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\centerline{\ttlfnt Approach to Equilibrium of Glauber Dynamics }
\centerline{\ttlfnt In the One Phase Region. I: The
Attractive Case}\vskip 1cm
\author{F. Martinelli\ddag\hskip 0.5cm and\hskip 0.5cm E.
Olivieri\dag}
\address{\ddag Dipartimento di Matematica
Universit\`a "La Sapienza" Roma, Italy \hfill\break{\rm e-mail:
martin@mercurio.dm.unirm1.it}} \address{\dag Dipartimento di
Matematica Universit\`a "Tor
Vergata" Roma, Italy \hfill\break{\rm e-mail:
olivieri@mat.utovrm.it}}
\abstract{Various finite volume mixing conditions in classical
statistical mechanics are reviewed and critically analyzed. In
particular some {\it finite size conditions} are discussed,
together with their implications for the Gibbs measures and for the
approach to equilibrium of Glauber dynamics in {\it arbitrarily
large } volumes. It is shown that Dobrushin-Shlosman's theory of {\it
complete analyticity} and its dynamical counterpart due to
Stroock and Zegarlinski, cannot be applied, in general, to the
whole one phase region since it requires mixing properties for
regions of {\it arbitrary} shape. An alternative approach, based on
previous ideas of Olivieri and Picco, is developed, which allows to
establish results on rapid approach to equilibrium deeply inside the
one phase region. In particular, in the ferromagnetic case, we
considerably improve some previous results by Holley and
Aizenamn and Holley. Our results are optimal in the sense
that, for example, they show for the first time fast
convergence of the dynamics {\it for any temperature} above the
critical one for the d-dimensional Ising model with or without an
external
field. In part II we extensively consider the general case (not
necessarily attractive) and we develop a new
method, based on renormalization group ideas and on an assumption of
strong mixing in a finite cube, to prove hypercontractivity of the
Markov semigroup of the Glauber dynamics.}
\vskip 1cm\noindent Work partially supported
by grant SC1-CT91-0695
of the Commission of European Communities
\numsec=0\numfor=1
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{\bf Section 0}\par\noindent
\centerline{\bf Introduction}\bigskip
Recently many efforts have been devoted, with increasing interest,
to analyze the precise connections between
i) mixing properties of Gibbs measures for lattice spin systems,
(typically expressed in terms of rapid decay of the truncated
correlations) and ii) the (properly defined) speed of approach
towards equilibrium of some associated spin flip
Glauber type dynamics. We have in mind, in particular, the basic
paper by Holley [H2], and the subsequent works by Aizenman and
Holley [AH] and Stroock and Zegarlinski [SZ], where such connections
were established, first for ferromagnetic Ising type models [H2],
[AH] and then for very general discrete or continuous spin systems
[SZ].\par The goal of those papers was to show,
under very general hypotheses on the flip rates, that a
Glauber dynamics
for values of
thermodynamical parameters (e.g temperature and magnetic
field) which do not give rise to a phase transition , must have a
rapid (typically exponentially fast in the sup-norm) approach to
equilibrium.\par In all these works the mixing properties of the
Gibbs state were expressed in term of some finite volume condition
similar if not equal (see [SZ]) to the famous Dobrushin and Shlosman
{\it complete analiticity} conditions [DS2] [DS3]. If such
conditions hold then the Glauber dynamics approaches its invariant
Gibbs measure $\mu$ exponentially fast either in the $L^2(d\mu)$
sense or in the $L^\infty$ norm in any finite or infinite volume and
with rates uniformly bounded in the volume and in the boundary
conditions. In the standard Ising model the various
conditions are easily verified at high temperature or large
external magnetic field.\par The first result referring to the
region of parameters really close to the line of first order phase
transition was proved few years ago by the authors of the present
paper in collaboration with E. Scoppola [MOS] while working on
metastability for the dynamical Ising model. By purely dynamical
arguments we established the rapid approach to equilibrium for the
standard 2D stochastic (=dynamical) Ising model for any value of the
magnetic field h, provided that the temperature T was low enough
(depending on h !). In the arguments of proof in [MOS] a crucial role
was played by the results of J. Neves and R. Schonmann [NS] on the
metastable behaviour of the 2D Ising model in finite volume. In
order to extend the result to 3D in the same region of the phase
diagram and since detailed results on metastability were ( and still
are )
not available in 3D, we tried to see whether one of the various
finite volume mixing conditions of the above mentioned papers could
be satisfied by our model. With our surprise only very weak results
could be deduced in our case since we could not verify the
majority of these conditions; moreover, thanks also to some
simple examples by R. Schonmann that are described in section 2, we
realized that, in general, any Dobrushin-Shlosman type of finite
volume mixing condition is probably bound to fail near to a first
order phase transtion line. The main reason for this surprising
result is that, in such conditions, one is required to control the
Gibbs state in a finite collection of sets of the lattice some of
which with a ratio surface/volume of order 1 (e.g. a layer of
thikness equal to one in 3D) . Clearly, close to a first order phase
transition line, one should expect to be able to establish mixing
properties of the Gibbs state only in sufficiently regular regions (
for example such that their surface/volume ratio tends to zero as
the volume tends to $\infty$).\par
A second important observation on the
approach to equilibrium of Glauber dynamics that comes out of the
present work is the following one: as in equilibrium statistical
mechanics where sometimes one is able to prove rapid decay of the
{\it infinite volume} correlation functions but {\it not} of the
{\it finite volume } ones, with bounds uniform in the volume and in
the boundary conditions, also for the dynamics one has to carefully
distinguish between {\it infinite volume} results and {\it finite
volume} ones (with bounds uniform in the volume and in the boundary
conditions). The reason is that even if there exists a unique
infinite volume Gibbs state with exponentially decaying correlation
functions, it may happen that, in arbitrarily large but finite volume
(e.g. a big cube), a sort of long-range order close to the boundary
occurs with a consequent decay of the correlation functions
non-uniform in the location of the observables; such a
non-uniformity must give rise to a global slowing down of the
dynamics in the whole volume (see the remark at the end of section
4).\par Such a "boundary phase transition" is known to occur in 3D
for the so called Czech models [Sh]; even for the Ising model at
low temperature and very small (depending on the temperature)
magnetic field a "layering phase
transition" is expected to take place (Basuev phenomenon [D1]).
However it is reasonable to conjecture that the above phenomenon
should never appear in 2D since, in this case, the boundary is
one-dimensional and, in this case, phase transitions for a short
range interaction can never take place. This is exactly what we
prove in a paper in preparation done in collaboration with R.
Schonmann [MOSh].
\par
As a consequence of the above discussion, if one is willing to
prove rapid convergence to equilibrium for the Glauber dynamics in
the whole one-phase region, one should try to envisage a method that
works directly for the infinite volume dynamics without any
assumptions on the finite volume one. In [AH] first and later on in
[SZ] the exponential convergence to equilibrium, directly in the
infinite volume in the $L^2(d\mu )$, is proved without requiring
anything on the finite volume dynamics. However in order to get
stronger result in which the $L^2(d\mu )$ norm is replaced by the
$L^\infty$ one, all the existing methods had to assume a uniform
lower bound on the gap of the generator of the dynamics in
a finite region uniformly in the size, in the shape of the region
as well as in the boundary conditions.\par
By the above discussion this seems a too strong
requirement in order to cover the whole one phase region.\par In
this paper we make what we believe is an important step toward the
solution of the above problems at least for discrete, finite range
spin systems.\par In the attractive (= ferromagnetic) case we show
that rapid approach to equilibrium in the {\it infinite volume} in
the {\it uniform norm} is equivalent to exponentially weak dependence
on the boundary conditions for the magnetization at the origin. In
the Ising model such condition is expected to be true in the whole
phase diagram outside the coexistence line but we are able to verify
it only for :
\item{i)}$T\,>\,T_c$ and any uniform magnetic field h
\item{ii)
}for $T$ small enough and any $h\,\neq\,0$.\par In order to obtain
stronger results, namely rapid approach to equilibrium in {\it
finite volume} in the uniform norm with bounds uniform in the volume
and in the boundary conditions, we make a mixing assumption (rapid
decay of two point truncated correlations) on the Gibbs state on a
given large enough cube $\Lambda_o$ (thus we have no arbitrariness
on the shape). Such a finite volume mixing condition was introduced
some years ago by Olivieri [O] and Olivieri and Picco [OP] in order
to derive, by renormalization group methods (decimation)
and cluster expansion, analyticity and decay of truncated
correlations of the Gibbs state in infinite and finite
volumes, provided that the latter are in some sense "multiple" of
$\Lambda_o$. Although their results may appear weaker than those of
Dobrushin and Shlosman, since not all possible geometric shape are
covered, they are certainly suited to deal with systems close (but
not too close because e.g. of the Checks models ) to a first order
phase transition. We show that under the hypothesis of [O] and [OP]
we can get the rapid approach to equilibrium, both in the $L^2(d\mu
)$ and in the $L^\infty$ sense, for volumes that are "multiple" of
$\Lambda_o$. Again we verify that for the Ising model our condition
holds for high temperature or low temperature and arbitrarily small
( not vanishing) magnetic field h but with $h/T\,>>\,1$. This in
particular covers the case of metastabilty in the 3D case.\par In
this paper we only give the proofs in the attractive case; the
general case requires proving the ipercontractivity of the Markov
semigroup generated by the dynamics, which in turn is equivalent to
proving the existence of a finite Logarithmic Sobolev constant for
the Gibbs state. This was provided for the first time by Stroock and
Zegarlinski [SZ] under a Dobrushin-Shlosman complete analiticity
assumption; in our case we found a different approach, based on
renormalization group methods. This new method is the argument of
a forthcoming paper [MO] \par
The present paper is organized as follows:
\item{} In section 1 we
define the models and the Glauber dynamics. \item{} In section 2 we
critically review the existing finite volume mixing conditions
together with their implications both for the equilibrium problem
and for the approach to equilibrium of the dynamics. \item{} In
section 3 for the attractive case (not necessarily reversible with
respect to a Gibbs measure) we prove rapid convergence to
equilibrium in the infinite volume under a weak dependence on the
boundary conditions of the magnetization in the origin. \item{} In
section 4 we establish finite volume results. \item{} In section 5
we discuss the implications of our results for the stochastic Ising
model.\bigskip Yau
and ShengLin Lu [SY],
starting from mixing properties of the Gibbs measure, proved
a very interesting lower bound on the spectral gap of the
generator of the Kawasaki dynamics. Their method, different
from ours, allows them to treat also Glauber dynamics and obtain
results similar to those
described in Theorem 4.2. After learning of our work they also
adopted the point of view of the present paper and considered the
strong mixing condition $SM(\Lambda, C,\gamma)$ only for cubes.
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{\bf Section 1.}\par\noindent
\centerline{\bf General Definitions and Notation.}
\bigskip
In this section we define the statistical mechanics models and the
associated Glauber dynamics
that we want to examine. \par
We will consider lattice spin systems.
We start giving a list of basic definitions.
\medskip
\noindent
\item{-}\ \ Configuration space of a single spin: finite set
$S\, =\, \{1,\dots, N\},\ \ N\in{\bf N}$.
\medskip
\indent
\item{-}\ \ Configuration space in a subset $\Lambda \subset
\Z$.
$$\Omega_\Lambda \, = \,S^\Lambda $$
\indent
\ thus an element $\sigma_\Lambda $ in $\Omega_\Lambda $ is a
function
$$\sigma_\Lambda : \Lambda \to S$$
\indent
\item{-}\ \ Configuration space in the whole $ \Z$:
$$\Omega \,=\, S^{\Z}$$
\indent
\ thus an element $\sigma$ in $\Omega$ is a function
$$\sigma : \Z\to S$$
\item{-}\ \ $\sigma_x \equiv\sigma(x)$ is called value of the
spin at the site $x\in\Lambda $ in the configuration
$\sigma$.\bigskip
\item{-}\ \ By $|X|$ we denote the cardinality of
$$X\subset \subset \Z$$\indent
\ \ (we write \ $X \subset \subset \Z$ to express that
$X $ is a {\it finite} subset of $ \Z$)
\bigskip
\item{-}\ \ Potential $U \,=\,$ family of functions indexed by finite
sets in $ \Z$
$$U \,=\, \{U_{X},\ X \subset \subset \Z\}$$
\indent
\ where, for every finite $X$,
$$U_{X}\ :\ \Omega_{X}\to \ R$$
On the potential U we will always assume the following hypotheses:
\bigskip\indent
{\bf H1}.\ \ {\it Finite range : $\exists \quad r > 0 :
U_{X}\equiv 0$ if diam$X > r$ (we use Euclidean distance).}
\bigskip\indent
{\bf H2}.\ \ {\it Translation invariance }
$$\forall X \subset \subset \Z\quad\forall k\in \Z\quad
U_{X+k} \,=\, U_{X}$$
\item{-}\ \ Given $\Lambda \subset \Z$ and
$\tau\in\Omega_{\Lambda ^c}
\;(\Lambda ^c \,=\, \Z\setminus\Lambda )$, for every
$\sigma\in\Omega$
we denote by $\sigma\tau$ the configuration obtained from
$\sigma$ by
substituting $\tau$ to $\sigma$ outside $\Lambda $:
$$\eqalign{(\sigma \tau)_x \,&=\, \sigma_x\quad \forall \; x\in\Lambda\cr (\sigma \tau)_x \,&=\,
\tau_x\quad \forall \; x\in\Lambda ^c} \Eq(1.1)$$
\item{-}\ \ Given a set $\Lambda \subset \subset \Z$, a
{\it boundary
condition}, (b.c.), is a configuration
$$\tau\in\Omega_{\Lambda ^c} $$
\item{-}\ \ Given $\Lambda \subset \subset \Z$ the
{\it energy} associated to a
configuration $\sigma\in\Omega_\Lambda $ when
the boundary condition outside $\Lambda $ is $\tau\in\Omega_
{\Lambda ^c}$ in given by :
$$H^\tau_\Lambda (\sigma) \,=\, H_\Lambda (\sigma | \tau) \,=\,
\sum
_{X:X\cap\partial\Lambda \ne\emptyset}U_{X}((\sigma\tau)_{X}) \Eq(1.2)$$
\indent\ \ because of the hypothesis H1,
$ H^{\tau}_{\Lambda}(\sigma)$ depends
only on $\tau_x$ for $x$ in $\partial^+_r\Lambda$ :
$$\partial^+_r
\Lambda \,=\, \{x\not\in\Lambda : \hbox{dist} (x, \Lambda )\leq
r\} \Eq(1.3)$$
\item{-}\ \ The {\it Gibbs measure} in
$\Lambda $ with b.c. $\tau\in\Omega_{\Lambda ^c}$ and inverse
temperature $\beta > 0$ is
$$\mu_\Lambda ^\tau(\sigma) \,=\, {\exp(-\beta H^\tau_\Lambda
(\sigma))\over Z^\tau_\Lambda } \Eq(1.4)$$
\ \ The normalization factor, called {\it partition function} is given
by
$$Z^\tau_\Lambda \,=\,\quad\sum_{\sigma\in\Omega_\Lambda }
\exp(-\beta H^\tau_\Lambda (\sigma)) \Eq(1.5)$$
\ \ If there exists a unique limiting
Gibbs measure for $\Lambda \to
\Z,$ independent on $\tau$, it will be denoted by $\mu$ .\bigskip
\item{-}\ \ The {\it variation distance} between two probability
measures
$P, Q$ on a finite set $Y$ is :
$$Var (P, Q) \,=\, {1\over 2} \sum_{y\in Y}\vert P(y) - Q(y)
\vert \,=\,\sup_{X \subset Y}\vert P(X) -
Q(X)\vert \Eq(1.6)$$
\item{-}\ \ Given a metric $\varrho(\cdot, \cdot)$ on a finite
space $Y$
(a much more general framework can also be considered) the
{\it Kantorovich - Rubinstein - Ornstein - Vasserstein distance with
respect to $\varrho$} between two probability measures $\mu_1,
\mu_2$ on $Y,$ that we denote by $ \hbox{KROV}_\varrho(\mu_1,
\mu_2)$, is defined as
$$\hbox{KROV}_\varrho(\mu_1, \mu_2) \,=\,
\inf_{\mu\in
K(\mu_1, \mu_2)}\sum_{y, y'\in Y} \varrho(y, y')\mu(y , y') \Eq(1.
