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%Versione del 4/11/92. Manca ancora la bibliografia.
\title{Finite volume mixing conditions for lattice spin
systems and exponential approach to equilibrium of Glauber
dynamics.}
\author{F. Martinelli\ddag and E. Olivieri\dag}
\address{\ddag Dipartimento di Matematica Universit\`a "La Sapienza"
Roma, Italy\newline
\dag Dipartimento di Matematica Universit\`a "Tor Vergata" Roma,
Italy}
\abstract{We critically review various finite volume conditions in
classical statistical mechanics together with their implications
both for the Gibbs state and for an associated Glauber type dynamics.
Moreover we considerably improve some old results by Holley and
Aizenamn and Holley on the relationship between mixing properties of
the Gibbs state and fast convergence of the Glauber dynamics.
Our results are optimal in the sense that, for example, they show
for the first time fast convergence of the dynamics above the
critical point for the d-dimensional Ising model with or without an
external field.}
\section {Introduction.}
In the recent years a series of works by several authors have been
dedicated to the so called "finite size conditions" to analyze the
properties of systems of classical statistical mechanics in the pure
phase region; they have been introduced in order to explicitely
show that, in the pure phase region, a system behaves as if it was
weakely coupled provided it is analyzed on the correct,
sufficiently large, scale.
To simplify the exposition let us consider Ising-like lattice spin
systems.
Finite size conditions or, in the language of Dobrushin and
Shlosman (DS) constructive conditions are mixing properties,
involving some parameters, of the Gibbs measure, corresponding to a
given interaction, in a finite volume $\Lambda$.
Let us call, for instance, such a condition $C(\Lambda,k,\gamma)$,
where
$\Lambda$ is the volume and $k$,$\gamma$ are (in this case two)
parameters; for example $\gamma$ is a rate of exponential decay of
truncated correlations, $k$ is a constant in front of the
exponential .
The importance of these conditions becomes clear once one is able
to prove a statement of the following type:
There exists a finite set of volumes $\Gamma = \{ \Lambda_1,
\dots,\Lambda_N\}$,
depending on $k$,$\gamma$ such that if one supposes true
$C(\Lambda,k,\gamma)$ for every $\Lambda\in \Gamma$ then the
infinite volume
Gibbs measure is unique and enjoys some properties of weak
dependence like exponential decay of truncated correlations.
For example DS in their theory of Completely Analytical
Interactions prove a theorem like above with $\Gamma$ given by {\it
all subvolumes} of a cube with suitable edge $L=L(k,\gamma)$. Whith
this hypothesis they prove exponential clustering not only for the
unique infinite volume Gibbs measure but also for Gibbs measure
relative to {\it arbitrary finite or infinite volumes}.
This result is extremely strong since it shows very good mixing
properties of the Gibbs state also in strange
(pathological) shapes.
This point will be discussed in Section 3 in
connection with another approach ( [12],[13]) giving weaker results
but with a wider applicability.
The main point of our discussion will be that DS conditions are too
strong to be verified near the coexisting curve corresponding to a
first order phase transition. This is because these conditions,
always assuming uniformity w.r.t. the boundary conditions (b.c.),
have to include cases of regions $\Lambda$ whose shapes are such
that the exterior boundary $\partial \Lambda$ " enters ", so to
speak, into the bulk of $\Lambda$ (think to a subset of a 2-D layer
in $\bf Z^3$ or to a cube with a regular array of holes).
Then, for some particular b. c. one can produce, in these
pathological $\Lambda$'s, exactly the situation corresponding to
values of thermodynamical parameters producing a first order phase
transition.
In [12], [13] for regular (say Van Hove) regions results similar to
the ones of DS are proven starting from weaker conditions involving
only "fat" regions (say cubes) and covering also the part of the
phase diagram near the coexistence line where DS condition fails.
Finite size conditions play a role also in some dynamical
problem.\par
Consider a Glauber dynamics namely a single spin flip
Markov process reversible w.r.t. the Gibbs measure corresponding to
a given interaction.
Some authors and especially R.Holley and more recently Zegarlinski
and Strook and Zegarlinski
investigated the connection
between mixing properties of the (unique) invariant Gibbs measure
and the speed of approach to equilibrium under very general
assumptions on the dynamics.
We want to quote, for example, the fundamental paper [6] by Holley
where the author, in particular, for the case of short range
translation invariant attractive (see Section 2) stochastic Ising
models introduces a strong finite size condition, referring to the
invariant Gibbs measure, that ensures exponential convergence to
equilibrium in a strong sense.\par
Before concluding this short
introduction we warn the reader that there are different types of
finite volume equilibrium mixing properties and different notions
of exponential approach to equilibrium. The first ones can be
divided into
{\it weak mixing} and {\it strong mixing} . Both notions
can be expressed as weak dependence, inside $\Lambda$, say in $x\in
\Lambda$, on the value of a conditioning spin, say in
$y\in \partial \Lambda$.
We have strong mixing if the influence of what happens in
$x$ decays with the distance $|x-y|$ of $x$ from $y$ whereas we
speak of weak mixing when the influence decays with the distance
of $x$ from the boundary $\partial \Lambda$ and not from $y$.
There are models, (the so called Czech models) satisfying weak
mixing but violating strong mixing also for regular domains.
They exhibit absence of phase transition in the bulk but a sort
of phase transition with long range order in a layer near to the
boundary takes place.
On the other hand we can also consider approach to equilibrium in
different possible senses depending on which norm we want to use
($L^2$ or $L^\infty$ ) and whether we want results directly for the
infinite volume dynamics or for the finite volume dynamics with
estimates uniform in the volume and in the boundary conditions.
\par
In the present paper, after some definitions concerning our model
(Section 2), we
\bigskip \item{i)} review and discuss the role of the different
notions of mixing conditions both at equilibrium and in dynamics
(Section 3)
\item{ii)} improve previous results by proving, for ferromagnetic
systems, exponential convergence in the uniform norm by only
assuming weak mixing (Section 4)
\item{iii)} prove uniform exponential approach to equilibrium in a
very strong sense for {\it any } (finite or infinite ) sufficiently
regular volume starting from strong mixing condition in a cube
(Section 5)
\item{iv)} give some applications (Section 6).\medskip
We only sketch the proofs and refer to [10] for more details.
\section {Definitions.}
\subsection{The model.}
Let us first describe the Hamiltonian of our spin
system. Given a subset $\Lambda$ of the lattice $\bf Z^d$ and
given a spin configuration $\tau$ outside $\Lambda$ we
set for any
spin configuration $\sigma\; \in\;
\Omega_{\Lambda}\; \equiv \; \{-1,1\}^{\Lambda}$ :
$$H_{\Lambda}^{\tau}(\sigma)\;=\; -\sum_{X\cap\Lambda\, \neq \,
0}J(X)\sigma_X \eqno(2.1) $$ where $\sigma_X$ is the product of the
values of the spins at the sites of the set X. It is
understood here that the values of the spins at sites
x not in
$\Lambda$ are those of the "boundary" configuration
$\tau \in \Omega_{\Lambda^c}$. The
potential J(X) is assumed to be translation invariant and
finite range. Some of our results are proved with an
additional hypothesis which in the sequel will be called
{\it attractivity}.
