)\}}^K]\}<\infty,
\label{ips2}
\end{equation}
Then there exists a constant $C_1$ such that if
\be
{\bf E}((\log(1+\frac{\lambda_{\langle x,y \rangle}}
{\delta_x}))^K)1$ so that
for every $10$ we can find
constants $0 C_2\}T_0$.
\end{theo}
\vspace{5mm}
\noindent{\it Proof}
Combining (\ref{hulp1}) and (\ref{hulp2}) in the same way as we did
for Theorem~\ref{0} gives a proof of Theorem~\ref{t0}.\\
\QED
{\it Acknowledgments:} We are grateful to L. Gray, G. Grimmett and E. Speer
for interesting discussions in an early stage of this work while C.M. was
visiting the Newton Institute (Cambridge). We also thank A. van Enter and
A. Klein for suggestions and careful reading of a first draft of this
paper. This work is partially supported by EC grant CHRX--CT93--0411.
\section*{Appendix A}
{\it Remark} In the Appendices A and B we give the necessary modifications
to the argument of Klein (1993) and Campanino--Klein (1991).
As we already mentioned, the proof of Proposition~\ref{2} is similar
to the one in
Klein (1993) and Campanino--Klein (1991). In this paragraph the
argument of (mainly)
Klein (1993) is summarized. The most important lemma's are presented,
with
emphasis on the necessary modifications due to the mutual dependence of the
$k_x$'s. \\
\noindent Let $ W,W'\subset {\Zbar}^{d+1}$. Define\\
\be
\partial(W,W')=\{(y,M)\in W \cap W'|\exists (x,N) \in W'\setminus W,
|(y,M)-(x,N)|=1\}
\end{equation}
\be
\partial (W,{\Zbar}^{d+1}) \equiv \partial W
\end{equation}
For $x\in{\Zbar}^d$ and $L > 0$ let
\be
\Lambda_L(x)=\{y\in{\Zbar}^d;|y-x|\leq L\}.
\end{equation}
For $(x,N), L>0, T>0$, we set
\be
B_{L,T}((x,N))=\Lambda_L(x)\times([N-T, N+T]\cap \Zbar)
\end{equation}
and
\be
B_L((x,N))=B_{L,e^{L^{1/v}}}((x,N)).
\end{equation}
Define $G_{\pi,B_L((x,N))}((y_1,N_1),(y_2,N_2))$ the probability to find an
open path from $(y_1,N_1)$ to $(y_2,N_2)$ in $B_L((x,N))$.
\begin{defi}\label{reg}
Let $m,L > 0$. We say that $x\in {\Zbar}^d$ is $(m,L)$-regular if
\be
G_{\pi,B_L((x,0))}((x,0),(y,M))\leq e^{-mL}
\end{equation}
for all $(y,M) \in \partial B_L((x,0))$. Otherwise $x$ is called
$(m,L)$-singular. A space--time point $(x,t) \in {\Zbar}^{d+1}$
will be called $(m,L)$-regular
if $x$ is $(m,L)$-regular. $\Lambda\subset {\Zbar}^d$ will be called
$(m,L)$-regular if every $x \in \Lambda$ is $(m,L)$-regular. Otherwise
$\Lambda$ is $(m,L)$-singular.\\
\end{defi}
Klein (1993) uses a multiscale analysis. The
proof is by induction. In Lemma~\ref{2.0}
the induction hypothesis is proven: for every $(m_0,L_0)$ we can
find constants $0C_1\}1$ and $m_{k+1}K$ the origin is $(m_\infty,L_k)$-regular. Klein (1993)
estimates $G_\pi^N (x,y)$ in such a $(m_\infty,L_k)$-regular
region.
\begin{lem}\label{2.0}
For every $m, L > 0$ there exist constants $0C_1\} 0$ there exist constants $0C_1\}{\overline L}$ we have
\be
{\bf Q }\{0 \mbox{ is\ } (m_0,L_0)-regular\}\geq 1- \frac{1}{L_0^p}.
\end{equation}
Taking $L_{k+1}=L_k^\alpha, k=1,2 \ldots$, we also have
\be
{\bf Q}\{0 \mbox{ is\ } (m_\infty,L_k)\mbox{-regular\ }\} \geq
1-\frac{1}{L_k^p},
\end{equation}
for all $k=0,1,2,\ldots$ .
\end{lem}
\vspace{5mm}
\noindent{\it Proof Lemma~\ref{2.1}}
To prove Lemma~\ref{2.1} we need
Sublemma's~\ref{2.2.2} and \ref{2.2.3}.
\begin{slem}\label{2.2.2}
Pick positive integers $R,\kappa$ and $b$ such that
\be
\alpha < \frac{(R +1)p}{p+(R+1)d},
\end{equation}
\be
1<\alpha(1-1/v)+1/v <\kappa <\frac{b}{d}< \frac{\alpha}{vd}.