7)$$ \noindent
where $K(\mu_1, \mu_2)$ is the set of joint representation of
$\mu_1 ,\mu_2$ namely the set of measures on the cartesian product
$Y \times Y$ whose marginals with respect to the factors are,
respectively, given by $\mu_1 , \mu_2$. Namely we have, $\forall B \subset
Y$:
$$\eqalign{
&\mu(B\times Y) \,=\, \sum_{y\in B\, y'\in Y}\mu(y , y')\,=\, \mu_1(B)\cr &\mu(Y \times B) \,=\, \sum_{y\in Y\,y'\in B}
\mu(y , y') \,=\, \mu_2(B)}$$
\noindent
For the particular case
$$\varrho(y , y') \,=\, 1\quad \hbox{iff }y\ne y'\;\hbox{ and }
0\; \hbox{ otherwise} \Eq(1.8)$$
\noindent
$\hbox{KROV}_\varrho(\cdot,\cdot)$ coincides with the variation
distance $ Var(\cdot,\cdot)$.\bigskip
\item{-}\ \ 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
$$\mu_{\Lambda , \Delta}(\sigma_\Delta) \,=\, \sum_{\sigma_
{\Lambda \setminus\Delta}}\mu_\Lambda
(\sigma_{\Lambda \setminus
\Delta}, \sigma_\Delta) \Eq(1.9)$$
\noindent \bigskip
Next we define the stochastic {\it jump} dynamics, given by
a continuons time Markov process on $\Omega \,=\, S^{ \Z},
$ that will be studied in the sequel. Discrete time versions
can also be
considered.\par
Given $\Lambda \subset \subset \Z$ let
$$D (\Lambda) \,=\, \{f: \Omega \to \ R :
f (\eta) \,=\, f (\sigma)
\ \hbox{if}\quad
\eta_x \,=\, \sigma_x \ \forall x \in \Lambda \} $$
be the set of {\it cylindrical functions} with support $\Lambda.$
The set
$$ D\,=\, \cup_{\Lambda} D (\Lambda)$$
is the set of cylindrical functions and by $C(\Omega )$ we denote the
set of all
continuons function on $\Omega \,=\, \Pi_x S_x$
with respect to the product topology of discrete topologies
on $S_x$.
\medskip
The dynamics is defined by means of its {\it generator} $L$ which is given, for
$f \in D$, by:
$$L f (\sigma) \,=\, \sum_{x,a} c_x (\sigma, a)
\bigl ( f (\sigma^{a,x})-f (\sigma) \bigr )\Eq(1.10)$$
where $\sigma^{a,x}$ is the configuration obtained from $\sigma$
by setting the spin at $x$ equal to the value $a$ and the
non-negative
quantities $c_x (\sigma, a) $ are called ``jump rates''.
\medskip
The general hypoteses on the jump rates, that we shall always
assume,
are the following ones.
\bigskip
{\bf H3.}\ {\it Finite range $r$. This means that if $\eta (y) \,=\, \sigma
(y) \quad \forall\, x,y :\; |y - x| \leq r$ then \indent \ $c_x (\sigma, a) \,=\,
c_x (\eta , a)$}
\bigskip
{\bf H4.}\ {\it Translation invariance. That is if $\eta (y) \,=\, \sigma
(y+x) \quad \forall \,y\;$ then $\;c_x (\sigma, a) \,=\, c_x (\eta, a)$
}\bigskip
{\bf H5.}\ {\it Positivity. There exists a positive constant $k$ such
that
$ \inf_{\sigma, x, a} \ c_x (\sigma, a )
\geq\, k > 0$ }
\bigskip
For reasons that will be clear in the sequel it will be important
for us to
consider also the Markov processes associated to the above described
jump rates
in a {\it finite volume} $\Lambda$ with boundary conditions
$\tau$
outside $\Lambda.$ By this we mean the dynamics on
$\Omega_\Lambda$
generated by $L^\tau_\Lambda$ defined as before starting from the
jump rates
$$c^{\tau, \Lambda}_x (\sigma, a) \equiv c_x (\sigma \tau, a)$$
where, given $\tau \in \Omega_{\Lambda ^c}\; \; \sigma \in
\Omega_\Lambda \; \hbox{and }\sigma \tau$
has been defined in \equ(1.1) .\par
It is well known (see $[L]$) that under the above conditions
$L$ ($L^\tau_\Lambda$) generates
a unique positive contraction semigroup on the space $C(\Omega )$ (
$C(\Omega_\Lambda )$) that
will
be denoted by $T_t$ or $T_t^{\Lambda ,\tau}$ .\par
Sometimes we will use the more
probabilistic notation
$E_\sigma f (\sigma_t)$ for $T_t f (\sigma)$
where $\sigma_t$ denotes the Markov process generated by $L$
at time $t$ and $E_\sigma (\cdot )$ denotes
expectation starting from the configuration $\sigma$.\par
It is also easy to see, using positivity, that in finite volume there
exists a unique invariant measure that will be denoted by $\nu_\Lambda^\tau$ .\par
\medskip
In this paper we will mostly consider attractive dynamics.
{\it Attractivity}
is an important property enjoied by some interesting spin dynamics and it can be formulated as follows:
\bigskip
{\bf H6.}\ {\it Attractivity: \ If $\sigma (x) \ge \eta (x)$
for all $x$ then:}\bigskip
\centerline{\it If a $\leq \eta (x) $ then $ \sum_{b \leq a}
c_x (\sigma, b) \leq \sum_{b \le a} c_x (\eta, b)$}\medskip
\centerline{\it If $a \geq \sigma (x) $ then $ \sum_{b \geq a}
c_x (\sigma, b) \geq \sum_{b \ge a} c_x (\eta, b)$}
\bigskip
It is easy to show (see [L]) that attractivity is equivalent to the following condition on the
semigroup
$T_t:$ 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
the Markov semigroup $T_t$ leaves invariant the set of increasing
(decreasing) functions w.r.t. the above partial order.
\medskip
Another important class of spin dynamics on $\Omega$,
generally called Glauber dynamics, are those which are
{\it reversible} with respect to an a priori given Gibbs measure $\mu$
(in finite
or infinite volume). We will say that the generator $L$ \equ(1.10)
is {\it reversible}
with respect to a Gibbs measure $\mu$ corresponding to a hamiltonian
$H(\sigma)$ iff:
$$\hbox{exp}(-\beta\sum_{ X \ni x}U_X(\sigma_X))c_x(\sigma ,a)\;=\;
\hbox{exp}(-\beta\sum_{X \ni x}U_X((\sigma^{x,a}_X)))
c_x(\sigma^{x,a} ,\sigma_x)\quad \forall\,x\in \L\Eq(1.11)$$
A similar equation holds in finite volume $\Lambda$ with
boundary conditions $\tau$,
provided that we replace in \equ(1.11) $\sigma$ with the configuration $
\sigma \tau$. It is immediate to check that in finite volume
\equ(1.11) implies that
the unique invariant measure of the dynamics coincides with the Gibbs
measure
$\mu^\tau_\Lambda$. This important fact holds also in infinite volume
provided that there exists a unique Gibbs measure in the
thermodynamic limit. In the sequel such kind of dynamics will be
referred to as Glauber dynamics.\par
\item{-}\ \ Finally we recall the definition of
{\it stochastic
Ising models} that will be analyzed in section 4.
They are stochastic processes on $\Omega$,
reversible with respect
to the Gibbs measure of an Ising--like spin system
$( S \,=\, \{-1,1\})$.
\medskip
To introduce them it is enough to define the class of their
Hamiltonians. They will be of the form given in \equ(1.2) with
$$
U_X\,=\,-J_X \Pi_{x \in X} \sigma_x \Eq(1.12)
$$
and $ J_X \in \ R $\par \bigskip
\item{-}\ \ We say that an Ising spin system is
{\it ferromagnetic} if
the local field at the origin
$$h(\sigma )\;=\;\sum_{X;\,0\,\in\,X}J_X\Pi_{x \in X\setminus\{0\}}\sigma_x
\Eq(1.13)$$
is an increasing function of the spins $\sigma_x\quad x\,\neq\,0$
\par
The condition to be ferromagnetic is easily
seen to be implied by the
following more usual condition on the interaction $J_X $
(see e.g. [FKG]) which ensures the
validity of the F.K.G. inequalities for the Gibbs state. \par
Let us
introduce the lattice gas variables $\rho _x\,=\,{1+\sigma_x\over
2}$ and write the Hamiltonian $H(\sigma)$ as
$$\bar H(\rho)\;=\;-\sum_{X\cap\Lambda\,
\neq \, \emptyset}\Phi_X \Pi_{x \in X} \rho_x\Eq(1.14)$$
If the new potential $\Phi _X$
is non-negative for any set $X$ consisting of more than one
points then
the system is ferromagnetic.\par
It is easy to check that
in the case of only two body interaction the
system is ferromagnetic iff $J_{(x,y)}\;\geq\;0$.
\numsec=2\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Section 2}\par\noindent
\centerline{\bf Critical Analysis of}
\centerline{\bf Finite Volume Mixing Conditions}\bigskip
In this section we will critically review the existing notions of
mixing for {\it finite volume} measures and their implications
for the Gibbs state in the thermodynamic limit as well as for the
rate of convergence to equilibrium of an associated Glauber
dynamics.\medskip
We will distinguish between {\it strong } and {\it weak}
finite volume mixing conditions.
Both notions
can be expressed as weak dependence, inside $\Lambda$, say in $x\in
\Lambda$, on the value of a conditioning spin, say in
$y\in \partial^+_r \Lambda$.
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 $y$.
\par
\bigskip
STRONG MIXING.
\medskip
Strong mixing properties of measures are naturally expressed in
terms of truncated expectation.\par
A mixing condition of {\it strong type} for a measure
$\mu_\Lambda $ on $\Omega_\Lambda $ is a relation of the
form:\medskip For every pair of {\it cylindrical functions} $f\, g$
with supports $S_f, S_g \subset\Lambda $ there exists a
constant $C_{ f, g}$ such that
$$\vert\mu_\Lambda (f, g)\vert \equiv\vert\mu_\Lambda
(fg) -\mu_\Lambda
(f)\mu_\Lambda (g)\vert\leq C_{f, g}\exp(-\gamma \hbox {dist}(S_f,
S_g))\Eq(2.1)$$
\noindent
for some $\gamma > 0$\par
For example $C_{f, g}$ can be given by
$$C_{f, g} = C\vert\vert f\vert\vert\ \vert\vert g\vert\vert\, |S_f|
|S_g|$$
with
$$\vert\vert f\vert\vert =\sup_{\sigma}\vert f(\sigma)
\vert$$
and $C$ independent of $f, g.$\par
A mixing condition in the present form is not particularly
meaningful;
the exponential function in the $r. h. s.$ of Eq. 10.
is just a way of parametrizing the dependence between the
variables.\par
Of course \equ(2.1) would become interesting if it was true for
arbitrarily large volumes $\Lambda $ with $\gamma$ independent of
$\Lambda , f, g$.\par
For instance, $\Lambda $ could be a generic element of a Van Hove
sequence in $\Z$. In this last case a condition like
the one given in \equ(2.1), uniform in $\Lambda $, is of corse at
least as strong as the corresponding infinite volume
analogue.\medskip\noindent \item{-}\ \ 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 $D, C, \gamma$ if for every
cylindrical functions $f, g$ with $S_f, S_g \subset \Lambda ,\ \
\hbox{diam} S_f, \hbox{diam} S_g \leq D$
$$\vert\mu_\Lambda (f, g)\vert\leq\ \ C\vert\vert f\vert\vert\ \
\vert\vert g\vert\vert\ e^{-\gamma \hbox{dist}(S_f, S_g)} \Eq(2.2)$$
\indent
and we denote it by $SMT(\Lambda , D, C, \gamma)$.\medskip
\noindent
\item{-}\ \ We
simply say that a Gibbs measure $\mu_\Lambda $ on $\Omega_\Lambda $
satisfies a {\it strong mixing} condition with constants $C, \gamma$
if for every subset $\Delta \subset \Lambda $:
$$ \sup_{\tau,\tau^{(y)} \in \Omega _{\Lambda^c}}
Var(\mu_{\Lambda , \Delta}^\tau\ \ \mu_{\Lambda ,
\Delta}^{\tau^{(y)}})\leq C e^{-\gamma \hbox{dist}(\Delta, y)}
\Eq(2.3)$$
where\quad $\tau^{(y)}_x = \tau_x$ for $ x\ne y$.\medskip
We denote this condition by
$ SM(\Lambda, C,\gamma)$\medskip
It is easy to prove ( see [SZ], proof of Eq. 3.4) that
$ SMT(\Lambda,r, C,\gamma)$ ($r$ is the range of the interaction)
implies that there exists $C' > 0$ such that
$ SM(\Lambda, C',\gamma)$ holds (notice that $\Lambda$ and $\gamma$
are unchanged).\par
As it has been initially discussed by Dobrushin and Pecherski [DP],
and more extensively by Dobrushin and Shlosmann [DS2], [DS3], the
assumption that $SM(\Lambda , C, \gamma)$ holds {\it for all (finite
or infinite)} volumes $\Lambda $ with uniform constants $C, \gamma$,
is equivalent to many other conditions of mixing as,
for instance, $SMT$
and analyticity
properties of the thermodynamical functions and correlation
functions always {\it for all} volumes.\par
To partially clarify
this point we give the main result of [DP]. \bigskip
{\bf Theorem 2.1.}\quad (Dobrushin-Pecherski [DP]) \par {\it If for some
metric $\varrho$ on $S$, every $\Lambda \subset \Z,\Delta \subset
\Lambda , \tau,\tau'\in\Omega_{\Lambda ^c}$
$$\hbox{KROV}_{\varrho_\Delta }(\mu^\tau_{\Lambda ,\Delta},
\mu^{\tau'}_{\Lambda ,\Delta})
\leq\,\sum_{x\in\Delta}\ \ \ \sum_{y\in\partial^+
_r\Lambda }\varphi(\vert x -
y\vert)\cdot\varrho(\tau_y , \tau'_y)\Eq(2.4)$$
\noindent
with
$\lim_{t\to\infty}\ \ \varphi(t) t^\alpha\to 0\quad,
\alpha > d^2$
\par
then it follows that there exist $C > 0,
\gamma > 0$ such that $SM(\Lambda , C, \gamma)$ holds {\it for
every $\Lambda $}.}\par\bigskip
Dobrushin and Shlosmann called {\it complete analytical interactions}
the class of potentials whose Gibbs measures in {\it any finite or
infinite volume } satisfies $SM(\Lambda , C ,\gamma)$ and proved a
result ( stronger than the above quoted Theorem DP)
of equivalence of $SM(\Lambda , C ,\gamma) , \; \forall \; \Lambda$
to some fifteen other mixing or analyticity conditions always
considering all (finite or infinite) volumes with arbitrary size and
shape . In their theory the arbitrariness of the volumes
involved seems to play a crucial role (see [DS2],[DS3]).