The system is said to be {\it attractive} or {\it ferromagnetic} if
the local field at the origin $$h(\sigma
)\;=\;\sum_{X;\,0\,\in\,X}J(X)\sigma_{X\setminus
\{0\}}\eqno(2.2)$$
is an
increasing function of the spins $\sigma_x\quad x\,\neq\,0$
\bigskip
{\bf Remark } It is easy to check that
in the case of only two body interaction attractivity coincides with
the requirement $J(x,y)\;\geq\;0$. \bigskip Given
$H_{\Lambda}^{\tau}$ we will denote by $\mu_{\Lambda}^{\tau}$ the
associated Gibbs state. If there exists a
unique Gibbs state in the infinite volume limit $\Lambda
\, \to \,
{\bf Z^d}$ independent of the boundary conditions $\tau$
then it
will be simply denoted by $\mu$. For shortness we will use
the
notation $\mu_{\Lambda}^{ \tau}(f)$ to denote the mean
of the
observable f under the Gibbs state $\mu_{\Lambda}^{
\tau}$
\subsection{The dynamics.} The stochastic
dynamics that will
be discussed in this paper will be one of the many
standard
stochastic Ising models for the hamiltonian (2.1) (see
[8]). We will
need to analyze the stochastic Ising model in finite
volume
$\Lambda$ with boundary conditions $\tau$ as well as in
the whole
lattice ${\bf Z^d}$. Both cases are defined through their
jump rates
that will always be denoted by $c_x(\sigma , a)$ , $x\,
\in \, {\bf
Z^d}$ or $x\, \in \,\Lambda$ and $a\, \in \, \{-1,+1\}$
whenever
confusion does not arise. Then the generator $L$ of the
dynamics
takes the form :
$$Lf(\sigma)\;=\; \sum_{x,a}c_x(\sigma ,a)(f(\sigma
^{a,x})\,-\,f(\sigma ) )\eqno(2.3)$$ where $\sigma^{a,x}$ is the
configuration obtained from $\sigma$ by putting
the spin at x
equal to the value a.\par
In order to simplify the exposition and the
computations we decided to take from the beginning
a precise form
for our jump rates : $$c_x(\sigma , a)\, =\,
\mu_{\{x\}}^{\sigma}(\eta (x)\,=\,a)\;=\;
{1\over 1+\hbox{exp}(-2a\sum_{X;\,x\,\in\,X}J(X)\sigma_
{X\setminus\{x\}})}\eqno(2.4)$$ where it is understood
that if we are in a
finite volume $\Lambda$ the configuration $\sigma$ agrees
with the
boundary configuration $\tau$ outside $\Lambda$. This choice
corresponds to what is known as the {\it heat bath} dynamics.
In the finite volume case our expression for
the jump rates
makes sure that the Markov process generated by the jump
rates on
$\{-1,1\}^{\Lambda}$ is reversible with
respect to the Gibbs state
$\mu_{\Lambda}^{ \tau}$. This implies that $\mu_{\Lambda}^{
\tau}$
is the unique invariant measure of the process. This important
fact
holds also in the infinite volume limit if the Gibbs state
is unique
. Finally if the interaction is attractive and if in
the space of spin configurations we introduce the partial
order
$\sigma\, \leq \, \eta$ iff $\sigma (x) \, \leq \, \eta (x)
$ for all x
then there exists
a coupling in
our probability space such that if $\sigma^\eta_t$ denotes
the
evoluted at time t of the above stochastic Ising model
starting from
the configuration $\eta$ then, using the coupling : $$\sigma^-_t,
\leq \,\sigma^\eta_t\, \leq \, \sigma^{\eta '}_t\, \leq \,
\sigma^+_t$$ if $\eta '\, \geq \, \eta$, where $\sigma^+_t$
is the
evolution starting from all spins equal to plus one
and analogously
for $\sigma^-_t$.
\section {Review and critical analysis.}
\subsection {Finite volume mixing conditions.}
-\ \ We first give the definition of {\it variation distance}
between two probability measures $P, Q$ on the finite set $Y$ :
$$Var (P, Q) = \frac 1 2 \sum_{y\in Y}\vert P(y) - Q(y)
\vert = \max_{X \subset Y}\vert P(X) -
Q(X)\vert \eqno(3.1)$$
\noindent
-\ \ Given a measure $\mu_\Lambda $ on $\Omega_\Lambda $ we call
{\it relativization} of $\mu_\Lambda $ to $\Omega_\Delta$ with
$\Delta \subset\Lambda $, the measure $\mu_{\Lambda , \Delta}$
on $\Omega_\Delta$ given by
$$\mu_{\Lambda , \Delta}(\sigma_\Delta) =
\sum_{\sigma_
{\Lambda \setminus\Delta}}
\mu_\Lambda (\sigma_{\Lambda \setminus \Delta},
\sigma_\Delta)\eqno(3.2)$$ \noindent
-\ \ We
say that a Gibbs measure $\mu_\Lambda $ on $\Omega_\Lambda $
satisfies a strong mixing condition with constants $C, \gamma$ if for
every subset $\Delta \subset \Lambda $:
$$\sup_{\tau\in \Omega _{\Lambda^c}}
Var(\mu_{\Lambda , \Delta}^\tau\ \ \mu_{\Lambda ,
\Delta}^{\tau^{(y)}})\leq C e^{-\gamma \hbox {dist}(\Delta, y)}
\eqno(3.3)$$
where\quad $\tau^{(y)}_x = \tau_x$ for $ x\ne y$.\par
\noindent
We denote this condition by
$ SM(\Lambda, C,\gamma)$\par\noindent
-\ \ We say that a {\it strong mixing} condition in the sense of
{\it truncated} expectations holds for the measure $\mu_\Lambda $
on $\Omega_\Lambda $, with constants $C, \gamma$ if for every
cylindrical functions $f, g$ with supports $S_f , S_g \subset
\Lambda $
$$\vert\mu_\Lambda (f, g)\vert\leq\ \ C\parallel f\parallel\ \
\parallel g\parallel\ e^{-\gamma \hbox {dist} (S_f, S_g)}
\eqno(3.4)$$ \noindent
and we denote is by $SMT(\Lambda , C, \gamma)$.
\par
Here $\mu_\Lambda (f, g)$ denotes the truncated expectation
of $f$ and $g$:
$$
\mu_\Lambda (f, g) = \mu_\Lambda (f g) -
\mu_\Lambda (f)\mu_\Lambda (g)
$$
\noindent
It is easy to show that, supposing that $SMT(\Lambda , C, \gamma)$
holds true, then there exists $C'$ such that $SM(\Lambda , C',
\gamma)$ holds.