\end{equation}
\noindent
Let $m_0 \geq m_k \geq 1/L_k^\theta$, with $0 < \theta < \theta_0 =
\min \{\alpha (1-1/v),1\}$.\\
$$
\mbox{Suppose there exist } x_1, \ldots ,x_R \in \Lambda_{L_{k+1}} (0)
\mbox{ such that } \Lambda_L
\setminus \bigcup\limits_{j=0}^{R}\Lambda_{2L_k+1}(x_j)
$$
\be
\mbox{is a\ } (m_k,L_k)\mbox{-regular region}.
\label{cond1}
\end{equation}
Define
\be
{\tilde \Lambda}=
\bigcup\limits_{j=1}^R\Lambda_{l^\kappa(x_j)}\cap\Lambda_L(0).
\end{equation}
Suppose
\be
\sum\limits_{x \in \tilde{\Lambda}}\log{(1-k_x)}^{-1}
\leq L_k^b.
\label{cond2}
\end{equation}
Then, if $L_k$ is large enough $\Lambda_{L_{k+1}} (0)$ is a
($m_{k+1},L_{k+1}$)-regular region with $m_{k+1} \geq 1/L_{k+1}^\theta$.
\end{slem}
\vspace{5mm}
\noindent{\it Proof }
The proof is analogous to the proofs of sublemmas $3.6$ and $3.7$ in
Campanino--Klein (1991) and the sublemmas $4.2$ and $4.3$ in Klein
(1993). However, there is
one necessary modification that should be mentioned.\\
Consider the sets
$V_1,\ldots,V_k\subset V_0\subset{\Zbar}^d$ with $V_i \cap V_j =
\emptyset$. \\
For $i:1,\ldots,k$ we define
\be
B_i=V_i\times \{(-T,T)\cap \Zbar \},
\end{equation}
\be
\partial B =\partial(B_i,B) \mbox{ and\
}B'=B\setminus\bigcup\limits_{i=1}^k B_i.
\end{equation}
Let $\{(x,N_1)\stackrel{B}{\longrightarrow}(y,N_2)\}$ the event that there is
an oriented open path in $B$ from $(x,N_1)$ to $(y,N_2)$.\\
For $(y,t)\in \partial B$
$$
\{(0,0)\stackrel{B}{\longrightarrow}(y,N)\}
$$
\be
\begin{array}{rl}
\subseteq \bigcup \limits_{r=0}^k
\bigcup\limits_{\{i_1,\ldots,i_k\}\subset\{1,\ldots,k\}}
&\{\mbox{There\ }\mbox{exist oriented open paths\ }
(0,0)\stackrel{B'}{\longrightarrow}\partial
B_{i_1},\\
&\partial B_{i_1} \stackrel{B'}{\longrightarrow}\partial
B_{i_2},\ldots,\partial B_{i_r}\stackrel{B'}{\longrightarrow}(y,N)\}\\
\subseteq \bigcup \limits_{r=0}^k
\bigcup\limits_{\{i_1,\ldots,i_k\}\subset\{1,\ldots,k\}}
&\{(0,0)\stackrel{B'}{\longrightarrow}\partial B_{i_1}\}\circ
\{\partial B_{i_1} \stackrel{B'}{\longrightarrow}\partial B_{i_2}\}
\circ\ldots\\
&\ldots\circ\{\partial B_{i_r}\stackrel{B'}{\longrightarrow}(y,N)\}.
\end{array}
\end{equation}
Note that we used the time--oriented character of the paths.\\
The van den Berg--Kesten inequality can be used to estimate the
probability of the event $\{(0,0)\stackrel{B}{\longrightarrow}(y,N)\}$
and we can complete the proof as in Klein (1993) and Campanino--Klein
(1991). \QED
\begin{slem}\label{2.2.3}
If
\be
{\bf Q}\{0 \mbox{\ is $(m_k,L_k)$-regular}\}\geq 1 - \frac{1}{L_k^p},
\end{equation}
then ${\bf Q}\{(\ref{cond1})\mbox{ and \ }(\ref{cond2})\}
\geq 1 - \frac{1}{L_{k}^p}$.