\par
An important concept introduced by Dobushin
and Shlosmann in [DS3] is the one of {\it constructive
condition}. Namely in suitable circumstances supposing only that a
condition like $S M (\Lambda , C, \gamma)$ is true for a suitable
{\it finite} family of regions $\Lambda $ is sufficient to garantee
that the same condition holds {\it for all} (finite or infinite)
$\Lambda $ that is it implies {\it complete analiticity}.\par
More generally, it is natural to introduce the notion of {\it
effectiveness } :
\medskip
\item{-}\ \ Given two
families $\Gamma,\Gamma'$ of subsets of $\Z$ a strong
mixing condition $SM(. , C, \gamma)$ is called $(\Gamma,
\Gamma')$-{\it effective} if, supposing that $SM(\Lambda , C,
\gamma)$ holds for any $\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\bigskip
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).\bigskip
WEAK MIXING. \bigskip
We want now to give an interesting notion of {\it weak mixing}.
\bigskip
\item{-}\ \ We say that a Gibbs measure $\mu^\tau_\Lambda $
satisfies a weak
mixing condition with constants $C , \gamma$ if for every subset
$\Delta \subset \Lambda $
$$\sup_{\tau, \tau'\in\Omega_{\Lambda ^c}}
\quad Var(\mu^\tau_{\Lambda , \Delta},
\mu^{\tau'}_{\Lambda , \Delta})\leq C\sum_{x\in\Delta ,\,y\in
\partial^+_r\Lambda }\hbox{exp}(-\gamma\vert x-y\vert)\Eq(2.5)$$
\noindent
We denote this condition by $WM(\Lambda , C, \gamma)$.\par\medskip
Condition \equ(2.5) implies:
$$\sup_{\tau , \tau'\in\Omega^c_\Lambda }\quad
Var(\mu^\tau_{\Lambda , x}\mu^{\tau'}_{\Lambda , x})
\leq C'\sum_{y\in\partial^+_r\Lambda }\ \
\exp(-\gamma'\vert x-y\vert)\Eq(2.5').$$
$\forall x\in\Lambda $ for suitable $C' > 0\ \gamma' > 0$.\par
It is easy to see that in the attractive (ferromagnetic) case
\equ(2.5) and \equ(2.5') are equivalent.\par A similar weak mixing
condition, that we call $WM_\varrho(\Lambda , C, \gamma)$, is given
in the following way.\par Suppose $\varrho(\cdot, \cdot)$ is a
metric on the single spin space $ S$.\par Given $\Lambda
\subset \subset \Z$, let $\varrho_\Lambda (\cdot, \cdot)$ be the
metric on $\Omega_\Lambda $ given by
$$\varrho_\Lambda (\sigma_\Lambda , \sigma'_\Lambda ) = \sum_{x
\in\Lambda }\varrho(\sigma_x, \sigma'_x)\Eq(2.5*)$$
\noindent
We say that $WM_\varrho(\Lambda , C,\gamma)$ holds if
$\forall\Delta \subset \Lambda $
$$\sup_{\tau, \tau'\in\Omega_{\Lambda ^c}}\ \
\hbox{KROV}_{\varrho_\Lambda }(\mu^\tau_{\Lambda ,\Delta},
\mu^{\tau'}_ {\Lambda , \Delta})\leq
C\sum_{x\in\partial^+_r\Lambda }\ \ \exp(-\gamma
\hbox{dist}(x, \Delta))\Eq(2.5'').$$ \noindent
It is immediate to see that, when $\varrho$ is given by (1.8)
$WM_\varrho(\Lambda , C, \gamma)$ implies the validity of
the bound given by \equ(2.5) with {\it the same} constant
$ \gamma$ (but with a different $C$) so that the validity of
$WM_\varrho(\Lambda , C, \gamma)$ implies, in that case, that :
$\exists C' : WM(\Lambda , C', \gamma)$ is satisfied.\par
\noindent
It is immediate to see that $SM(\Lambda , C, \gamma)$ implies
$WM(\Lambda , C, \gamma)$ . The converse is not true.
There exist potentials, the so called Czech
potentials,(see [DS1], [Sh]) which satisfy
$WM(\Lambda , C, \gamma)$
but do not satisfy $SM(\Lambda , C, \gamma)$, uniformly
on $\Lambda $ for any $C > 0\ \ \gamma > 0$.\par
These models, that in Dobrushin-Shlosmann's language are not
completely
analytical, exhibit a sort of boundary phase transition even thought
the phase in the bulk is unique.\par
It is expected that also for the standard Ising model for $d \geq 3$
at very low temperature and for special values of the magnetic field
(depending on the temperature) some ``layering phase transition''
involving long range order along the boundary takes place. This
analysis is due to Basuev [D1] .\par Nothing similar is expected in d
= 2 since, in that case, the boundary is one-dimensional [MOSh].
.\par
In the sequel, while critically analyzing the concept of
complete analyticity in the Dobrushin-Shlosmann's sense we will
exhibit some counterexamples, involving ``pathological'' shapes,
violating complete analyticity namely the validity of $SM(\Lambda , C
,\gamma)$ {\it for every }$\Lambda $. These models, however, satisfy
as we will see, some weaker form of strong mixing involving only
sufficiently regular shapes (see below) .\par
We can say that the way in
which the Czech models or the Ising model in the Basuev situation
violate complete analyticity is more ``intrinsic'' and it is
related to a real phase transition that, however, is not detected
inside the bulk.\par
In a paper in preparation, [MOSh], the authors of the present paper,
in collaboration with R.Schonmann, analyze the relations between
strong and weak mixing conditions and show that in two dimensions,
given $C > 0,\gamma> 0$ if $WM(\Lambda,C,\gamma)$ holds for a
sufficiently large square then $SM(\Lambda,C',\gamma')$ for
some $ C' > 0,\gamma' > 0$
holds {\it for all} cubes. \par
We want to notice, at this point, that $
WM(\Lambda,C,\gamma)$ implies not only uniqueness of limiting Gibbs
measure but also decay of infinite volume correlations (see [DS1])
. However, the example of Czech models shows that the
notion of exponential decay of finite volume correlations (uniformly
in the volume) namely, for instance, the validity of $
SM(\Lambda,C,\gamma)$ for some fixed $C>0,\gamma>0$ and {\it for any
cube} $\Lambda$ is {\it strictly stronger} than the corresponding
infinite volume property.\par
Finally we remark
that the above definitions can be extended to the
case of non-Gibbsian measures for which there is a natural notion of
imposing boundary conditions outside $\Lambda $ (See sect. 3)\par
\noindent \bigskip
{\bf Review of known results: the Gibbs state}\par
\medskip
Let us now review some of the known results concerning
finite size conditions and mixing properties of Gibbs measures.
We begin with a result by Dobrushin and Shlosmann concerning
uniqueness of infinite volume Gibbs measures.\par
This result generalizes previous results by Dobrushin based
on a ``one point condition'' on Gibbs conditional distribution.
(see [D2]).\par
First we need a definition.\medskip
\noindent
\item{-}\ \ Given a metric $\varrho$ on the single
spin space $ S$ we say that
condition $DSU_{\varrho} (\Lambda _0, \delta)$ is
satisfied if:\medskip
\ \ there exists a finite set $\Lambda _0
\subset \subset \Z$, a $\delta > 0$ such that:
$ \forall \tau , \tau' \in \Omega ^c_{\Lambda _0}$ with
$\tau'_x =
\tau_x \forall x \neq y$ and $\forall y
\in \partial ^+_r \Lambda _0$
there is a number $\alpha _y$ such
that :
$$\sup_{\tau , \tau'}\ \
\hbox{KROV}_{\varrho_{\Lambda_0}}(\mu^\tau_{\Lambda _0},
\mu^{\tau'}_{\Lambda _0})\leq\ \ \alpha_y\varrho(\tau_y, \tau'_y)
\Eq(2.6)$$ \noindent
where
$$\sum_{y\in\partial^+_r\Lambda _0}\alpha_y\ \ \leq
\delta\vert
\Lambda _0\vert\Eq(2.7)$$
\item{-}\ \ We simply say that $DSU (\Lambda _0, \delta)$ is
satisfied if \equ(2.6), \equ (7) hold with $\varrho$ given by Eq.
(1.8). We observe that, for this choice of $\varrho$, in the
ferromagnetic case we can substitute, in \equ(2.6) KROV with $Var$.
\bigskip
{\bf Theorem 2.2}\quad(Dobrushin - Shlosman [DS3])\par
{\it Let $DSU_{\varrho} (\Lambda _0, \delta)$ be satisfied for some
$\varrho$, $\Lambda _0$ and {\bf $\delta < 1$}; then $\exists\ \ C >
0, \gamma > 0$ such that condition $WM_\varrho(\Lambda , C, \gamma)$
holds {\it for every $\Lambda $}.}\par
\noindent
\bigskip
Notice that the result of the above theorem is valid {\it for
every} $\Lambda$ but obviously it loses interest when $\Lambda$ is
such that any point of $\Lambda$ is near to some point of
$\partial \Lambda$ (One can say that, in this case, the boundary
"penetrates" inside the bulk).
Examples of $\Lambda$'s with this kind of shapes will be analyzed
later on. One can apply Theorem 2.2 to, say, Van Hove sequences of
regions $\Lambda$.\par
\noindent
\bigskip
{\bf Remark }\quad Theorem 2.2 implies, in particular,
the uniqueness of infinite volume Gibbs measure. Then
\equ(2.6),\equ(2.7) provide an example of finite size condition : one
supposes true some properties of {\it finite volume} Gibbs measure
and deduces properties for {\it infinite volume} distributions.\par
\noindent
\bigskip
{\bf Remark }\quad One can see that $SM(\Lambda , C, \gamma)$
for {\it every} $\Lambda $ implies $DSU (\Lambda _0, \delta)$ with
$\delta < 1$ for a sufficently large $\Lambda _0$ (depending
on $C, \gamma$). It can be shown for the above mentioned
Czech models that even though they satisfy $WM(\Lambda , C, \gamma)$
for all cubes $\Lambda$, they violate $DSU (\Lambda _0, \delta)$
with $\delta < 1$ for any cube $\Lambda_0$
\bigskip {\bf Theorem 2.3}\quad
(Dobrushin-Shlosmann [DS3])\par {\it 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 $
\Z$.}\par \bigskip
{\bf Remark }\quad The above theorem requires to verify a strong
mixing condition for regions of {\it arbitrary} shape with
given maximal diameter and insures the validity of the strong
mixing {\it for any volume} (finite or infinite).
One can ask oneself whether or not it is reasonable to expect the
validity of the above notion of complete analyticity in the
Dobrushin-Shlosman's sense for the Ising model either in the whole
pure phase region or at least when Basuev phenomena are excluded:
for example for any given positive magnetic field, for all
sufficiently large inverse temperature $\beta$.\par The simplest
example where no phase transition of any kind takes place which,
however, violates complete analyticity (in its strong form) is
simply given by the usual $3D$ Ising model with coupling constant
$J=1$ \ $\beta$ larger than $2D$ critical value $\beta^{(2)}_c$
and $h=2$.\par
Consider a (horizontal) squared layer of size $L$ namely a
parallelepiped $(\equiv box)\;\Lambda$ with dimensions $L, L, 1$ in
the directions 1, 2, 3, respectively. Suppose to introduce -1
boundary conditions on the sites contiguous to $\Lambda$ from
direction 3 (namely the sites belonging to the $L \times L$ squared
layer adjacent
to $\Lambda$ from above and below). The effective field inside
$\Lambda$ is zero and since $\beta > \beta^{(2)}_c$ the spins
inside $\Lambda$ are very sensible to the value of the conditioning
external spins belonging to the same horizontal layer as
$\Lambda$.\par
Certainly for these value of thermodynamical parameter both
strong and weak mixing conditions are violated for these flat
regions.
However, as we will see later on and as it is very reasonable, one
can prove strong mixing for every $h > 0$ and
$\beta$ sufficient large {\it for any (arbitrarily large) cube}
and even {\it for a very wide class
of ``sufficently fat''} regions.\par
Another even more interesting example has been found by Roberto
Schonmann ([S]).\par
Consider a 2D ferromagnetic Ising model with nearest neighbours
and next nearest neighbours interactions whose hamiltonian in
the finite region $\Lambda$, with open b.c.