It is possible to give some weaker converse statement (see below)
so that, in a proper sense, $SM$ and $SMT$ are equivalent.\par
\noindent
-\ \ We say that a Gibbs measure $\mu^\tau_\Lambda $ satisfies a
{\it weak mixing condition} with constants $C , \gamma$ if for every
subset $\Delta \subset \Lambda $
$$\sup_{\tau, \tau'\in\Omega_{\Lambda ^c}}
\quad Var(\mu^\tau_{\Lambda , \Delta},
\mu^{\tau'}_{\Lambda , \Delta})\leq C\sum_{{x\in\Delta } , {y\in
\partial^+_r\Lambda }}\exp(-\gamma\vert x-y\vert)\eqno(3.5)$$
where
$$
\partial^+_r \Lambda = \{x\not\in\Lambda : \hbox{dist}
(x, \Lambda )\leq r\}$$
\noindent
We denote this condition by $WM(\Lambda , C, \gamma)$.\par
\noindent
An important notion, already quoted in the Introduction, is what we
call {\it effectiveness}.\par \noindent
-\ \ Given two families $\Gamma,\Gamma'$ of subsets of $\bf Z^d$ a strong
mixing condition is $(\Gamma, \Gamma')$-effective if $C, \gamma$
are such that by supposing that $SM(\Lambda , C, \gamma)$ holds
for every $\Lambda $ in the class $\Gamma$ we have that there exist
$C' , \gamma'$ such that $SM(\Lambda ', C', \gamma')$ holds for
every $\Lambda '$ in $\Gamma'$.\par
Of course the interesting cases correspond to a {\it finite}
family $\Gamma$ and an {\it infinite} $\Gamma'$\ \ (finite size condition
for exponential decay of truncated correlations on arbitrarily large
volumes).\par \noindent
\subsection {Some known equilibrium results.}
The following theorems DS1,DS2,OP hold in the general (finite
range, translation invariant, not necessarily attractive) case.
Theorem H1 is restricted to the attractive case.\par \noindent
We start from a
result by Dobrushin and Shlosman concerning weak mixing [3].
\bigskip
{\bf Theorem DS1}\quad(Dobrushin - Shlosmann [3])\par
Suppose that a suitable finite size condition $DSU(\Lambda _0,
\delta)$ on a finite volume $\Lambda_0$ is satisfied for some
$\Lambda _0$ and {\bf $\delta < 1$}; then $\exists\ \ C > 0, \gamma
> 0$ such that condition $WM(\Lambda , C, \gamma)$ holds
{\it for every $\Lambda $}.\par \noindent
\bigskip
In the general case to give condition $DSU (\Lambda _0, \delta)$ we
need some more definitions. In the attractive (ferromagnetic) case
the condition is :
$$\sup_{\tau\in \Omega_{\Lambda^c_0}} Var(\mu_{\Lambda_0
,x}^{\tau}\ \ \mu_{\Lambda_0 , x}^{\tau^{(y)}})\leq\alpha_{x,y}
\eqno(3.6)$$
$$\sum_{x\in \Lambda_0 ,y\in\partial^+_r\Lambda
_0}\alpha_{x,y}\ \ \leq \delta\vert
\Lambda_0 \vert\eqno(3.7)$$
\bigskip
In [3],[4] Dobrushin and Shlosman introduced the concept of
{\it complete analytical interactions}.They are those potentials
whose corresponding Gibbs measure {\it for every (finite or
infinite) } volume $\Lambda$ satisfy $SMT(\Lambda , C, \gamma)$ for
some $ C > 0 , \gamma > 0$. DS show that $SMT(\Lambda , C,
\gamma)$, when supposed true for every volume $\Lambda$ is
equivalent to many other mixing conditions ( in particular
$SM(\Lambda , C, \gamma),\; \forall \Lambda$ ) and to analyticity
properties of thermodynamical and correlation functions. \par
\noindent The main result, in the framework of the {\it constructive
description} of completely analytical interactions is the following
\par \noindent
\bigskip {\bf Theorem DS2}\quad
(Dobrushin-Shlosmann [4])\par There exists a function $L = L(C,
\gamma)$ such that $SM(\cdot, C, \gamma)$ is $(\Gamma,
\Gamma')$-effective with $\Gamma$ given by the set of all subsets
of a cube of edge $L(C, \gamma)$ and $\Gamma'\equiv$ the set of
{\it all} ({\it finite or infinite}) subsets $\Lambda $ of $\bf
Z^d$.\par \bigskip
As remarked in the Introduction this result is extremely strong (
it covers volumes of arbitrary sizes and shapes ).However there are
situations (especially near a first order coexisting line) to
which Theorem DS2 does not apply. In these cases one needs a
somehow different approach where only quite regular ( but
arbitrarily large) volumes appear.\par \bigskip
{\bf Theorem
OP}\quad(Olivieri, Picco [12], [13], [10]).\par There exists $\
L=L(C,\gamma)$ such that $SM(\cdot, C,\gamma)$ is $(\Gamma,\Gamma')$
effective whith $\Gamma$ given {\it only} by the cube $Q_L$ of side
$L$ and $\Gamma' $ contains all "sufficiently fat" regions.\par
\bigskip
$\Gamma'$ can be taken as the set of all {\it multiples} of $Q_L$.
(We say that $\Lambda$ is a multiple of $Q_L$ if it is
partitionable into cubes, with disjoint interior
, obtained from $Q_L$ by translating its center by vectors of the
form $ y = xL$ with $x \in \bf Z^d$).\par \noindent
We would like to quote, at this point, an example due to
R. Schonmann
[14]. Consider a ferromagnetic Ising system with nearest neighbours
and next to the nearest neighbours interactions at low temperature.
It is easy to see that for some particular value ( $\ne 0 $)
of the magnetic field h the corresponding potential is not
completely analytical (in Dobrushin-Shlosman's sense ) so that no
finite volume strong mixing condition can be simultaneously
verified {\it for all } subvolumes of any cube in an {\it
effective} manner. More subtle examples of this phenomenon can be
found in a recent paper by A.v.Enter, R.Fernandez and A.Sokal
[5]
\par\noindent
On the other hand exactly for the same models in the same region of
parameters Theorem OP applies so that one obtains uniform
exponential decay of truncated correlations for any sufficiently
"fat" volume.(see Section 6). \par \noindent
Finally we want to quote
a result by Holley (valid for the attractive case) referring
only to some particular shapes: the boxes. A box in $\bf Z^d$ is
the cartesian product of $d$ finite intervals.