\end{slem}
\vspace{5mm}
\noindent{\it Proof }
We call two points $x_1$ and $x_2$ non $l$-touching if it is not possible to
walk from the box $\Lambda_l (x_1)$ to $\Lambda_l (x_2)$ without passing a
point that does not belong to any of the sets.\
If $R+1$ points $x_1,\ldots,x_{R+1} $ are non l-touching,
the events \{$x_i$ is
($m,l$)-regular\} i:1,2,... are independent.\
\be
\begin{array}{rl}
{\bf Q}\{ \exists x_1, x_2, \ldots,x_{R+1}& \in \Lambda_{L_{k+1}}(0)
\mbox{ that are non $L_k$-touching and\ }\\
&(m_k,L_k)\mbox{\ -singular\ } \} \\
&\leq \frac{{(2L_{k+1}+1)}^{d(R+1)}}{L_k^{p(R+1)}}
< \frac{1}{L_k^p}
\end{array}
\end{equation}
for $L_k$ sufficiently large by the choice of $R$. Hence
\be
\begin{array}{rl}
{\bf Q }\{\exists x_1, x_2,\ldots,x_R& \in \Lambda_{L_k}(0) \mbox{
such that } \Lambda_{L_{k+1}}(0) \setminus
\bigcup\limits_{j=0}^{R}\Lambda_{2L_k+1}(x_j) \\
&\mbox{is a\ }(m_k,L_k)-\mbox{regular region} \}\\
&\geq 1-\frac{1}{2L_{k+1}^p}
\end{array}
\end{equation}
For the {\bf Q}-probability of (\ref{cond2}), we have that
\be
\begin{array}{rl}
{\bf Q }&\{\sum\limits_{x \in \tilde{\Lambda}} \log{(1-k_x)}^{-1}>
L_k^b \}\\
&\leq {\bf Q}\{\exists x \in \tilde{\Lambda}: \log{(1-k_x)}^{-1} >
\frac{L_k^b}{{(2L_k^\kappa +1)}^d}\}\\
&\leq {(2L_k^\kappa +1)}^d {\bf Q}\{ \log{(1-k_x)}^{-1} >
\frac{L_k^b}{{(2L_k^\kappa +1)}^d}\}\\
&\leq {(2L_k^\kappa +1)}^d {\bf Q}\{\log(1+ \log{(1-k_x)}^{-1}) >
\log( 1+\frac{L_k^b}{{(2L_k^\kappa +1)}^d})\}
\end{array}
\end{equation}
\be
\begin{array}{rl}
{\bf Q}&\leq {(2L_k^\kappa +1)}^d {\bf Q}\{\log(1+ \log{(1-k_x)}^{-1}) >
\frac{L_k^b}{{(2L_k^\kappa +1)}^d}\}\\
&\leq \frac{{(2L_k^\kappa +1)}^d {(2L_k^\kappa +1)}^{Kd}}{ L_k^{K b} }
{\bf E}({[\log(1+\log{(1-k_x)}^{-1})]}^K) \\
&\leq \frac{ {\bf E}({[\log(1+\log{(1-k_x)}^{-1})]}^K) }
{L_k^{K(b-\kappa d)-\kappa d}}\\
&\leq \frac{1}{2L_{k+1}^p}
\end{array}
\end{equation}
To prove the sublemma, note that both (\ref{cond1}) and (\ref{cond2}) are
decreasing events.
\QED
\section*{Appendix B}
The only thing to be done is to observe that the events (1.6), (4.1) and
(4.2) in Klein (1993) have the same {\bf Q} probability.
Choose $\kappa ,b, \gamma, \tau$ such that
\be
\alpha(1-1/v)+1/v<\kappa< \frac{b}{d}<\frac{\alpha}{v d}, 0< \gamma <
b-\kappa d, 1/v<\tau<\alpha(1-1/v).
\end{equation}
Consider the following events:
\begin{eqnarray}
8d\rho_{L_0} &<& e^{2m_0} \label{vw1} \\
\delta_{L_0} &>& e^{-1/2L_0^{1/v}}\label{vw2}\\
\lambda_{L_0} &<& e^{L_0^{1/v}}\label{vw3},
\end{eqnarray}
with
\be
\rho_L=\sup\limits_{x \in \Lambda_L(0)}
\{\frac{1}{\delta_x}\max\limits_{y\sim x}\lambda_{\langle
xy \rangle}\}
\end{equation}
\be
\delta_L = \inf\limits_{x \in \Lambda_L(0)}\delta_x
\end{equation}
\be
\lambda_L=\max\limits_{(x,y)\in \partial(\Lambda_L(0),{\Zbar}^d)}
\lambda_{\langle xy \rangle}
\end{equation}
\be
\partial(\Lambda_L(0),{\Zbar}^d)= \{(x,y)\in ({\Zbar}^d)^2:x
\in \Lambda_L(0), y \in \Lambda_L^c(0):y\sim x \}
\end{equation}
and the events
\be
\lambda_{L_{k+1}}