(no interaction with the exterior), is given by
$$H= -J\sum_{x, y\in\Lambda:\vert x-y\vert=1}\sigma_x
\sigma_y - K\sum_{x, y\in\Lambda:\vert x-y\vert=\sqrt 2}
\sigma_x\sigma_y - h\sum_{x\in\Lambda}\sigma_x$$
\noindent
where\quad $J = K = 1;\quad h = 4$\par
\bigskip
Consider the partition of $ {\bf Z^2}$ into two sublattices
$ E_o\ \equiv\ {\bf Z_{\sqrt 2, o}^2}, E_e\ \equiv
{\bf Z_{\sqrt 2, e}^2}
$ of spacing $\sqrt 2$ and directions at 45
degrees with respect to the original lattice directions
$$ E_o = \{x\equiv(x_1, x_2)\in {\bf Z^2} : x_1 + x_2 = \hbox
{odd}\}$$ \bigskip
$$ E_e = \{x\equiv(x_1, x_2)\in {\bf Z^2} : x_1 + x_2 = \hbox
{even}\}$$
consider the square $\Lambda$ with (oblique) edges parallel to the
axes of $ E_e, E_o$ {\it contained in $ E_e$} and
containing $(2L+1)^2$ points:
$$\Lambda=\{x\equiv(x_1, x_2)\in {\bf Z^2}:x_1+x_2=
\hbox {even},\ -L\leq
x_1+x_2\leq L, -L\leq x_1-x_2\leq L\}$$
\bigskip\noindent
The set of sites in $ {\bf Z^2}$ exterior to $\Lambda$ but
conditioning $\Lambda$, namely $\partial^+_{\sqrt 2}\Lambda$,
is given by
$$\partial^+_{\sqrt 2}\Lambda = \partial^+_o \cup\partial^+_e$$
\bigskip\noindent
$$\partial^+_{o, e}\Lambda = \partial^+_{\sqrt 2}\Lambda\cap
E_{o, e}$$
\bigskip\noindent
Notice that $\partial^+_o$ ``penetrates in the bulk''of $\Lambda$
whereas $\partial^+_e$ contains just the sites of $ E_e$
adjacent (at distance $\sqrt 2$) from the exterior to $\Lambda$.\par
Consider any boundary condition $\tau$ with -1 in $\partial^+_o
: \tau_{\partial^+_o} = -\underline 1$ .\par
In this way we reduce ourselves to a usual nearest neighbours Ising
model in a oblique square with zero effective field and boundary
condition simply given by $\tau_{\partial^+_e}$.\par
If $\beta$ is large enough our system will be, for every $L$,
sensitive to the boundary condition $\tau_{\partial^+_e}$
(first order phase transition) and then strong mixing condition
for this
particular sequence of regions $\Lambda$ will certainly fail.\par
Other interesting examples violating, for some pathological shapes,
DS complete analyticity (without exhibiting any
real phase transition) are provided by A.van Enter, R.Fernandez and
A.Sokal in the framework of their critical analysis of
renormalization group transformations [EFS].\par
We want to stress that for these counterexamples to the DS complete
analyticity it is essential to have chosen ``strange''
(pathological) shapes.\par Again one can see that in the above
examples strong mixing $SM(\Lambda, C, \gamma)$ holds true for some
$C > 0, \gamma > 0$ {\it for every regular} (without holes) box in
$ {\bf Z^2}$.\medskip In the context of studying properties of
approach to equilibrium of Glauber dynamics several authors:\par
Holley [H2], Aizenmann and Holley [AH], Stroock and Zegarlinski [SZ]
have considered relations between finite size conditions and
different types
of mixing conditions as those considered by Dobrushin and Shlosman
or similar ones.\par
We want to quote first a result by Holley:
one among many other results contained in the basic paper [H2].\par
Holley considers Ising spin systems enclosed in a particular kind
of regions: the boxes where:\medskip
\centerline{$\Lambda\subset\subset\Z$ is a {\it box} if it is the
cartesian
product of $d$ finite intervals in $ {\bf Z}$.}\par
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 $DSU(\Lambda_0,\delta)$
and it is given by: \par\bigskip
for every $x \in \Lambda_0$, $y
\in \partial^+_r\Lambda_0$ ,there exists $\bar \alpha _{x,y} >0$
such that for every box $\Lambda \subset \Lambda_0$ :
$$\sup_{\tau , \tau^{(y)}\in \Omega_{\Lambda^c}}
Var(\mu_{\Lambda}^{\tau}\ \ \mu_{\Lambda }^{\tau^{(y)}})\leq
\,\sum_{x\in \L}\bar \alpha_{x,y} $$ with
$$\sum_{x\in \Lambda_0 ,y\in\partial^+_r\Lambda
_0}\bar \alpha_{x,y}\ \ \leq \delta\vert
\Lambda_0 \vert$$
\par
\bigskip
{\bf Theorem 2.4}\quad . (Holley [H2])\par
{\it Consider a ferromagnetic Ising model.Then 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
A similar statement is contained in [AH]. A generalization
of Theorem 4.2 is due to Stroock and Zegarlinski and it is based on
a condition that we call $SZ(\Lambda_o,\delta)$. This condition
refers to an arbitrary finite subset $\Lambda_o\subset \subset
\Z$; it is exactly like $H(\Lambda_o,\delta)$ with " for every
box $\Lambda\subset \Lambda_o$" replaced by "for every
$\Lambda\subset \Lambda_o$". \bigskip
{\bf Theorem 2.5}.\quad (Stroock, Zegarlinski [SZ])\par {\it In the
general case (potentials satisfying hypotheses H1, H2) the
existence of a region $\Lambda_0$ such that $SZ (\Lambda _0,
\delta)$ holds with $\delta < 1$ is equivalent to the existence of
$C>0,\gamma>0$ such that $SM(\Lambda, C,\gamma)$ holds {\it for
every set} $\Lambda$}. \bigskip
{\bf Remark} Condition $SZ (\Lambda _0,
\delta)$, $\d\,<\,1$, is called in [SZ] condition DSM($\L_o$) and
it is erroneously attributed to Dobrushin-Shlosman. We want to
notice, at this point, that also Theorem 1 in [AH] (and the same in
[SZ]), even though it is attributed to Dobrushin-Shlosman, differs
both in the hypothesis and in the thesis from the analogous Theorem
2.2 (see Theorem 3.1 of [DS1]). The difference in the hypothesis is
the use of $Var$ (in Theorem 1 of [AH]) instead of $KROV$ (in Theorem
2.2). The difference in the thesis is in a prefactor, corresponding
to the boundary of the volume, in front of the
exponential (in
Theorem 1 of [AH]) and absent in Theorem 2.2 . We refer to [AH] and
[DS1] for more details.\par The proof of
Theorems 2.4 and 2.5 uses dynamical arguments similar, in spirit, to
the ``surgery'' methods of [DS1], [DS2], [DS3], which are based on
subsequent local modifications of joint representations of Gibbs
measures in a big volume $\Lambda$.\par It provides a very simple
way to deduce $SM(\L,C,\gamma )$ for every $\L$, starting from a
finite size condition, $SZ (\Lambda _0,
\delta)$, $\d\,<\,1$, that is easily seen to be implied by the
validity of $SM(\L,C,\gamma )$ for every $\L\subset \L_o$ for a
sufficiently large cube $\L_o$. Thus it provides an alternative proof of
Theorem 2.3.\par
Holley's argument of
proof takes into account all the translates $\Lambda _o(x)$ of
the basic cube $\Lambda _0$ (of edge $L$ ) in $\Lambda $ ; here $x$
is a vector in $\Z$ not necessarily of the form: $x=Ly,\; y \in
\Z$. In this case sometimes it happens that $\Lambda_o (x) \cap
\Lambda$ is not a cube but, rather, a box and this leads to the
consideration of properties of a Gibbs measure in an arbitrary box
subset of $\Lambda_0$.\bigskip
Now we want to introduce a last condition that we call
$K(\L_o,\d)$, very similar to $SZ(\L_o,\d)$ (and also to
$H(\L_o,\d)$). It is exactly $SZ(\L_o,\d)$ with the substitution of
$Var$ with $KROV$:\bigskip
\item{-} \ \ Condition $K(\L_o,\d)$:\par
for $\L\subset \subset \Z$ let $\rho_\L$ be given by (2.7) where,
for simplicity, we choose $\rho$ as in (1.8). Then for every $x \in
\Lambda_0$, $y \in \partial^+_r\Lambda_0$, there exists $\bar
\alpha _{x,y} >0$ such that for every $\Lambda \subset \Lambda_0$ :
$$\sup_{\tau ,\tau^{(y)}\in \Omega_{\Lambda^c}}
KROV_{\rho_{\L}}(\mu_{\Lambda
}^{\tau}\ ,\ \mu_{\Lambda }^{\tau^{(y)}})\leq
\,\sum_{x\in \L}\bar \alpha_{x,y} $$
with
$$\sum_{x\in \Lambda_0 ,y\in\partial^+_r\Lambda
_0}\bar \alpha_{x,y}\ \ \leq \delta\vert
\Lambda_0 \vert$$
Then we have:\bigskip
{\bf Theorem 2.5'}.\par {\it In the
general case (potentials satisfying hypotheses H1, H2) the
existence of a region $\Lambda_0$ such that $K (\Lambda _0,
\delta)$ holds with $\delta < 1$ is equivalent to the existence of
$C>0,\gamma>0$ such that $SM(\Lambda, C,\gamma)$ holds {\it for
every set} $\Lambda$.} \bigskip
Theorem 2.5' is very similar to Theorem 2.5 and it can also be
considered as another proof of theorem 2.3.\par
In Appendix 3 we give a
proof of Theorem 2.5' which hopefully will shed some light of why
hypothesis $K(\L_o,\d)$ with $\d\,<\,1$,
or the similar
conditions $H(\L_o,\d)$ and $SZ(\L_o,\d)$, are
so natural within the Dobrushin-Shlosman'approach to
complete analiticity. We will show that Theorem 2.5', and in a
proper sense also Theorem 2.4 and 2.5, can be reduced to a corollary
of Theorem 2.2: one finds in fact that $K(\Lambda,\delta)$ is the
correct strengthening of $DSU(\Lambda,\delta)$ that is needed in
order to show $SM$ instead of $WM$. The price to pay, in this way,
is to consider regions with arbitrary shape. \par \bigskip In the
work of Olivieri [O] and Olivieri and Picco, [OP], an approach to
the same problem, substantially different with respect to the one of
Dobrushin and Shlosman, was developed; it uses a block decimation
procedure and the theory of cluster expansion; it can be considered
as the analogue, for a suitable class of regular domains, of the DS
theory of complete analytical interactions ( that, we repeat, is
intrinsically formulated in terms of arbitrary shapes).
\par Here (see Appendix 2) we propose a further
simplification of the assumptions and statements of [O], [OP]. In
this formulation it is sufficient to assume Strong Mixing {\it only}
for a suitable {\it cube} in order to ensure the same property for
{\it any multiple} of this cube. Let us give the corresponding
definitions.\par Given the odd integer $L$ let
$$Q_L(x)=\{y\in\Z;|x_i-y_i|\leq\,{L-1\over 2} , \; i=1\dots ,
d\}$$
be the cube of edge $L$ centered at $x$.\par
We say that $\Lambda$ is a {\it multiple} of the cube $\Lambda_0 =
Q_L(0)$ if it is a union of translated cubes $Q_L(x)$ with disjoint
interior :
$$\Lambda = \cup_{y\in Y} Q_L(L\ y)$$
for some $Y\subset \Z$\par
\bigskip
{\bf Theorem 2.6}\quad(Olivieri, Picco [O], [OP]).\par
{\it In the general case (hypotheses H1, H2 satisfied) $\exists\
L=L(C,\gamma)$ such that $SM(\cdot, C,\gamma)$ is
$(\Gamma,\Gamma')$-effective where $\Gamma$ consists just in the cube
$\Lambda_0 = Q_L(0)$ and $\Gamma' $ is the class of all {\it
multiples} of $\Lambda_0$. }\bigskip
A proof of the theorem in this form (a corollary of
Propositions 2.5.1,...,2.5.4 of [OP]) can be found in Appendix 2.
For an alternative dynamical proof see section 4.
\bigskip
{\bf Remark}\par It is easy to see that $\Gamma'$ can be extended
to contain all properly defined ``sufficiently fat'' regions.\par
\bigskip
{\bf Remark}\par
The approach in [OP] makes use of a somehow complicated geometrical
construction and of a suitable polymer expansion; it proves not
only effectiveness but also analyticity properties (similar to the
ones proved by DS in the case of their completely analytical
interactions) by expressing any quantity of interest, referring to
an arbitrary volume $\Lambda$ multiple of $\Lambda_0$, in terms of
a series expansion which is convergent by virtue of the assumed
finite size condition on $\Lambda_0$
.
It is remarkable that the proof of the effectiveness alone can be
given by avoiding this complicated approach and relying only on
simple dynamical arguments.
\par
\bigskip
The OP theory, by omitting the consideration of arbitrary regions (
practically excluding {\it only} pathological shapes), can be
successfully applied near to the coexisting line corresponding to a
first order phase transition where the previous DS theory
failed.
In particular for the Schonmann's example one can immediately
show complete analyticity in the above (weaker) sense
(other examples will be discussed in section 5).\par
\par \bigskip
{\bf Review of known results: the dynamics}\par \bigskip
In what follows we define various different notions of exponential
convergence to equilibrium for the stochastic
spin dynamics defined in section 1. As we have already explained in
the introduction one has to carefully distinguish among the various
notions if one wants to derive results in a region of the phase
diagram very close to a phase transtion line. In what
follows we will assume that there exists a unique
invariant measure $\mu$\medskip \item {1)} Exponential convergence
in $L^2$ for the infinite volume dynamics. We denote it by $$EC,
L^2 (d \mu), \Z$$ It means that there exists $\gamma > 0$ such that
$\forall f \in L^2(d\mu ) :$
$$\Vert T_t f - \mu (f) \Vert_{L^2
(\mu)} \leq \Vert f - \mu (f) \Vert_{L^2 (\mu)} \ e^{- \gamma t}$$
\item {2)} Uniform $(L^\infty)$ exponential convergence
for infinite
volume dynamics, denoted by
$$UEC, \Z$$
It means:
$$\exists \gamma > 0: \ \forall f \in D \
\exists \ \ C_f >0:$$
$$\Vert T_t f - \mu (f) \Vert_u \leq C_f \ e^{- \gamma T}$$
namely
$$\sup_ {\sigma} \ \vert E_\sigma
f (\sigma_t)
- \mu (f) \vert \le C_f \ e^{- \gamma t}$$
\item {3)} Exponential convergence in $L^2$ for finite
volume dynamics in
$\Lambda$ uniformly in $\Lambda\,\in\,\Gamma$ and in the b.c.
$\tau $ namely:
$$\eqalign{ \exists \gamma > 0 \ : \ \forall \ \Lambda \in \Gamma,
& \forall \tau \in \Omega^c_{\Lambda},\forall f \in
\,L^2(d\mu_\Lambda^\tau ):\cr \Vert T^{\Lambda, \tau}_t f -
\mu^\tau_\Lambda (f) \Vert_{L^2(\mu^\tau_\Lambda)} &\leq
\Vert f - \mu^{\tau}_{\Lambda} (f) \Vert_{L^2(d\mu_{\Lambda}^{\tau}
)} \ e^{-\gamma t}}$$ We denote it by
$$EC, L^2 (\mu^\tau_\Lambda)\quad \forall
\Lambda \in \Gamma$$
\medskip
\item {4)} Uniform exponential convergence for finite
volume dynamics in
$\Lambda$ uniformly in $\Lambda$ varying in a class
$\Gamma$ and in the
b.c. $\tau$; namely :
$$\exists \gamma > 0\ \ : \ \forall f
\in D(\Lambda) \ \exists C_f > 0 :$$
$$\sup_{\tau\in \Omega_{\Lambda^c}}\Vert T^{\Lambda, \tau}_t f -
\mu^\tau_\Lambda (f) \Vert_u \leq C_f \ e^{- \gamma t} $$
We denote it by
$$UEC, \ \ \forall \Lambda \in \Gamma$$\par
Many authors and in particular Holley investigated the
relationshisp between
the above (and other) notions of convergence;
on the other hand, for the
case of dynamics reversible with respect to Gibbs
measures like Stochastic
Ising Models, they studied the
relations between the speed of approach to
equilibrium and mixing properties
of invariant Gibbs measure.\par
In particular the problem of deducing exponential
approach to equilibrium
(in the different above senses) from {\it finite size}
condition
involving properties of finite volume Gibbs measure
has been recently the object of many studies.\par
The following theorems 2.7, 2.8 hold for the attractive case.\par
\bigskip
{\bf Theorem 2.7}\quad (Holley [H2])\par
{\it Suppose that there exists a cube $\Lambda_0$ such that
$H (\Lambda_0, \delta)$, with $\delta<1$, holds; then $UEC,
\Z$ 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 2.7
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 2.8}\quad (Aizenmann,Holley [AH])\par
{\it If there is a cube $\Lambda_0$ such that
$DSU (\Lambda _0, \delta)$ is satisfied with $\delta<1$, then
$EC, L^2 (d \mu), \Z$ holds.}\par\bigskip
\noindent
For the general, not necessarily attractive, case we want to quote
the following Theorem, due to Stroock and Zegarlinski, obtained in
the framework of the theory making use of logarithmic Sobolev
inequalities.\par \bigskip
{\bf Theorem 2.9} \quad ( Stroock, Zegarlinski [SZ])\par
{\it The following statements are equivalent
\item {i)} There exists a finite region $\Lambda_0$
such that $SZ(\Lambda_0,\delta)$ holds with $\delta < 1$.