Holley introduces a finite size condition referring to a
cube $\Lambda_0$ that we call condition $H(\Lambda_0, \delta)$; it
can be considered as a stronger version of $DSU(\Lambda,\delta)$
and it is given by: \par
for every $x \in \Lambda_0$, $y
\in \partial^+_r\Lambda_0$ ,there exists $\bar \alpha _{x,y} >0$
such that for every $\Lambda \subset \Lambda_0, \Delta \subset
\Lambda$ : $$\sup_{\tau\in \Omega_{\Lambda^c}} Var(\mu_{\Lambda
,x}^{\tau}\ \ \mu_{\Lambda , x}^{\tau^{(y)}})\leq \bar
\alpha_{x,y} \eqno(3.8)$$
with
$$\sum_{x\in \Lambda_0 ,y\in\partial^+_r\Lambda
_0}\bar \alpha_{x,y}\ \ \leq \delta\vert
\Lambda_0 \vert\eqno(3.9)$$
\par \bigskip
{\bf Theorem H1}\quad (Holley [6])\par
The existence of a cube $\Lambda_0$ such that
$H (\Lambda _0, \delta)$ holds with $\delta < 1$ is equivalent to
the existence of $C>0,\gamma>0$ such that $SM(\Lambda, C,\gamma)$
holds {\it for every box} $\Lambda$.\par\bigskip
{\bf Remark} Consider a 3-D n.n. ferromagnetic Ising system with
$h=2J$ ($h=$ magnetic field, $J=$ coupling constant) at inverse
temperature $\beta$ larger then the 2-D critical temperature
$\beta^{(2)}_c$.
Consider a particular class of boxes: the squared two-dimensional
layers; for suitable boundary conditions we get zero effective
field ( $-1$ b.c. compensate for $h$ in the interior of
$\Lambda$ ).So it is impossible to get $SM(\Lambda,C,\gamma)$ for
some $C >0,\gamma > 0$ and for every (even thin) $\Lambda$ because
otherwise we would contraddict the
existence of long range order for 2-D Ising system at $h=0$,
$\beta$ large.
Then, in this case,the hypotheses of Theorem H1 cannot be satisfied
.
On the other hand, as it is easy to verify,the hypotheses of
Theorem DS1 are satisfied ( $DSU(\Lambda_0,\delta)$ holds with
$\delta <1$ for a sufficiently large {\it cube} $\Lambda$
if $\beta$
is large enough); we conclude that it is the {\it arbitrariness} of
the subsets $\Lambda$ in $\Lambda_0$ to create problems (as it is
easy to verify directly by considering, again, two-dimensional boxes
$\Lambda \subset \Lambda_0$ and trying to verify (3.8) ).\par
Finally it is easy to see that Theorem OP applies to the above case
(the corresponding finite size condition on a cube with sufficiently
large edge is satisfied), giving $SMT(\Lambda, C,\gamma)$ for every
sufficiently "fat" region $\Lambda$ ( "thin" regions are excluded!).
\subsection{Notions of exponential approach to equilibrium.}
Let us now analyze the Glauber dynamics associated to our
generalized Ising models.\par\noindent
We want first to define some different notions of exponential
approach to equilibrium ( we always suppose that there exists a
unique infinite volume Gibbs measure $\mu$).
\par
\noindent
In what follows $T_t$ will denote the Markov semigroup generated by
$L$.
\par \bigskip
\item {1)} $EC, L^2 (d \mu), \bf Z^d$
: exponential convergence in $L^2$ for the
infinite volume dynamics.
It means that there exists $\gamma > 0$ such that
$\forall f \in D$ (the space of cylindrical functions), $\quad
\exists \ C_f >0:$
$$\Vert T_t f - \mu (f) \Vert_{L^2 (\mu)} \le C_f
\ e^{- \gamma t}$$
\item {2)} $UEC, \bf Z^d$: Uniform $(L^\infty)$ exponential
convergence for infinite
volume dynamics.
It means:
$$\exists \gamma > 0: \ \forall f \in D \
\exists \ \ C_f >0:$$
$$\Vert T_t f - \mu (f) \Vert_u \le C_f \ e^{- \gamma T}$$
namely
$$\sup_{\sigma} \ \vert E_\sigma
f (\sigma_L)
- \mu (f) \vert \le C_f \ e^{- \gamma t}$$
where $E_\sigma$ denotes expectation over the process starting from
$\sigma$.
\item {3)} $EC, L^2 (\mu^\tau_\Lambda)\quad \forall \Lambda
\in \Gamma$: exponential convergence in $L^2$ for finite volume
dynamics in $\Lambda$ uniformly in $\Lambda$ varying in a class
$\Gamma$ and in the b.c. $\tau ;$ namely:
$$\exists \gamma > 0 \ : \ \forall f \in
D \ \exists \
C_f > 0:
\forall \ \Lambda \in \Gamma, \Lambda \supset S_f , \ \forall \tau
\in \ \Omega_{\Lambda^c} $$
$$\Vert T^{\Lambda, \tau}_t f - \mu^\tau_\Lambda (f)
\Vert_{L^2(\mu^\tau_\Lambda)} \le C_f \ e^{-\gamma t}$$
\medskip
\item {4)} $UEC, \ \ \forall \Lambda \in \Gamma$: uniform
exponential convergence for finite volume dynamics in
$\Lambda$ uniformly in $\Lambda$ varying in a class
$\Gamma$ and in the
b. c. $\tau$; namely :
$$ \exists \gamma > 0\ \ : \ \forall f
\in D \ \exists C_f > 0 :
\forall \Lambda \in \Gamma \ : \ \Lambda \supset S_f,
\forall \tau \in
\Omega_{\Lambda ^c}:$$
$$
\Vert T^{\Lambda, \tau}_t f - \mu^\tau_\Lambda (f) \Vert_u \le
C_f \ e^{- \gamma t} $$
\bigskip \noindent
Of course $UEC, \ \ \forall \Lambda $ in a Van Hove sequence
imply $UEC, \bf Z^d$ and $EC, L^2 (\mu^\tau_\Lambda)\quad \forall
\Lambda $ in a Van Hove sequence imply $EC, L^2 (d \mu), \bf Z^d$;
a less obvious statement, due to Holley, says that, in the
attractive case, $EC, L^2 (\mu^\tau_\Lambda)\quad \forall$ box
$\Lambda$ implies $UEC, \bf Z^d$.
\noindent
Recently some papers have been devoted to the
relations between
the different notions of exponential approach to equilibrium
and between this speed of approach to
equilibrium and mixing properties
of the invariant Gibbs measure.\par\noindent
In particular the problem has been studied of deducing exponential
approach to equilibrium
(in the different above senses) from {\it finite size}
conditions.