\item {ii)}
$UEC$ {\it for every} $\Lambda$ holds.
\item{iii)} $EC$, $L^2(d\mu_\Lambda^\tau)$ {\it for every} $\Lambda$
holds.}\par \bigskip
Notice that, by Theorem 2.5, points i), ii), iii) of theorem 2.9
are also equivalent to the existence of $C >0,\gamma > 0$ such
that $SM(\Lambda,C,\gamma)$ holds {\it for every set} $\Lambda$.
\par
Following the previously developed critical analysis it is
reasonable to try to prove a theorem being the analogue of Theorem
9.2 for some class of sufficiently regular regions.
In particular, giving up with the consideration of {\it every} shape,
one would like to substitute point i) of Theorem 2.9 with a finite
size condition referring {\it only} to a cube (for example
$SM(Q_L,C,\gamma)$ for $L$ chosen sufficiently large in terms of $C
,\gamma$) and, moreover, to substitute "for every set" in Theorem
2.5 and in ii), iii) of Theorem 2.9 with: "for every multiple of
$Q_L$".\par
Finally, in analogy with the case of equilibrium
statistical mechanics, it is reasonable to expect that $UEC \;
\forall \; \Lambda$ or even $UEC \; \forall \; \Lambda \; \in
\Gamma$ with $\Gamma \; \equiv $
class of regular domains ( for example van Hove) is a strictly
stronger notion than $UEC \; \Z$. Thus it is conceivable to look for
some theorem stronger than, for example, Theorem 2.7, and such that
the statement: "validity of $UEC \; \Z$ " follows {\it only} from
some hypothesis {\it strictly weaker} than $H(\Lambda,\delta) , \,
\delta <1$: this hypothesis should not imply $EC\,
L^2(d\mu_\Lambda^\tau )\quad \forall\;\hbox{box }\Lambda$
otherwise $H(\Lambda,\delta) , \,
\delta <1$ would follow via Lemma 3.1 of [SZ].\par In the present
paper and in [MO] we develop the above sketched program by
positively answering to the above quoted open questions.
\numsec=3\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Section 3}\par\noindent
\centerline{\bf Exponential convergence to
equilibrium}
\centerline{\bf under a weak mixing condition}\bigskip
In this section we state and prove the first one of our main
results, namely
that for an attractive spin systems, not necessarily reversible with
respect to a Gibbs measure, a {\it weak mixing} condition on the
invariant measure of the dynamics implies exponential
convergence in a {\it strong sense} for the dynamics in the whole
lattice $\bf Z^d$ to its equilibrium measure.\par
The infinitesimal generator L of our spin dynamics on $\Omega$ is
given by (1.10) and the hypotheses on the jump rates are those
discussed in section 1 : H3, H4, H5, H6 namely finite range,
positivity, translation invariance and attractivity. However we
{\it do not} require the detailed balance condition with respect to
some Gibbs measure (1.11). For notation convenience we will denote by
$\mu_{\Lambda}^+$ and $\mu_{\Lambda}^-$ the invariant measures
obtained using as boundary conditions the extreme configurations
identically equal to the maximum value N and to the minimal value 1
of the spins, respectively; moreover any given realization of our
Markov process on $\Omega$ at time t will always be denoted by
$\sigma_t$ independently of the starting point. The latter will
always be specified in the expectation value of the observables over
$\sigma_t$ e.g. $E_\xi f(\sigma_t)$ if the starting point was the
configuration $\xi$. We will also denote the average of an arbitrary
function f with respect to a measure $\nu$ on $\Omega$ or
$\Omega_{\Lambda}$ by $\nu (f)$.\par It is easy to check, using
attractivity, that if f is an increasing cylindrical function with
support inside the finite set $\Lambda$ then the following
inequalities holds:\bigskip \item{a)} If $\tau\,\leq\,\tau '$ then
$T_t^{\Lambda ,\,\tau}(f)\;\leq \; T_t^{\Lambda ,\,\tau '}(f)$
\item{b)} If $\Lambda \;\subset \; \Lambda '$ and if $\tau
(x)\,=\,N$ for all x in $\Lambda '\setminus \Lambda$ then
$T_t^{\Lambda ',\,\tau}(f)\;\leq \; T_t^{\Lambda ,\,\tau}(f)$
\item{c)} $T_t^{\Lambda ,\,+}f(\xi )\;\geq \;T_tf(\xi )$ if f is an
increasing function.\bigskip {\bf Remark} Clearly by taking the
limit as $t\,\to\,\,\infty$ analogous inequalities hold for the
invariant measures. The invariant measure in finite volume is unique
because of the positivity of the jump rates.\bigskip
We
formulate now a condition on
the finite volume invariant measure which ensures the ergodicity of
the infinite volume Markov process and the exponential convergence
of its distribution at time t to the unique invariant measure as t
$\to \, \infty$. Such a condition, in analogy to the weak mixing
condition for Gibbs states, will also be called {\it weak
mixing}.\par We recall that $Q_L(x)$ is the cube of side L,
L odd, centered at x; we will write $Q_L$ for $Q_L(0)$.\bigskip {\it
Weak mixing}\par There exist two positive constants C and $\epsilon$
such that for any integer L $$\mu_{Q_L}^{+}(\sigma
(0))\;-\;\mu_{Q_L}^{-}(\sigma (0)) \;\leq
\;C\hbox{exp}(-\epsilon L)\Eq(3.1)$$ {\bf Remark } One sees
immediately that the above mixing condition implies that there
exists a unique invariant measure for the Markov process on $\Omega$
that will be denoted by $\mu$.\bigskip Our main result then reads as
follows: \bigskip {\bf Theorem 3.1}\par {\it The following are
equivalent: \item{\bf i)} {\it weak mixing}
\item{\bf ii)} There exists a positive constant $m$ and for any
cylindrical function f there exists a constant $C_f$ such that:
$$\sup_{\xi} \vert T_t(f)(\xi )\, -\, \mu (f)\vert \; \leq \;
C_f\hbox{exp}(-mt)$$
namely $UEC\, ,\,\Z$ holds.}
\bigskip
{\bf Proof}\par
{\bf i) $\Rightarrow $ \bf ii)}\par
Let us define
$$\rho (t) \;=\; E_+(\sigma_t(0))\;-\;E_-(\sigma_t(0))\Eq(3.2)$$
where
$E_+()$ and $E_-()$ denote the expectations over the Markov process
starting from the configurations identically equal to N and to 1
respectively. It is easy to see that if $\rho (t)$ decays
exponentially fast to zero then the theorem follows. It is an
important result by Holley [H2] (see also [AH] for a different
derivation ) that the exponential decay of $\rho
(t)$ follows once one is able to show that $\rho
(t)$ goes to zero faster than $1\over t^d$ . In order to prove
such a weaker decay of $\rho (t)$ the main new technical tool is a
recursive inequality satisfied by $\rho (t)$ that for
convenience we state as a proposition:\bigskip {\bf
Proposition 3.1}\bigskip {\it Under the hypotheses of theorem 3.3
there exist two finite
positive constants C and $\epsilon$ such that for any
integer L :
$$\rho (2t) \, \leq \,
2(L)^d\rho (t)^2\; +\; 2C\hbox{exp}(-\epsilon L)$$}
\par
{\bf Proof}\par
We write $\rho (2t)$ as:
$$\rho (2t)\;= \; \int d\mu (z)[
E_+(\sigma_{2t}(0))\;-\;E_z(\sigma_{2t}(0))]\;+\;
\int d\mu (z)[
E_z(\sigma_{2t}(0))\;-\;E_-(\sigma_{2t}(0))]\Eq(3.3)$$
and we show that each one of the two integrals is bounded by a half
of the r.h.s. of the recursive inequality.\par
Because of the attractivity assumption the distribution of the
process at time t starting from the "+" configuration is
stochastically larger than the one starting from a generic
configuration z. Therefore, using the results of [H1], there exists a
joint representation $\nu_t^{+,z}$ of the two distributions
$E_+(\cdot)$ and $E_z (\cdot)$ which is above the diagonal, i.e.
$\nu_t^{+,z}((\xi ,\eta ):\; \xi\,\geq \, \eta)\;=\;1$. In what
follows $\xi$ and $\eta$ represent the evoluted at time t of the
configurations $+$ and $z$ respectively. Let now $\chi_{L}$ be the
characteristic function of the event that $\xi (j)\,=\,\eta (j)\;
\forall \, j\in Q_L$. Then, using the Markov property, we can write:
$$\int d\mu (z)
[E_+(\sigma_{2t}(0))\;-\;E_z(\sigma_{2t}(0))]\;=\;
$$
$$\int d\mu (z)\int d\nu_t^{+,z}(\xi ,\eta )\chi_{L}
[E_\xi(\sigma_{t}(0))\;-\;E_\eta(\sigma_{t}(0))]\;+\;$$
$$+\;\int d\mu (z)\int d\nu_t^{+,z}(\xi ,\eta )(1\,-\,\chi_{L})
[E_\xi (\sigma_{t}(0))\;-\;E_\eta(\sigma_{t}(0))]\Eq(3.4)$$
Again by using attractivity and traslation invariance, the second
term in the r.h.s. of \equ(3.4) can be bounded by:
$$(L)^d\rho (t)
\int d\mu (z)\nu_t^{+,z}(\xi (0)\,\neq \, \eta (0))\;\leq \;
(L)^d\rho (t)^2\Eq(3.5)$$
If we now denote by $\tau$ the common projection in $Q_L$ of
the configurations $\xi$ and $\eta$ and we denote by $\hat
\chi_{L,\tau}$ the characteristic function of the event : $$\xi
(j)\,=\,\eta (j)\,=\,\tau (j)\; \forall \, j\in
Q_L$$
then $\chi_L$ is equal to
$$\chi_L\;=\;\sum_{\tau \in \Omega_{Q_L}}\hat
\chi_{L,\tau}$$ and therefore
the first term in the r.h.s. of \equ(3.4) can be written as:
$$\int d\mu (z)\sum_{\tau \in \Omega_{Q_L}}\int
d\nu_t^{+,z}(\xi ,\eta )\hat\chi_{L,\tau}
[E_\xi (\sigma_{t}(0))\;-\;E_\eta(\sigma_{t}(0))]\Eq(3.6)$$
Attractivity allows us to bound the quantity
$[E_\xi (\sigma_{t}(0))\;-\;E_\eta(\sigma_{t}(0))]$
by imposing extra "+" and "$-$" boundary conditions outside the cube
$Q_L$. More precisely :
$$E_\xi (\sigma_{t}(0))\;-\;E_\eta(\sigma_{t}(0))\;\leq \;
E^{Q_L,\,+}_\xi (\sigma_{t}(0))\;-\;
E^{Q_L,\,-}_\eta(\sigma_{t}(0))
\Eq(3.7)$$
where in general $E^{Q_L,\,\zeta}_\xi ()$ denotes the expectation
over the process starting from the configuration $\xi$ and evolving
in the box $Q_L$ with jump rates
$c_x^{\zeta ,\, \Lambda}(\xi\, ,\, a)$. Thus \equ(3.6) is bounded above by : $$\int d\mu (z)\sum_{\tau \in
\Omega_{Q_L}}\int d\nu_t^{+,z}(\xi ,\eta )\hat\chi_{L,\tau}
[E^{Q_L,\,+}_\tau(\sigma_{t}(0))\;-\; E^{Q_L
-}_\tau(\sigma_{t}(0))] \;\leq $$ $$\leq \; \int d\mu (z)\sum_{\tau
\in \Omega_{Q_L}}\nu_t^{+,z}(\eta (j)\,=\,\tau (j)\; \forall \,
j\in Q_L)[E^{Q_L,\,+}_\tau(\sigma_{t}(0))\;-\;
E^{Q_L,\,-}_\tau(\sigma_{t}(0))]\;=$$ $$=\;\int d\mu
(z)E_z(E^{Q_L,\,+}_{z_t}(\sigma_{t}(0)))\;-\; \int d\mu
(z)E_z(E^{Q_L,\,-}_{z_t}(\sigma_{t}(0)))\Eq(3.8)$$ where, by an
abuse of notation, $z_t$ is the value of the process at time t
starting from the configuration $z$. Since $E^{Q_L
+}_{z}(\sigma_{t}(0))$ is increasing in $z$ ( because of
attractivity ) $E_z(E^{Q_L,\,+}_{z_t}(\sigma_{t}(0)))$ is smaller
than $E^{Q_L,\,+}_z(E^{Q_L,\,+}_{z_t}(\sigma_{t}(0)))$.