\subsection{Some known results on Glauber dynamics}
The following theorems H2, AH hold for the attractive case.\par
\bigskip
{\bf Theorem H2}\quad (Holley [6])\par
Suppose that there exists a cube $\Lambda_0$ such that
$H (\Lambda_0, \delta)$, with $\delta<1$, holds; then $UEC, \bf Z^d$
holds; moreover $EC, L^2 (\mu^\tau_\Lambda)$ holds
{\it
for every box} $\Lambda$.\par\bigskip
\noindent
Notice that, as previously remarked, the hypotheses of Theorem H2
do not apply to situations (like the previously discussed 3-D
Ising system with $h=2J$) for which, however, the thesis is
certainly expected to be true provided that we replace {\it
for every box} $\Lambda$ with {\it
for every cube} $\Lambda$.
\par
\bigskip
{\bf Theorem AH}\quad (Aizenmann,Holley [2])\par
If there is a cube $\Lambda_0$ such that
$DSU (\Lambda _0, \delta)$ is satisfied with $\delta<1$, then
$EC, L^2 (d \mu), \bf Z^d$ holds.\par\bigskip
\noindent
For the general, not necessarily attractive case we want to quote
the following Theorem, due to Stook and Zegarlinski, obtained in
the framework of the theory making use of logarithmic Sobolev
inequalities.\par \bigskip
{\bf Theorem SZ} \quad ( Strook, Zegarlinski [15])\par
The following statements are equivalent
\item {i)} There exists a finite region $\Lambda$
such that $H(\Lambda,\delta)$ holds with $\delta < 1$.
\item {ii)}
There exist $C>0,\gamma>0$ such that $SMT(\Lambda,C,\gamma)$ holds
{\it for every} volume $\Lambda$; namely complete analyticity, in the
Dobrushin-Shlosman's sense, holds.
\item {iii)} $UEC$ {\it for every} $\Lambda$ holds.
\item{iv)} $EC$, $L^2(d\mu_\Lambda^\tau)$ for every $\Lambda$ holds.
\section {Exponential convergence under a weak mixing condition}
In this section we discuss the first one
of our main results, namely
that for an attractive stochastic Ising model a weak mixing
condition on the Gibbs state implies the ergodicity of the infinite
volume Markov process and the exponential convergence in the strong
$UEC , \bf Z^d$ sense of its distribution at time $t$ to the unique
invariant measure as $t \,\to \, \infty$.\par
Let us first reformulate
our {\it weak mixing} condition in the context of the attractive
systems.\par
We recall that by $Q_L$ we denote the cube
$\{y\in {\bf Z^d}\,;\;\vert\,y_i\vert \;\leq \; {L-1\over 2} \;
\forall\,i=1...d\}\quad$, for $L$ odd. $Q_L(x)$
will denote the
translated by the vector $x \, \in \, {\bf Z^d}$ of
$Q_L$
\bigskip
{\it Weak
mixing} $WMA(C,\gamma)$\par
There exist two positive constants $C $ and
$\gamma$ such for every integer L
$$\mu_{Q_L}^{+}(\sigma
(0))\;-\;\mu_{Q_L}^{-}(\sigma (0)) \;\leq
\;C\hbox{exp}(-\gamma L)$$
\noindent
It is immediately seen that $WM(Q_L,C,\gamma),\; $ for every $L$
implies $WMA(C,\gamma)$.\par \bigskip
{\bf Remark } One sees
immediately that the above mixing condition implies that there
exists a unique Gibbs state in the thermodynamic limit that will be
denoted by $\mu$ . \bigskip
As already mentioned in the introduction
an important question is what kind of implication has a mixing
condition on the Gibbs state for the convergence to equilibrium of
the associated stochastic Ising model. A first
partial result (by Aizenmann and Holley) is the above quoted
Theorem AH where exponential convergence, in the $L^2(d\mu)$ sense,
follows from $DSU(\Lambda_0,\delta), \delta < 1$ (recall
that, by Theorem DS1, $DSU(\Lambda_0,\delta), \delta < 1$ implies
$WM(\Lambda,C,\gamma)$ for some $C > 0,\gamma > 0$ and for every
$\Lambda$ which, in turn, implies $WMA(C,\gamma$).\par
Here we will prove a much stronger
result:
\bigskip {\bf Theorem 1}\par
{\it Weak mixing} implies that
there is a positive constant $m$ and for any cylindrical function f
there exists a constant $C_f$ such that: $$\sup_{\sigma} \vert
T_t(f)(\sigma )\, -\, \mu (f)\vert \; \leq \;
C_f\hbox{exp}(-mt)$$
In other words if $WMA(C,\gamma)$ holds for some $C >0, \gamma > 0$
then $UEC,\bf Z^d $ holds. \bigskip
{\bf Sketch of the Proof}\par
Let us define
$$\rho (t) \;=\;
\sup_xP(\sigma^+_t(x)\neq\sigma^-_t(x))\eqno(4.1)$$ where
$\sigma^+_t$ and $\sigma^-_t$ are the configurations
starting from
all pluses and all minus respectively.\par Given $\rho (t)$
we
estimate the quantity appearing in Theorem 1 by:
$$\sup_{\sigma}\vert T_t(f)(\sigma)\;-\;\mu (f)\vert\;\leq\
\vert\vert\vert f\vert\vert\vert\rho
(t) \eqno(4.2)$$
where $\vert\vert\vert f\vert\vert\vert\;=\;\Sigma_x\sup_{\sigma}\vert\vert
f(\sigma^x)\,-\,f(\sigma)\vert\vert$ and $\sigma ^x$ is the configuration
obtained from $\sigma$ by flipping the spin at x.\par
Thus we have to show that $\rho (t)$ decays exponentially to
zero. Actually thanks to an important result by Holley
[6] (see
also [2] for a different derivation of the same result)
it is
sufficient to show that $\rho (t)$ goes to zero faster than
$1\over t^d$ . The standard way to try to get this input
is the following: one increases $\rho (t)$ by imposing
extra plus (minus)- boundary
conditions on the boundary of a box centered at the origin
and of side L for the evolution which starts from all
pluses (all minuses). Then one gets very easily that:
$$\rho (t)\; \leq \;
\mu_{Q_L}^+(\sigma (o)\,=\,+1)\;-\;
\mu_{Q_L}^-(\sigma (o)\,=\,+1)\;+\;$$
$$+\;\hbox{exp}(c_1L^d\,-\, \hbox{gap}(Q_L
,+)t)\;+\;\hbox{exp}(c_1L^d\,-\, \hbox{gap}(Q_L
,-)t)\eqno(4.3)$$
where gap$(Q_L
,+)$ is the gap in the spectrum of the (selfadjoint) generator
of
the stochastic Ising model in the cube $Q_L$ of side
L with plus boundary
conditions and analogously for gap$(Q_L
,-)$. If we assume {\it weak mixing} then we have that the first
difference in the r.h.s. of (4.3) is smaller than :
$$C\hbox{exp}(-\gamma L)$$
If we now assume a lower bound on the gap uniform
in the
volume $Q_L$, then, by choosing L = $\delta t^{1/d}$
with $\delta$ sufficiently small, we get that $\rho (t)$
decays
faster than exp(-const $t^{1/d}$) $<<\;{1\over t^d}$ and
therefore, thanks to the Holley's theorem, $\rho (t)$ will
decay exponentially fast.\par
However, as one can easily
check, the rather strong input
that the gap in the spectrum of the
generator of the process in finite volume is bounded away
from zero uniformly in the volume implies in some sense
that one is
able to prove fast convergence to equilibrium (in the
$L^2(d\mu_{\Lambda} )$ sense ) not only in
the bulk but also close
to the boundary. From a static point of view this is equivalent
to a
{\it local }weak dependence on the boundary conditions,
where local
means that the change of one boundary spin does not affect
far away
spins even if they are located close to the boundary. As
it has been discussed in the Introduction and in Section 3
this in general is not implied by the {\it weak mixing}
condition above but requires {\it strong mixing}.\par
Under only the
weak mixing condition it might very well be that the gap is no
longer bounded away from zero uniformly in the volume and one is
left with a very rough and rather trivial bound of the form:
$$\hbox{gap}(\Lambda,+/-)\; \geq \; \hbox{exp}(-c(J)\vert
\Lambda\vert )\eqno(4.4)$$
where c(J) is a suitable positive constant depending only on the
interaction J. \par Such a weak bound forces us to choose the side L
as : L = L(t) = $const.\{log(t)\}^{1\over d}$. By plugging L(t) into
(4.3) we get :
$$\rho (t) \, \leq \, \hbox{exp}(-\bar \gamma log(t)^{1\over
d})\eqno(4.5)$$
which is certainly not sufficient to apply Holley's theorem.\par
Thus we
need to find a new method that allows us to improve the above very
rough bound. The main new technical tool for our analysis
is
the following recursive inequality satisfied by $\rho (t)$
that for
convenience we state as a proposition (see [10] for a
proof):\bigskip {\bf Proposition 1}\par
Under the hypotheses of Theorem 1 there
exist two finite
positive constants C and $\gamma$ such that for any
integer L :
$$\rho (2t) \, \leq \,
(2L+1)^d\rho (t)^2\; +\; C\hbox{exp}(-\gamma L)$$
It is not difficult to see that the above recursive inequality
allows
us to transform the bound (4.5) into a bound of the form :
$$\rho (t) \, \leq \, \hbox{exp}(-\hbox{exp}(+\bar \gamma
log(t)^{1\over
d}))\eqno(4.6)$$
$( \bar \gamma > 0)$ which is clearly faster than the inverse of any
power of t (see [10] for details). \vskip 3cm
\section{Exponential convergence in "fat" finite volumes} We
discuss in this section
the exponential convergence to equilibrium in finite volumes with
rates that are estimated uniformly in the volume by assuming a
finite volume condition of strong type. All the
results of this section can be proved without the assumption
of the attractivity of the dynamics (see [11]); some of the
proofs are however much simpler in the attractive case. The
mathematics involved in the non attractive case becomes much
more sophisticated and relies upon the theory of logarithmic Sobolev
inequalities applied to Gibbs measures as it has been developed in
an important series of papers by Zegarlinski
and Zegarlinski and Strook (see, for instance, [16],[15] and
references quoted there). An independent proof based on
renormalization
group ideas of the existence of a finite logarithmic Sobolev
constant for the Gibbs state under a finite volume mixing condition
(see below) can be found in [11].\par \noindent
In our proof it turns out to be convenient to use a particular
finite volume condition which does not contain parameters
(like $C,\gamma$). In the sequel we will refer to it as
$L_o$-mixing.It is the following one:\par\bigskip
{\bf
$L_o$-mixing} :\hskip 1cm Let $\Lambda_o \equiv Q_{2L_o + 1}$ be the
cube of side $2L_o+1$ with sides parallel to the coordinate axes
and let for any $V \subset \Lambda_o$ $\mu^{\sigma}_{\Lambda_o,V}$
be the relativization of $\mu^{\sigma}_{\Lambda_o}$ to the set V.
Then for any k outside $\Lambda_o$ and any V in $\Lambda_o$ with
dist(V,k) $\geq\;L_o^{1\over 2}$ we must have: $$
Var(\mu^{\sigma}_ {\Lambda_o,V} ,
\mu^{\sigma^{(k)}}_{\Lambda_o,V})\; \leq \; {1\over
\hbox{dist}(k,V)^{2(2d+1)}}\quad \forall \, \sigma\, \in
\{-1,+1\}^{\Lambda_o^c}\quad \eqno(5.1)$$
Notice that $L_o$-mixing easily follows from
$SM(\Lambda_o,C,\gamma), SMT(\Lambda_o,C,\gamma) $ once, given
$C>0,\gamma>0 ,L_o$ is taken sufficiently large.
We emphasize again
that our condition has to hold only in a definite geometric shape ,
in our case a cube, contrary to what assumed by Aizenman and
Holley [2] or Zegarlinski and Strook [15] where the arbitrariness
of the geometric shape of the finite volume plays an important role.
Of course, as already remarked, in weakening the condition there is
a price to pay: we will prove our results only in volumes that are
"multiple" of the elementary volume $\Lambda_o$. However this has to
be the case if we want to apply our condition to systems like the
Ising model at low temperature in the presence of a positive
external field where it can be proved (see Section 3) that the
previous conditions of Aizenman-Holley or Strook-Zegarlinski can
fail.\par Let us now state our main results (see [10] and [11] for
details). In what follows we will call $L_o$-compatible any subset
of the lattice $\bf Z^d$ which is the union of translates of the
cube $\Lambda_o$ such that their vertices lay on the rescaled
lattice $(2L_o+1){\bf Z^d}$.
\bigskip {\bf Theorem 2 } (Effectiveness)\par There
exists
a positive constant $\bar L\,\geq \,R$ such that if
$L_o$-mixing holds with $L_o\,\geq \, \bar L$ then
there exists positive constants
$\gamma$ and C such that for any $L_o$-compatible set
$\Lambda$, any
$L\,\geq \, {L_o^{1\over 2}}$, any $\sigma$ and any
site k outside
$\Lambda$ we have : $$\vert\vert\mu^{\sigma}_{\Lambda}\,
-\,
\mu^{\sigma^{(k)}}_{\Lambda}\vert\vert_{\sigma ,k,L}\; \leq \;
C\hbox{exp}(-\gamma L)\eqno(5.2)$$
where $\vert\vert\mu^{\sigma}_{\Lambda}\,
-\, \mu^{\sigma^{(k)}}_{\Lambda}\vert\vert_{\sigma ,k, L}$ is
the
variation distance between the relativization of the
Gibbs states in
$\Lambda $, with boundary conditions $\sigma$ and $\sigma^{(k)}$
,respectively, to the maximal subset $\bar \Lambda (L)$ of $\Lambda$
which is at distance
greater than $L$ from k.\bigskip {\bf
Remark} Thus the Theorem says that, provided $L_o$ is large
enough, $L_o$-mixing propagates to all larger scales
that are multiple of the basic length scale $L_o$. In particular
it
implies the exponential decay of correlations in any
$L_o$-compatible volume uniformly in the volume
and thus also of the
unique infinite volume Gibbs state.\par
The content of the above Theorem 2 is similar to the one of
Theorem OP. However in [10] a simple dynamical proof of it is
provided, avoiding the complicated geometrical constructions and
the theory of the cluster expansion that where at the basis of the
arguments of proof in [12], [13] where, on the other hand, also
analyticity properties where proved.