Analogously $E_z(E^{Q_L,\,-}_{z_t}(\sigma_{t}(0)))$ is
larger than
$E^{Q_L,\,-}_z(E^{Q_L,\,-}_{z_t}(\sigma_{t}(0)))$. Thus
$$\int d\mu (z)E_z(E^{Q_L,\,+}_{z_t}(\sigma_{t}(0)))\;\leq\;
\int d\mu (z)E^{Q_L,\,+}_z(E^{Q_L
+}_{z_t}(\sigma_{t}(0)))\;\leq$$
$$\leq\;
\int d\mu_{Q_L}^+ (z)E^{Q_L,\,+}_z(E^{Q_L
+}_{z_t}(\sigma_{t}(0)))\;=\;\mu_{Q_L}^+(\xi (0))\Eq(3.9)$$
and
$$\int d\mu (z)E_z(E^{Q_L,\,-}_{z_t}(\sigma_{t}(0)))\;\geq\;
\int d\mu (z)E^{Q_L,\,-}_z(E^{Q_L
-}_{z_t}(\sigma_{t}(0)))\;\geq$$ $$
\int d\mu_{Q_L}^- (z)E^{Q_L,\, -}_z(E^{Q_L, \,
-}_{z_t}(\sigma_{t}(0)))\;=\;\mu_{Q_L}^-(\xi (0))\Eq(3.10)$$
In order to derive the last two equalities we used the fact that
$\mu_{Q_L}^+$ is the invariant measure of the process in
$Q_L$ with "+" boundary conditions and analogously for
$\mu_{Q_L}^-$. Thus the
r.h.s of \equ(3.8) is bounded from above by : $$\vert
\mu_{Q_L}^{+}(\xi (0))\, -\, \mu_{Q_L}^{-}(\sigma
(0))\vert \, \leq C\hbox{exp}(-\epsilon L)\Eq(3.11)$$ because of the
weak mixing assumption.\par
Exactly the same steps
show that also the second term in the r.h.s. of \equ(3.8) is bounded
from above by \equ(3.11). Thus combining together \equ(3.8)
and \equ(3.5)
we get the proposition.\bigskip The main idea at this stage is to use
the recursive inequality as a tool to transform a very rough and
weak decay in time of $\rho (t)$ of the form :
$$\rho (t) \, \leq \, \hbox{exp}(-\gamma log(t)^{1\over
d})\Eq(3.12)$$
into a
much better decay of the form :
$$\rho (t) \, \leq \, \hbox{exp}(-\hbox{exp}(+\gamma log(t)^{1\over
d}))\Eq(3.13)$$
Once the above bound is established, then one has that $\rho
(t) $
decays for large times faster than the inverse of any
power of t
and therefore, thanks to Holley's theorem (see theorem
0.1 of [H2]),
$\rho (t)$ has to decay exponentially fast.\par
Let us first prove the rough bound \equ(3.12).\bigskip
{\bf Proposition 3.2}\bigskip
{\it There exists a finite time $t_o$ and a positive constant
$\gamma$
such that :
$$\rho (t) \, \leq \, \hbox{exp}(-\gamma log(t)^{1\over
d})$$
for all t greater than $t_o$ .} \bigskip
{\bf Proof}\par
Using the attractivity
of the dynamics, we have that for any cube $Q_L$ : $$\rho
(t)\, \leq \, E^{Q_L,\,+}_+(\sigma_t(0))\, - \,
E^{Q_L,\,-}_- (\sigma_t(0))\Eq(3.14)$$ By adding and
subtracting $\mu_{Q_L}^{+}(\xi (0))\, +\,
\mu_{Q_L}^{-}(\xi (0))$ the
r.h.s. of \equ(3.14) becomes equal to :
$$\eqalign{&E^{Q_L,\,+}_\xi (\sigma_t(0))\,
- \,\mu_{Q_L}^{+}(\xi
(0))\;+\cr &\mu_{Q_L}^{-}(\xi (0))\,-\,
E^{Q_L,\,-}_\xi (\sigma_t(0))\;+\cr &\mu_{Q_L}^{+}(\xi (0))\, -\,
\mu_{Q_L}^{-}(\xi (0))}\Eq(3.15)$$ The {\it weak mixing}
condition implies that the third term in \equ(3.15)
is bounded from above
by : $$C\hbox{exp}(-\epsilon L)\Eq(3.16)$$
The estimate of the first and of the second term is identical
and
one gets :
$$E^{Q_L,\,+}_+ (\sigma_t(0))\,
- \,\mu_{Q_L}^{+}(\xi
(0))\;\leq \; C\hbox{exp}(-\epsilon L) \Eq(3.17)$$
provided that :
$$t\; \geq \; \hbox{exp}(c_oL^d )\Eq(3.18)$$
where $c_o$ is a suitable positive constant depending only
on the
jump rates. The above one is a very poor estimate which uses
only the fact that the jump rates are uniformly positive. This fact
implies that starting from an arbitrary pair of configurations
$\xi$ and $\eta$ and coupling them together with e.g. the basic
coupling (see [L]) there is a positive probability, at
most exponentially small in the volume of $Q_L$, that at time
t=1 they have become identical in the cube $Q_L$. This
fact immediately implies the above rough bound on the first and
second term of \equ(3.15).\par We now choose the side L of the cube
as L = L(t) = 2[$\{{1\over 4c_o}log(t)\}^{1\over d}$] . With this
choice we have that $t\;\geq \; \hbox{exp}(c_o\vert \Lambda\vert)$
and thus : $$\rho (t)\; \leq \;3C\hbox{exp}(-\epsilon L(t))
\Eq(3.19)$$
The proposition is proved.\bigskip
We now use Proposition 3.1 to transform the weak decay of
$\rho
(t)$ given by \equ(3.19) into a fast decay. The key point is the
following
lemma :\bigskip
{\bf Lemma 3.1}{\it Let R(t) be a positive increasing function
of t
tending to plus infinity as $t\, \toJ\, \infty$ and such
that for
some $B\, <\, 2$ : R(2t) $\leq $ BR(t) for all sufficiently
large
times t. Then there exists a finite time $t_o$ and a positive
constant A such that if for some time $t_1\, \geq \,
t_o$ it happens
that : $$\rho (t_1)\, \leq \, AR(t_1)^{-d}$$ then there
exists a
time $t_2\, \geq \, t_1$ such that :
$$ \rho
(t_2)\, \leq \, \hbox{exp}(-{\epsilon\over 4}R(t_2))$$
where $\epsilon$ is the constant appearing in proposition
3.1}\bigskip
{\bf Proof}\par
Let us choose $t_o$ be so large that for all $t \, \geq \,
t_o$ the following conditions are satisfied:
\item{i)}$[12R(t)]^d$
2Cexp(-$\epsilon$R(t)) $\leq$ exp(-$ \epsilon\over 2$R(t));
\item{ii)} R(t) $>$
1;
\item{iii)} \equ(3.19) holds;
\item{iv)} R(2t) $\leq $ BR(t)\par where C appears in
proposition 3.1 .\par
We then set $x(t)\, =\, (6R(t))^d\rho (t)$ .
Then, using proposition 3.1, the assumption on R(t) and the
definition of $t_o$, we have : $$ x(2t)\, \leq \, x(t)^2
\; +\;
\hbox{exp}(-{ \epsilon\over 2}R(t))\Eq(3.20)$$ Let us now take the
constant A of the lemma equal to $1\over 3(6^d)$. Then
by hypothesis
there exists a time $t_1\, \geq \, t_o$ such that $x(t_1)\,
\leq \,
{1\over 3}$. Let $x_n\, =\, x(2^nt_1)$; we will show that, by
assuming
$$x(2^nt_1)\, \geq \, \hbox{exp}(-{ \epsilon\over
4}R(2^nt_1))\quad \,\forall \, n\Eq(3.20bis)$$
we would get a contradiction. For, from \equ(3.20) we
get : $$x_{n+1}\, \leq \, 2x_n^2 \Eq(3.21)$$
which implies :
$$x_n \,\leq \, {1\over 2}(2x_0)^{2^n}\, \leq \,({2\over
3})^{2^n}\Eq(3.22)$$
On the other hand the assumption R(2t) $\leq $ BR(t) with
B less
than 2 implies that :
$$\hbox{exp}(-{ \epsilon\over
4}R(2^nt_1))\; \geq \; \hbox{exp}(-{ \epsilon\over
4}B^nR(t_1))\Eq(3.23)$$
which clearly contradicts \equ(3.20bis). Thus there exists $n_o$ such
that $x(2^{n_o}t_1)\,
\leq \, \hbox{exp}(-{ \epsilon\over
4}R(2^{n_o}t_1))$ . We then take $t_2\, =\,
2^{n_o}t_1$ . The
lemma is proved.\bigskip
We can finally conclude the proof of the
theorem. Let R(t) = $\hbox{exp}({\gamma \over
2d}(log(t))^{1\over d})$ . Clearly R(t) satisfy
the hypotheses
of Lemma 3.1 . Moreover, using proposition 3.2, for all $t_1\,
\geq \, t_o$ and sufficiently large :
$$\rho (t_1)\, \leq \, AR(t_1)^{-d}$$
where A and $t_o$ are the constants appearing in Lemma 3.1
. Thus,
thanks to the Lemma, there exists a
time $t_2\, \geq \, t_1$ such that :
$$ \rho
(t_2)\, \leq \, \hbox{exp}(-{\epsilon\over 4}R(t_2))\Eq(3.24)$$
Since $t_1$ can be taken arbitrarily large, the above bound
implies
that for any finite N there exists an arbitrarily large time
T such that:
$$\rho (T)\, \leq \, {1\over T^N}$$
Thanks to Holley's theorem this implies that $\rho
(t)$ decays
exponentially fast in time . \bigskip
{\bf i) $\Leftarrow$ ii)} This was proved years ago by Holley and
Strook [HS] for the stochastic Ising model. For completeness we give
the proof also in the more general case of non reversible spin
dynamics. Clearly ii) implies that the infinite volume dynamics is
ergodic with a unique invariant measure $\mu$. Thus we write:
$$\mu_{Q_L}^{+}(\xi (0))\;-\;\mu_{Q_L}^{-}(\xi
(0)) \;= \;\mu_{Q_L}^{+}(\xi (0))\;-\;\mu (\xi
(0))\;+\;\mu (\xi
(0))\;-\;\mu_{Q_L}^{-}(\xi
(0))\Eq(3.25)$$
Let us estimate $\mu_{Q_L}^{+}(\xi (0))\;-\;\mu (\xi
(0))$. By adding and subtracting $E_+(\sigma_t(0))$ and using the
exponential convergence to equilibrium together with attractivity, we
get : $$0\;\leq\;\mu_{Q_L}^{+}(\xi (0))\;-\;\mu (\xi
(0))\;\leq\; C\hbox{exp}(-\gamma t)
\;+\;\mu_{Q_L}^{+}(\xi (0))\;-\;E_+(
\sigma_t(0))\Eq(3.26)$$
We now choose the time t as t = $\delta L$ .
Since the jump rates are finite range it easily follows (see
e.g. [H2], Lemma 1.1 ) that if $\delta$ is small enough one has:
$$E_+^{\L ,\,+}(\xi (0))\;-\; E_+(\sigma_t(0))\;
\leq \; \hbox{exp}(-L)$$
Thus the r.h.s. of \equ(3.26) can be bounded by :
$$ C\hbox{exp}(-\gamma \delta L)\;+\;
\mu_{Q_L}^{+}(\xi (0))\;-\;E_+^{\L
,\,+}(\sigma_t(0))\;+\;\hbox{exp}(-L)\;\leq$$ $$\leq \;
C\hbox{exp}(-\gamma \delta L)\;+\;\hbox{exp}(-L)\Eq(3.27)$$ since by
attractivity $\mu_{Q_L}^{+}(\xi (0))\;-\;E_+^{\L
,\,+}(\sigma_t(0))$ is negative. In conclusion we have shown that
$\mu_{Q_L}^{+}(\xi (0))\;-\;\mu (\xi (0))$ is smaller
than $ C\hbox{exp}(-\gamma \delta L)\;+\;\hbox{exp}(-L)$. \par The
same argument applies also to the other term in the r.h.s. of
\equ(3.25) $\mu (\xi (0))\;-\;\mu_{Q_L}^{-}(\xi
(0))$. \par
The theorem is proved.\bigskip
\numsec=4\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Section 4}\par\noindent \centerline{\bf Exponential convergence
in finite volumes:}
\centerline{\bf the stochastic Ising model}\bigskip We prove in this
section the exponential convergence to equilibrium in finite
volumes with rates that are estimated uniformly in the
volume for the stochastic Ising model under a strong mixing
condition on the Gibbs state. \par
The Hamiltonian $H_{\Lambda}^\tau$ of our spin
system satisfies hypotheses H1 and H2 of section 1 but for simplicity
we assume that the spins can take only the two values +1 or $-$1.
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 $\mu$.\par
Later on, in order to simplify some of the proofs, we will make the
assumption that the Hamiltonian is ferromagnetic or attractive
(see section 1); we emphasize, however, that all the results of this
section can also be proved without the assumption of
ferromagnetism by using the logarithmic Sobolev
inequalities (see [MO]). \par
The stochastic
dynamics that will
be the object of study in this section will be one of the Glauber
dynamics associated to the hamiltonian $H^\tau_{\Lambda}$ (see
(1.2) (1.12)). 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. In order to simplify the exposition and the
computations we decided to take from the beginning
a precise form
for our jump rates : $$c_x(\sigma , a)\, =\,
\mu_{\{x\}}^{\sigma}(\eta (x)\,=\,a)\;=\;
{1\over 1+\hbox{exp}(-2\beta a\sum_{X;\,x\,\in\,X}J_X\prod_
{y\in X\setminus\{x\}}\s_y )}\Eq(4.1)$$ 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.
\bigskip
{\bf
Remark} 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 means that the generator of the process L
becomes a non positive selfadjoint operator in the
Hilbert space $L^2(\Omega_\Lambda \,,\,d\mu_{\Lambda}^{ \tau})$ and 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
(see [L]). Moreover if the interaction is ferromagnetic then
automatically the above defined jump rates become attractive in the
sense made precise in section 1.\bigskip
For
reader's convenience we recall now our finite volume strong
mixing condition (see section 1) that
in the sequel we will refer to as {\it SM($L_o$,C,$\gamma$)} :\bigskip\noindent
Let $\Lambda_o$ be the cube of side $2L_o+1$ with sides parallel
to the coordinate axes and let for any $V\subset \Lambda_o$
$\mu^{\tau}_{\Lambda_o,V}$ be the relativization of
$\mu^{\sigma}_{\Lambda_o}$ to the set V. Then for any $y$
outside $\Lambda_o$ and any V in $\Lambda_o$ we must have:
$$ Var(\mu^{\tau}_{\Lambda_o,V}\,
,\, \mu^{\tau^y}_{\Lambda_o,V})\; \leq \;
C\hbox{exp}(-\gamma \hbox{dist}(y,V))\quad \forall \, \tau\,
\in
\{-1,+1\}^{\Lambda_o^c}$$ \bigskip
{\bf Remark} It is easy to check that the above condition
implies that if for two given configurations $\tau$ and $\tau '$
we denote by $V_{\tau ,\tau '}$ the set $\{x \notin \Lambda_o
\, ; \; \tau (x) \neq
\tau ' (x)\}$ and by Q the maximal
subset of $\Lambda_o$ which is at distance greater than
$L_o^{1\over 2}$ from $V_{\tau ,\tau '}$, then we have:
$$Var(\mu^{\tau}_{\Lambda_o,Q}\,
,\, \mu^{\tau '}_{\Lambda_o,Q})\; \leq
\;
c(r) {1\over L_o^{d+2}}\Eq(4.2)$$
where $c(r)$ is a numerical constant depending only on the
range $r$. \bigskip
Obviously since our condition has to hold only in a definite
geometric shape , in our case a cube, contrary to what
assumed by
Aizenman and Holley or Zegarlinski and Strook, we will
prove our
results only in volumes that are multiple of the
elementary volume $\Lambda_o$ (see the definition before Theorem
2.6). As already discussed in the introduction this has to be the
case if we want to apply our condition to a system at low temperature
near a first order phase transition for which it can be proved
(see section 1) that the Dobrushin-Shlosman complete analiticity
fails.\par Let us now state our main results.