\bigskip
The next result
says that $L_o$-mixing implies exponential convergence to
equilibrium in the strongest possible sense namely in any
$L_o$-compatible finite volume both in the $L^2$-norm
and in
the uniform norm. \bigskip
{\bf Theorem 3}\par There exists
a positive constant $\bar L\,\geq \,R$ such that if
$L_o$-mixing holds with $L_o\,\geq \, \bar L$ then there exist
two positive
constants $m_o$ and $m$ such that for any $L_o$-compatible set
$\Lambda$, any boundary configuration $\tau$ and
for any function f in $L^2(d\mu_{\Lambda}^{\tau})$ : \item{i)}
$$\quad \vert\vert
T^{\Lambda ,\tau}_t(f)\, -\,
\mu_{\Lambda}^{\tau} (f)\vert\vert_{L^2(d\mu_{\Lambda}^{\tau})}
\;
\leq \;
\vert\vert\,
f\,\vert\vert_{L^2(d\mu_{\Lambda}^{\tau})}\hbox{exp}(-m_ot)\eqno(5.3)
$$
\item{ii)} $$\quad\sup_{\sigma}
\vert
T^{\Lambda ,\tau}_t(f)(\sigma)\, -\, \mu (f)\vert \; \leq \;
\vert\vert\vert f\vert\vert\vert\,\hbox{exp}(-mt)\eqno(5.4)$$
where $T^{\Lambda ,\tau}_t$ denotes the Markov semigroup
of the
process evolving in $\Lambda$ with boundary conditions
$\tau$.\bigskip
{\bf Sketch of the proof in the attractive case}\par
Let us fix an
$L_o$-compatible set $\Lambda$ and a boundary configuration
$\tau$ and let $\{Q_i\}$ be a covering of the set $\Lambda$
with the
following two properties:\bigskip \item{a)} Each element of
the covering is a
a cube of side $2L_o+1$ with sides parallel to the coordinate
axes.
\item{b)} If two different cubes $Q_i$ and $Q_j$
overlap then
necessarily each one of them is the translated by
$L_o$ along
at least one coordinate axes of the other.\bigskip
It is very easy to check that for any $L_o$-compatible set
$\Lambda$
such a covering always exists.\par
Next we introduce a new dynamics (Gibbs sampling) on
$\{-1,+1\}^{\Lambda}$ by defining its generator $L_Q$ as :
$$L_Qf(\sigma) \;=\; \sum_{\eta ,i}c_{Q_i}(\sigma
,\eta)(f(\eta)\,-\,f(\sigma))\eqno(5.5)$$
where the new jump rates $c_{Q_i}(\sigma
,\eta)$ are a generalization of those of the heat bath
dynamics and are given by :
$$c_{Q_i}(\sigma,\eta)\;=\; \mu_{Q_i}^{\sigma}(\eta)\eqno(5.6)$$
if $\eta$ agrees with $\sigma$ outside the cube $Q_i$ and
zero
otherwise. It is understood that outside $\Lambda$ the
configurations $\sigma$ and $\eta$ agree with
$\tau$.\bigskip
{\bf Remark} The above version of the Gibbs sampling is
different
from the one employed by Holley [6] Aizenman and Holley [2]
and
Strook and Zegarlinski [15]. In these previous works the
updating
was as follows: each site x is chosen in $\bf Z^d$ with rate
one and then the configuration in $\Lambda_o(x)\cap\Lambda$
is put
equal to $\eta$ with probability
$\mu_{\Lambda_o(x)\cap\Lambda}^{\sigma}(\eta)$
, where $\Lambda_o(x)$
is the cube of side $2L_o+1$ centered at x. This dynamics
has however
the incovenience of updating sometimes regions that are not
squares
$\Lambda_o$ but rather boxes (= intersection between
two cubes). Contrary to what happen for cubes $\Lambda_o$,
not only we have no
control at all on the mixing properties of the Gibbs states
associated to such geometric regions, but there are situations
(see Section 3 ) in which our mixing condition while being true for
cubes fails for certain boxes !\bigskip
It is rather
simple to show that the above Gibbs sampling is still
reversible
with respect to the Gibbs state in $\Lambda$ with boundary
conditions $\tau$ ; more important one easily proves
(see Lemma 2.3 of
[15]) that if gap($L_Q$) and gap(L) denote the gap in the spectrum
of
the generators $L_Q$ and L respectively as operators in
$L^2(d\mu_{\Lambda}^\tau )$, then there exists a
positive constant c independent of $\Lambda$ and $\tau$
such that:
$$\hbox{gap}(L)\;\geq \;
\hbox{exp}(-cL_o^d)\hbox{gap}(L_Q)\eqno(5.7) $$
Thus in order to
prove the theorem we need only to estimate from below gap($L_Q$)
uniformly in $\Lambda$ and $\tau$ .\par For this purpose
in [10] we adopt a
scheme very similar to the one already used in section
2. We couple the Gibbs sampling dynamics starting from different
initial conditions and we define the quantity
$\rho
(t)$ as : $$\rho_{\Lambda}^{\tau}(t)\, =\, \sup_{\sigma
,\eta\, x\in \Lambda}P(\,
\sigma_t(x)\, \neq \, \eta_t(x)\,)\eqno(5.8)$$
It is easy to check
Holley's criterium :
if there exists a finite time $t_o$ such
that $\rho_{\Lambda}^{\tau}(t_o)
\,
<<\,{1\over t_o^d}$ then $\rho_{\Lambda}^{\tau }(t)$
decays exponentially fast. The
idea then is to verify the existence of the basic time scale
$t_o$
by just using our $L_o$-mixing condition. In fact if
{\it $L_o$-mixing} holds with $L_o$ large enough then the
updating of each single cube $Q_i$ of the covering becomes
almost independent
of the value of the spins in the other
cubes since their influence dies out outside a thin layer of width
$L_o^{1/2}$ around $\partial Q_i\setminus\partial\Lambda\cap\partial
Q_i$. In some sense the Gibbs sampling behaves as a high temperature
almost independent stochastic Ising model for which the
exponential convergence to equilibrium is a very well established
result.\bigskip
{\bf Remark} It is absolutely crucial for the whole
argument to work that the the influence of the neighboring cubes
around $Q_i$ dies out outside a layer only around $\partial
Q_i\setminus\partial\Lambda\cap\partial Q_i$ and {\it not } around
the whole boundary $\partial Q_i$; in other words there cannot be
propagation of information along the boundary of $\Lambda$.\bigskip
Part i) of the theorem then follows immediately.