\item{-}\ 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 $(2L_o+1){\bf
Z^d}$, and we will denote by gap($L_{\Lambda}^{\tau}$) the lowest
positive eigenvalue of -$L_{\Lambda}^{\tau}$ in
$L^2(\Omega_\Lambda \,,\,d\mu_{\Lambda}^{ \tau})$ \bigskip
{\bf Remark} For simplicity the
next three results are stated only
for volumes $\Lambda$ that are
$L_o$-compatible. It is however relatively easy to check that once
they hold for this rather restricted class of volumes, then they hold
also for the larger class of sets $\Lambda$ such that for
any x in $\Lambda$ it is possible to find a suitable translated $\Lambda_o(y)$
of the cube $\Lambda_o(y)$ entirely contained in $\Lambda$ and such
that $\hbox{dist}(\,x,\partial \Lambda\setminus\partial
\Lambda\cap\partial\Lambda_o(y) \,)\,\geq\,{L_o\over 2}$.\bigskip
The next result says that {\it SM($L_o$,C,$\gamma$)} implies
exponential convergence to equilibrium in any
$L_o$-compatible finite volume in the $L^2$-norm . \bigskip
{\bf Theorem 4.1}\par {\it There exists a positive constant $\bar L$
depending only on the range of the interaction and on the dimension
d such that if {\it SM($L_o$,C,$\gamma$)} holds with $L_o\,\geq \,
\bar L$ then there exists a positive constant $m_o$ such that for
any $L_o$-compatible set $\Lambda$ and
for any function f in $L^2(d\mu_{\Lambda}^{\tau})$ : $$\vert\vert
T^{\Lambda ,\, \tau}_t(f)\, -\,
\mu_{\Lambda}^{\tau} (f)\vert\vert_{L^2(d\mu_{\Lambda}^{\tau})}
\;
\leq \;
\vert\vert\,
f\,-\,\mu_{\Lambda}^{\tau} (f)\,\vert\vert_{L^2(d\mu_{\Lambda}^{\tau})}\hbox{exp}(-m_ot)$$
where $T^{\Lambda ,\, \tau}_t$ denotes the Markov semigroup
of the
process evolving in $\Lambda$ with boundary conditions
$\tau$.}\bigskip
{\bf Proof of Theorem 4.1}\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: \item{i)} Each element of
the covering is a
a cube of side $2L_o+1$ with sides parallel to the coordinate
axes.
\item{ii)} 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 axis, of the other.\par
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 :
$$L_Qf(\sigma) \;=\; \sum_{\eta ,i}c_{Q_i}(\sigma
,\eta)(f(\eta)\,-\,f(\sigma))\Eq(4.3)$$
where the new jump rates $c_{Q_i}(\sigma
,\eta)$ are a generalization of those of the heat bath
dynamics and are given by :
$$c_{Q_i}(\sigma,\eta)\;=\; \mu_{Q_i}^{\sigma}(\eta)\Eq(4.4)$$
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 [H2], Aizenman and Holley [AH]
and
Strook and Zegarlinski [SZ]. 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 $\Lambda_o(x)\cup\Lambda$
is put
equal to $\eta$ with probability
$\mu_{\Lambda_o(x)\cup\Lambda}^{\sigma}(\eta)$
, where $\Lambda_o(x)$
is the cube of side $2L_o+1$ centered at x. This dynamics
has however
the incovenient to update, sometimes, regions that are not
squares
$\Lambda_o$ but rather boxes (= intersection between
two cubes) on
which, contrary to what happen for cubes $\Lambda_o$,
we have no
control at all and for which our mixing condition may
very well fail !\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
[SZ]) that if gap($L_Q$) and gap(L) denote the gap in the spectrum
of
the generators $L_Q$ and L respectively, then there exists a
positive constant c independent of $\Lambda$ and $\tau$
such that:
$$\hbox{gap}(L)\;\geq \;
\hbox{exp}(-cL_o^d)\hbox{gap}(L_Q)\Eq(4.5)$$ 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
we adopt a
scheme very similar to the one already used in section
3 even if we are working with a very general,
not necessarily ferromagnetic, system .\par Given two initial
configurations $\sigma$ and $\sigma '$ we couple
their dynamics by
defining the generator $\tilde L_Q$ of the coupled process
as:
$$\tilde L_Qf(\sigma ,\sigma ') \;=\; \sum_{\eta ,\eta
' ,i}\tilde
c_{Q_i}(\sigma ,\sigma ' ,\eta, \eta ')(f(\eta,\eta
')\,-\,f(\sigma
, \sigma '))\Eq(4.6)$$ where the jump rates $\tilde c_{Q_i}(\sigma
,\sigma '
,\eta, \eta ')$ are given by : $$\tilde c_{Q_i}(\sigma
,\sigma '
,\eta, \eta ')\;=\; \mu_{Q_i}^{\sigma ,\sigma '}(\eta
, \eta ')\Eq(4.7)$$ if
the pair $\eta ,\eta '$ agrees with the pair $\sigma ,\sigma
'$
outside the cube $Q_i$ and zero otherwise. Here the measure
$\mu_{Q_i}^{\sigma ,\sigma '}$ is an element of the set $K$ of
the joint representations of the two Gibbs states
$\mu_{Q_i}^{\sigma}$ and
$\mu_{Q_i}^{\sigma '}$ and it is such that it realizes the
minimum:
$$Var(\mu_{Q_i,\tilde Q_i}^{\sigma},\mu_{Q_i,\tilde Q_i}^{\sigma '})\,=\,
\min_{\nu \in K}\sum_{\sigma ,\sigma'\in \Omega_{Q_i}}
\nu (\sigma , \sigma ')\rho_{\tilde Q_i}(\sigma , \sigma ')$$
In
the above formula $$\rho_{\tilde Q_i}(\sigma , \sigma ')\,=\,1$$ if
$\sigma (x)\,\neq\, \sigma '(x)$ for some x in $\tilde Q_i$ and
zero otherwise and $\tilde Q_i$ is the maximal subset of $Q_i$ which
is at distance greater than $L_o^{1\over 2}$ from the set V = $\{x
\notin Q_i \, ; \; \sigma (x) \neq \sigma ' (x)\}$.\bigskip
{\bf Remark} It is well known that in the
attractive case the joint representation
$\mu_{Q_i}^{\sigma ,\sigma '}$ is above the diagonal i.e.
$\mu_{Q_i}^{\sigma ,\sigma '}(\eta \,\leq
\,\eta ')\;=\;0$ if $
\eta \,\geq \,\eta '$\bigskip
A concrete
way to realize the coupled process is to attach an exponential
clock of
parameter one to each cube $Q_i$; then when a clock
rings, say at
time t and at the cube $Q_i$, one updates the pair $\sigma_t
,\sigma_t'$ inside $Q_i$ to the pair $\eta ,\eta '$ with
probability $\mu_{Q_i}^{\sigma ,\sigma '}(\eta ,
\eta ')$.\bigskip
Once the coupling has been established we define the quantity
$\rho
(t)_\L^\t$ as : $$\rho_{\Lambda}^{\tau}(t)\, =\, \sup_{\sigma
,\eta\, x\in \Lambda}P(\,
\sigma_t(x)\, \neq \, \eta_t(x)\,)\Eq(4.8)$$
It is elementary to verify that if $\rho_{\Lambda}^{\tau}(t)$
decays exponentially
with a rate $m_Q$ bounded away from zero, uniformly in the
volume $\Lambda$ and in the boundary conditions $\tau$,
then gap($L_Q$) $\geq\, m_Q$.\par
In order to prove the exponential decay of $\rho_{\Lambda}^{\tau}(t)$
we would like,
at this point, to apply to $\rho_{\Lambda}^{\tau}(t)$ the usual
Holley's criterion :
if there exists a large enough 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 {\it SM($L_o$,C,$\gamma$)} condition. In fact the
above described coupling
is such that after the updating at
time t of, say, the
cube $Q_i$, the probability to see a difference between
$\eta$ and
$\eta '$ at a site x in $Q_i$ at a distance greater than
$L_o^{1\over
2}$ from the set V = $\{x \notin Q_i \, ; \; \sigma_t (x)
\neq
\sigma_t ' (x)\}$ is smaller then $ {1\over L_o^{d+2}}\,
<<\, 1$ uniformly in the configurations $\sigma_t$ and $\sigma_t
'$, provided $L_o$ is large enough.
Thus, under this coupling, two arbitrary configurations $\sigma
$,
$\sigma '$ should become equal everywhere in $\Lambda$
in a short
time and, in some sense, the Gibbs
sampling behaves as a high temperature, almost independent,
stochastic
Ising model.\par In order to implement this program we
first prove
the Holley's recursive inequality (see [H2]) for
$\rho_{\Lambda}^{\tau}(t)$ : $$\rho_{\Lambda}^{\tau}(2t) \,
\leq \, (C(2L_o+1+r)t+1)^d\rho_{\Lambda}^{\tau}(t)^2\; +\;
\hbox{exp}(-\gamma t)\Eq(4.9)$$ for
suitable positive constants C and $\gamma$ independent of t,
$\Lambda$ and $L_o$.\par To prove \equ(4.9) let x, $\sigma$
and $\sigma '$
be fixed, let $A(x)$ be the box of side $C(2L_o+1+r)t+1$ centered
at
the site x where C is a constant to be fixed later and let
$\chi_{t,x}$ be the characteristic function of the
event, for the
coupled process $\{\sigma_t ,\sigma_t'\}$, that
$\sigma_t(j)\,=\,\sigma_t'(j)\;
\forall \, j\in A(x)\cap \Lambda$.
Then we can write:
$$P(\sigma_{2t}(x)\,\neq\,\sigma_{2t}'(x))\;\leq\;
E\chi(\sigma_{2t}(x)\,\neq\,\sigma_{2t}'(x))(1-\chi_{(t,x)}))\;
+\;E\chi(\sigma_{2t}(x)\,\neq\,\sigma_{2t}'(x))\chi_{(t,x)})\Eq(4.10)$$
The
first term in the r.h.s. of \equ(4.10), using the Markov property
and the definition of $\rho
(t)$, is bounded from above by :
$$(C(2L_o+1+r)t+1)^d\rho_{\Lambda}^{\tau}(t)^2\Eq(4.11)$$
In order to bound the second term we observe that the Gibbs
sampling has " finite speed of propagation
of information" since one single updating can influence spins
in a
region with diameter not larger than $r +2L_o+1$ . It is
then easy
to check (see e.g. Lemma 1.1 of [SZ] ) that if the constant
C is taken
large enough independently of $L_o$ and $t$ then
there exists another constant $\gamma$, e.g. larger than
one for C
large enough, such that the second term is bounded by :
$$\hbox{exp}(-\gamma t)$$
Using now \equ(4.9) it follows immediately from Lemma 2.4 of
[H2] that
there exist two numerical constants $\delta$ and $\bar t$
depending on the constant C such that, if for some
time $t_o\, \geq
\, \bar t$ $$(C(2L_o+1+r)t_o+1)^d\rho_{\Lambda}^{\tau}(t_o)\,\leq
\, \delta\Eq(4.11bis)$$
then there exists a positive constant $m_Q(t_o,\delta)$
such that
$$\rho_{\Lambda}^{\tau}(t)\, \leq
\, \hbox{exp}(-m_Q t)\quad \forall \, t\,
\geq \, t_o\Eq(4.12)$$
We finally verify the existence of such a time $t_o$ uniformly
in $\Lambda$ and in the boundary conditions $\tau$. Let x $\in
\,
\Lambda$ , $\sigma$ , $\sigma '$ and $t_o$ be fixed, let
$Q_i$ be a cube such that $x\in Q_i$ with \hbox{dist}(x,$\partial Q_i\setminus
(\partial
\Lambda \cap \partial Q_i )\,)\;\geq \,{L_o\over 2} \,)$
and let $Q_{i_1}...Q_{i_n}$ be the other elements of the covering
which intersect $Q_i$. The number n is clearly dependent on the
geometry of $\Lambda$ but can be bounded by a constant n(d) dependent
only on the dimension d. Let also $\nu (Q_i ,t)$ be the number
of ringings of the exponential clock of
parameter one attached to the cube $Q_i$ within time
t and analogously for the other cubes $Q_{i_1}...Q_{i_n}$. Then
for any integer N smaller than
$ L_o^{1\over 2}\over 10$
we estimate $\rho_{\Lambda}^{\tau}(t_o)$ by :
$$\rho_{\Lambda}^{\tau}(t_o)\,\leq
\, P(\nu (Q_i,t_o)\,=\,0)\;+\;
\sum_{j=1}^{n}P(\nu
(Q_{i_j},t_o)\,\geq \, N)\;+\;P(\nu
(Q_{i},t_o)\,\geq \, N)\;+$$
$$\sum_{j=1}^{n}\sum_{k=1}^{N}P(\hbox{the k}^{th}
\hbox{ updating of the cube }Q_{i_j}\hbox{ was "bad"}\,)\;+$$
$$\sum_{k=1}^{N}P(\hbox{the k}^{th}
\hbox{ updating of the cube }Q_{i}\hbox{ was "bad"}\,)\Eq(4.13)$$
where an updating $\{\sigma ,\sigma '\,\to\,\eta ,\eta
'\}$ of a cube $Q_{i_j}$
is "bad" if $\eta (x)\,\neq \, \eta '(x)$ for some x in $Q_{i_j}$
with dist(x,$\{y\in Q_{i_j}^c;\,\s_y\,\neq \,\s '_y\;\}\;\geq \,L_o^{1\over
2} \,)$.\par
Let us in fact assume that within time $t_o$ the cube $Q_i$
has been updated at least once and that all the updatings within
time $t_o$ of the cubes $Q_i$ ,$Q_{i_1}...Q_{i_n}$ have been
"good" and not more than N. Then, for $L_o$ large enough, since right
after the last update of the cube $Q_i$, say at time t, there is no
difference in the two configurations $\sigma_t$ and $\sigma '_t$ in
a cube $\bar Q_i\subset Q_i$ of side $L_o\over 4$ containing x
with dist(x,$\partial \bar Q_i\setminus
(\partial
\Lambda \cap \partial \bar Q_i )\,)\;\geq \,{L_o\over 8} \,)$and since a
"good" updating of one of the neighbors cubes $Q_{i_j}$ can only
bring a difference in the two configurations $\sigma_t$ and
$\sigma_t'$ inside $\bar Q_i$ at a distance from
$\partial \bar Q_i\setminus
(\partial
\Lambda \cap \partial \bar Q_i )$ smaller than $L_o^{1\over 2}$, after k updatings between times
t and $t_o$ of the cubes $Q_{i_1}...Q_{i_n}$ we have that the two
configurations $\sigma$ and $\sigma '$ are still equal for all
$y\in Q_i$ at distance from x less or equal than
${L_o\over 8}\,-\,kL_o^{1\over 2}$ . Thus if
$k\,\leq \,N\,\leq\,{L_o^{1\over 2}\over 10}$ at the final time
$t_o$ one still has $\sigma_{t_o}'(x)\,=\,\sigma_{t_o}(x)$.\par The
first term in \equ(4.13) is equal to exp($-t_o$). The second and
third term can also be bounded by exp($-t_o$) if N=$at_o$ for $a$
large enough but
$at_o\,\leq\,{L_o^{1\over 2}\over 10}$.
Finally the sum of the fourth and fifth term is bounded
by : $$(n(d)+1)N\sup_{\sigma ,\sigma
'}Var(\mu^{\sigma}_{\Lambda_o,Q}\,
,\, \mu^{\eta}_{\Lambda_o,Q})\; \leq
\;
c(r)(n(d)+1)N {1\over L_o^{d+2}} $$
where $Q$ is the maximal
subset of $\Lambda_o$ which is at distance from $V_{\sigma
,\sigma '}\,=\,\{x \notin \Lambda_o
\, ; \; \sigma (x) \neq
\sigma ' (x)\}$ greater than $L_o^{1\over 2}$. Here c($r$) is a
numerical constant depending only on the range $r$. \par
Thus, by choosing $t_o\,=\,{L_o^{1\over 2}\over 10a}$,
N=$at_o$ with $a$ large enough, we have that the
quantity $(C(2L_o+1+r)t_o+1)^d\rho_{\Lambda}^{\tau}(t_o)$
is bounded above by:
$$(C(2L_o+1+r)t_o+1)^d\rho_{\Lambda}^{\tau}(t_o)\,\leq \,$$
$$(C(2L_o+1+r)t_o+1)^d\{2\hbox{exp}(-t_o)\,+\,c(r)(n(d)+1)at_o
{1\over
L_o^{d+2}}\}\, \leq\, \delta$$
provided that $L_o$ is large enough (depending of $C,\, r,$
$\delta$ ) . The Theorem is proved.