For attractive systems part ii) follows from the argument used in
Theorem 1 once we know that there is a lower bound on the gap of
$L_{\Lambda}^{\tau}$ if $\Lambda$ is $L_o$-compatible which is
uniform in $\Lambda$. For non attractive systems part ii) follows
by proving a logarithmic Sobolev inequality for the Gibbs state
via a decimation procedure which uses in a crucial way the
"effectiveness" of the mixing condition.
\bigskip{\bf Remark} One may wonder why even for attractive
systems we needed in this section a
condition like $L_o$-mixing which is much stronger than the weak
mixing condition used in the previous section. The reason is that
under only the weak mixing condition we cannot prove in finite
volume $\Lambda$ the result of Proposition 1. In fact if we take
a site x close to the boundary of the set $\Lambda$ and we
consider, as we did in Proposition 1 the cube $\Lambda_o(x)$ of
side $2L_o+1$ centered at x, then x can be very close to the
boundary of $\Lambda_o(x)\cap\Lambda$. If this happens then, by
changing the boundary conditions on $\partial \Lambda_o(x)\cap
\Lambda$,
we may considerably affect the Gibbs state at x ( this phenomenon
occurs for instance in the Czech models [3] which are however non
ferromagnetic ); this fact is compatible with the weak mixing
condition which requires only a control of
the effects of changing the boundary conditions inside the bulk! Of
course in the infinite volume case this problem never occurs since
x is always in the bulk.
\section{Applications}
In this section we discuss some applications of our results.
In particular we prove the exponential convergence to equilibrium
for the infinite volume stochastic Ising model for all temperatures
above the critical one and for low temperature and non zero external
field.\par The model that we will consider is the standard
nearest neighbor Ising model in an external non negative field h
and at inverse temperature $\beta$. If we consider the associated
stochastic Ising model discussed in the previous sections, then
for $h\neq 0$ or
$\beta\,<\,\beta_c$ it
will be an ergodic Markov process on $\{-1,+1\}^{\bf Z^d}$ with
$\mu^{\beta ,\,h}$, the unique infinite volume Gibbs state, as
unique invariant measure. In the following theorem we will
strenghthen
this result. Let us denote by $ E_{\sigma }^{\beta
,\,h}(f(\sigma_t))$ or by $ E_{\sigma }^{\Lambda ,\,\tau ,\,\beta
,\,h}(f(\sigma_t))$ the expected value at time t of the function f
with respect to the distribution of the process evolving in the
infinite lattice $\bf Z^d$ or in the finite set $\Lambda$ with
boundary conditions $\tau$. Then we have :\bigskip
{\bf Theorem 4}\par \item{\bf a)} Assume that $\beta\,<\,\beta_c$.
Then for any $h\,\geq \, 0$ there exists a positive constant $m$
and for any cylindrical function f there exists a constant $C_f$
such that: $$\sup_{\sigma} \vert E_{\sigma }^{\beta
,\,h}(f(\sigma_t))\, -\, \mu^{\beta ,\,h} (f)\vert \; \leq \;
C_f\hbox{exp}(-mt)$$
\item{\bf b)} There exists a positive constant $\beta_o$ such that
for any
$\beta \;\geq\;\beta_o$ and $h\;>\;0$ there exists a positive
constant $m$ and for any cylindrical function f there exists a
constant $C_f$ such that: $$\sup_{\sigma} \vert E_{\sigma }^
{\beta ,\,h}(f(\sigma_t))\,
-\, \mu^{\beta ,\,h} (f)\vert \; \leq \;
C_f\hbox{exp}(-mt)$$
\item{\bf c)} Given $h\;>\;0$ there exist two positive constants
$\beta_o(h)$ and $L_o(h)$ such that for any
$\beta \;\geq\;\beta_o$
there exists a positive
constant $m$ such that for any $L_o(h)$-compatible set
$\Lambda$ and
for any function f on $\{-1,+1\}^{\Lambda}$ : $$\sup_{\sigma}
\vert E_{\sigma }^{\Lambda ,\,\tau ,\,\beta ,\,h}(f(\sigma_t))
\, -\, \mu_\Lambda^{\tau ,\, \beta ,\,h}(f)\vert \; \leq \;
\vert\vert\vert f\vert\vert\vert\hbox{exp}(-mt)$$
{\bf Proof}\par
Thanks to Theorem 1 {\bf a)} and {\bf b)} follow immediately
once we are able to verify our
{\it weak mixing} condition. In case {\bf a)} {\it weak mixing}
follows
from i) of Theorem 2 of a recent paper by Higuchi [7] which
exploits in a crucial way the results by Aizenman, Barski and
Fernandez on the absence of the third phase [1]. In case {\bf b)}
one uses the fact that for low enough temperature and any
positve h if one considers a large enough cube of side L with
minus boundary conditions, then within a distance from the boundary
smaller than Clog(L) and with very high probability there
exists a large contour of plus spins which screens the effect of
the negative boundary conditions. Such a result has been proved
many
years ago by Martyrosian [9].\par
Part {\bf c)} In this case we verify that for any $h\;>\;0$ there
exist $L_o(h)$ such
that the configuration identically
equal to +1 is
the unique ground state configuration of the
Hamiltonian $H_{\Lambda_{L_o(h)}}^\tau(\sigma)$ for any boundary
condition $\tau$.
This implies, as it is easy to verify, that there exists $\beta_o =
\beta_o(h)$ such that if $\beta > \beta_o$ our $L_o(h)$-mixing
condition is satisfied.
\nonumsection{Acknowledgments}
During the evolution of this work we took advantage
of many clarifying
discussions with some colleagues.
We want to thank, in particular, R.L.Dobrushin,
A.v.Enter and G.B.Giacomin.
It is a pleasure to thank especially R. Schonmann for many
valuable comments, suggestions and for some of the examples of non
complete analyticity for the Ising model at low temperature in the
presence of an external field. We are also in debt with M.
Aizenman for pointing out that the results on the absence of the
intermediate phase that he and his collaborators obtained few years
ago prove our weak mixing condition for Ising model for all
temperatures above the critical one. Few days after the discussion
we received a preprint by Higuchi where this and many other new
interesting results were proved for the Ising model. Finally we
would like to thank R.Kotecky , P.Picco and A.Bovier with F.Koukiou
for having organized three stimulating meetings in Prague , Les
Houches and Marseille which certainly helped to improve the quality
of the present work. \par
\noindent
\bigskip
Work partially supported by grant SC1-CT91-0695
of the Commission of European Communities
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\end
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