\bigskip
The first result that we derive from the above Theorem is
Theorem 2.6, namely the ($\Gamma$, $\Gamma '$)-effectiveness of
{\it $SM(\cdot ,C,\gamma$)}
with $\Gamma$ consisting only of the cube $\Lambda_{L_o}$ and
$\Gamma '$ the family of all $L_o$-compatible sets of ${\bf Z^d}$
provided that $L_o$ is large enough. The proof is based on the
following nice result due to Strook and Zegarlinski : \bigskip
{\bf Proposition 4.1} (see Lemma 3.1 of [SZ] )\par {\it Let us assume
that gap($L_{\Lambda}^{\tau}$) $\geq\,m\,>\,0$ uniformly in
$\Lambda$ and $\tau$. Then there exist positive constants $C'$ and
$\gamma '$ independent of $\Lambda$ such that for any subset V of
$\Lambda$ any site k outside $\Lambda$ and any function f with
support in V : $$\sup_{\tau}\vert \mu_{\Lambda}^{\tau^{(k)}}(f)\;-\;
\mu_{\Lambda}^{\tau}(f)\vert\;\leq\;C\hbox{exp}(-\gamma
\hbox{dist}(V,k)) \{\vert\vert\vert f \vert\vert\vert\,\wedge
\,\sup_\sigma\vert f(\sigma) \vert \,\}$$ where $\vert\vert\vert f
\vert\vert\vert\;=\; \sum_x\vert\vert \nabla_xf \vert\vert$ and
$\vert\vert \nabla_kf\vert\vert\; =\; \sup_{\sigma}\vert
f(\sigma^{(k)})\, -\,f(\sigma)\vert$}\bigskip {\bf Remark} Actually in
Lemma 3.1 in [SZ] the dependence of the estimate on f was only
through the seminorm $\vert\vert\vert f \vert\vert\vert$. That may
be not so convenient if f depends on a large number of spins (e.g.
f is the characteristic function of the event that all the spins in
$\Lambda$ at distance from k larger than L are +1) since one may
introduce a spurious volume factor. A little effort shows however
that the dependence on f can be improved to that of the
proposition.\bigskip Clearly the above result proves the theorem
since the variation distance between the relativization of the
Gibbs states in $V$ with boundary conditions outside $\Lambda$
given by $\tau$ and $\tau^{(k)}$ respectively, is equal to :
$$\sup_{A\,\subset\,
\Omega_{Q(L)}}\vert\mu_{\Lambda}^{\tau^{(k)}}(A)\;-\;
\mu_{\Lambda}^{\tau}(A)\vert$$
The last result strengthens the result given in Theorem 4.1 :\bigskip
{\bf Theorem 4.2}\par {\it There exists a positive constant $\bar L$
depending only on the range of the interaction and on the dimension
d such that if {\it SM($L_o$,C,$\gamma$)} holds with $L_o\,\geq \,
\bar L$ then there exists a positive constant $m$ such that for
any $L_o$-compatible set $\Lambda$ and
for any function f on $\{-1,+1\}^{\Lambda}$ : $$\sup_{\sigma}
\vert
T^{\Lambda ,\tau}_t(f)(\sigma)\, -\, \mu (f)\vert \; \leq \;
\vert\vert\vert f\vert\vert\vert\hbox{exp}(-mt)$$
where $T^{\Lambda ,\, \tau}_t$ denotes the Markov semigroup
of the
process evolving in $\Lambda$ with boundary conditions
$\tau$.}\bigskip
{\bf Proof}\par In this paper we prove the theorem only in
the attractive case. The proof for the general case can be found in
[MO].\par We proceed as in the proof of Theorem 3.1 . We define
$$\rho_{\Lambda}^{\tau}(t) \;=\; \sup_{x\in \Lambda}
E^{\Lambda_L,\,\tau}_+ (\sigma_t(x))\, - \, E^{\Lambda_L,\,\tau}_-
(\sigma_t(x))$$ As in section 2 we need only to show that
$\rho_{\Lambda}^{\tau}(t)$ decays
exponentially to zero with a rate independent of the volume
$\Lambda$ and of the boundary conditions $\tau$. One
easily checks
that also the finite volume definition of $\rho_{\Lambda}^{\tau}(t)$
obeys the
Holley's alternative: there exists a positive constant
$\delta_o$ independent of the boundary conditions $\tau$
such that if there
exists a sufficiently large finite time $t_o$ such that : $$t_o^d\rho_{\Lambda}^{\tau}(t_o)\,\leq
\,
\delta_o\Eq(4.14)$$ then there exists a finite constant $m$
depending on
$t_o$ and $\delta_o$ such that : $$\rho_{\Lambda}^{\tau}(t)\,\leq
\, \hbox{exp}(-mt)\quad \forall
\, t\,\geq \, t_o\Eq(4.15)$$ Thus, in order to prove the theorem, we
need only
to show that there exists a time $t_o$, independent of the
volume
$\Lambda$ and of the boundary condition $\tau$, such that the above
condition holds. In turn this will
follow from a computation similar to those of section 3 since we know
from the
previous theorem that the mass gap of the stochastic
Ising model can
be bounded from below uniformly in the boundary conditions
and in the
volume $\Lambda$ provided that $\Lambda$ is $L_o$-compatible.\par
More precisely let, for any x $\in \, \Lambda\;$, $\Lambda_N(x)$
be an
$L_o$-compatible subset of $\Lambda$ such that
: \item{a)} x is
contained in $\Lambda_N(x)$ \item{b)} \hbox{dist}(x,$\partial
\Lambda_N(x)\setminus \partial \Lambda
\cap \partial \Lambda_N(x))$
$\geq \,NL_o$ \par Then, by attractivity, we can
bound
$\rho_{\Lambda}^{\tau}(t)$ by : $$
E^{\Lambda_N(x),\,+}_+(\sigma_t(0))\, - \,
E^{\Lambda_N(x),\,-}_- (\sigma_t(0))\Eq(4.16)$$ where
$E^{\Lambda_N(x),\,+}_+()$ is the expectation
over the process which starts from all
pluses and evolves with + boundary conditions on $\partial
\Lambda_N(x)\setminus \partial \Lambda
\cap \partial \Lambda_N(x)$
and the given $\tau$-boundary conditions on $\partial \Lambda
\cap
\partial \Lambda_N(x)$ and analogously for
$E^{\Lambda_N(x),\,-}_- ()$. Thus, as in proposition 3.2, the r.h.s.
of
\equ(4.16) is bounded from
above by :
$$\eqalign{&E^{\Lambda_N(x),\,+}_{+} (\sigma_t(0))\,
- \,\mu_{\Lambda_N(x)}^{+}(\sigma
(0))\;+\cr&\mu_{\Lambda_N(x)}^{-}(\sigma (0))\,-\,
E^{\Lambda_N(x),\,-}_{-} (\sigma_t(0))\;+\cr
&\mu_{\Lambda_N(x)}^{+}(\sigma (0))\, -\,
\mu_{\Lambda_N(x)}^{-}(\sigma (0))}\Eq(4.17)$$ where the Gibbs states
$\mu_{\Lambda_N(x)}^{+}$
and
$\mu_{\Lambda_N(x)}^{-}$ have + and
$-$ boundary conditions on
$\partial \Lambda_N(x)\setminus \partial \Lambda \cap
\partial
\Lambda_N(x)$ and the given $\tau$-boundary conditions
on $\partial
\Lambda \cap \partial \Lambda_N(x)$.
Using Proposition 4.1 the third term is bounded from
above
by : $$\hbox{exp}(-\gamma NL_o)\Eq(4.18)$$
The estimate of the first and second term is the same and
we get
that each of them ,e.g. the first one, is bounded from
above by :
$$\hbox{exp}(-\gamma NL_o)\Eq(4.19)$$
provided that :
$$t\; \geq \; {1\over \hbox{gap}(\Lambda_N(x) ,+)}[log (
\mu_{\Lambda_N(x)}^{+}(+)^{-1}) \; +\;
\gamma NL_o)\Eq(4.20)$$
(see for instance [Si]) where $\mu_{\Lambda_N(x)}^{-}(+)$ is the
$\mu_{\Lambda_N(x)}^{-}$-measure of
the configuration in $\Lambda_N(x)$ identically equal to
plus one and
$\hbox{gap}(\Lambda_N(x) ,+)$ is the gap in the spectrum of the
(selfadjoint) generator of the stochastic Ising model in
$\Lambda_N(x)$ with + boundary conditions on
$\partial \Lambda_N(x)\setminus \partial \Lambda \cap
\partial
\Lambda_N(x)$ and the given $\tau$-boundary conditions
on $\partial
\Lambda \cap \partial \Lambda_N(x)$.
Using the result of proposition 4.1 we have that
$\hbox{gap}(\Lambda_N(x) ,+)$ ( and the same for
$\hbox{gap}(\Lambda_N(x) ,-)$ ) is bounded below by
$m_o$ uniformly in N.
Therefore, since $\hbox{log}(\mu_{\Lambda_N(x)}^{-}(+))$
$>$ -A$(NL_o)^d$ for
some constant A , if we take $NL_o$ = $c_ot^{1\over d}$
then, if
$c_o$ is sufficiently small depending on A, we get that
\equ(4.20) is satisfied and therefore
$$\rho_{\Lambda}^{\tau}(t) \; \leq \;
3\hbox{exp}(-\gamma c_ot^{1\over d})$$
Thus $\rho_{\Lambda}^{\tau}(t) $ decays faster than $1\over
t^d$ uniformly in the
volume $\Lambda$ and the theorem follows.\bigskip
{\bf Remark} One may wonder whether the rates $m_o$ and
$m$ of the
exponential convergence to equilibrium in the $L^2$-sense
and
in the
uniform norm are equal. The proof that we give of Theorem
4.3 in the atttractive case,
which basically rephrases in finite volumes the usual
Holley's argument , does not allow us to derive
any
conclusion about this question. However if instead of the
Holley's argument one uses logarithmic Sobolev inequalities
(see
[MO] ) then one can conclude that the two rates are actually
the
same.
\bigskip{\bf Remark} One may wonder why we needed in this section
a condition like
{\it SM($L_o$,C,$\gamma$)} which is much stronger than the weak mixing
condition used in the previous section. A first simple answer to this
question is the following: since exponential convergence to
equilibrium in finite volume (in $L^2$ or $L^\infty$ sense) implies
the exponential decay of truncated correlations in the given volume
(see proposition 4.1), certainly such convergence cannot take place
for those systems, like the Czech models or the 3D Ising model at
low temperature at very small magnetic fields (Bassuev phenomenon)
in which truncated correlations do not decay exponentially fast
uniformly in the locations of the two observables.\par
Another explanation which seems to be reasonable even in the
attractive case, is the following: let us suppose that we have only
a weak mixing property of the Gibbs state and let us consider the
quantity $E_\sigma^{\Lambda_L ,\tau} (f(\sigma_t))$, where $f$ is an
observable located well inside a box $\Lambda_L$ of side L. If t
$\leq \,\delta L$, where $\delta $ is a suitable small constant,
then, because of the finite speed of propagation of information,
$E_\sigma^{\Lambda_L ,\tau} (f(\sigma_t))$ is exponentially close (
in t) to its infinite volume version $E_\sigma (f(\sigma_t))$ which
is indeed, because of heorem 3.1, exponentially ( in t) close to
$\mu (f)$ which, in turn, because of weak mixing, is exponentially (
in L ) close to $\mu_{\Lambda_L}^{\tau}(f)$. Since $L\, \geq\,
{t\over \delta}$ it follows that for times t up to $\delta L$ we
have : $$\vert E_\sigma^{\Lambda_L ,\tau} (f(\sigma_t))\,-\,
\mu_{\Lambda_L}^{\tau}(f)\vert \; \leq\; C\hbox{exp}(-mt)$$ for
suitable C and m.\par Let us now consider times t much larger than L
and let us suppose that for these times the probability distribution
of $\sigma_t(x)$ for x close to the boundary of $\Lambda_L$ is not
exponentially (in t) close to the invariant measure. That it is not
an unreasonble assumption if our system exhibits a kind of phase
transition at the boundary as apparently does the 3D Ising model at
low temperature for some very small magnetic fields (Basuev
phenomenon). Let us now analyze the influence of this slow approach
to equilibrium at the boundary on $E_\sigma^{\Lambda_L ,\tau}
(f(\sigma_t))$. Certainly, because of weak mixing, the effect will
not be larger than a suitable negative exponential in L but we
cannot exclude that it will be precisely of this order. If this is
the case then, since $t\, >>\, L$, the influence on
$E_\sigma^{\Lambda_L ,\tau} (f(\sigma_t))$ of the slow convergence
to equilibrium at the boundary will be much larger than a negative
exponential of the time t and thus, even in the bulk, we will have a
convergence slower than exponential.\par
Finally, from a technical point of view we observe that in finite
volume, as the reader can easily check, we cannot repeat the proof of
Proposition
3.1 because for some site x the cube of side L and centered at x
intersects the boundary of the cube. Thus we are forced to choose
the stronger hypothesis ($SM(L_o,C,\gamma)$).
\numsec=5\numfor=1
\vskip 1cm
{\bf Section 5}\par\noindent \centerline{\bf Applications}
\bigskip
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 Ising
model whose Hamiltonian in a finite volume $\Lambda$ of the
lattice $\bf Z^d$ with boundary conditions $\tau$ is given by:
$$H_{\Lambda}^\tau(\sigma)\,=\,-{1\over 2}\sum_{x,y\in
\Lambda :\; \vert x-y\vert\,=\,1}\sigma (x)\sigma (y)
\;-\;{1\over 2}\sum_{x\in
\Lambda}[\;h\;+\;\sum_{y\notin \Lambda :\; \vert x-y\vert\,=\,1}\tau
(y)\;]\sigma (x)$$ The associate finite volume Gibbs state at
inverse temperature $\beta$ will be denoted by $\mu_\Lambda^{\tau
,\, \beta ,\,h}(\sigma )$. It is well known that if the dimension d
is greater or equal than 2 there exists a critical value of $\beta$,
denoted in the sequel by $\beta_c $, such that there exists a unique
infinite volume Gibbs state $\mu^{\beta ,\,h}$ iff $h\neq 0$ or
$\beta\,<\,\beta_c$. Thus, if we consider the associated stochastic
Ising model discussed in the previous section, then it will be an
ergodic Markov process on $\{-1,+1\}^{\bf Z^d}$ with $\mu^{\beta
,\,h}$ as unique invariant measure only for $h\neq 0$ or
$\beta\,<\,\beta_c$. 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 with
external field h and inverse temperature $\beta$ in the infinite
lattice $\bf Z^d$ or in the finite set $\Lambda$ with boundary
conditions $\tau$. Then we have :\bigskip
{\bf Theorem 5.1}\par {\it
\item{\bf a)} Assume that $\beta\,<\,\beta_c$. Then 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
\item{\bf a)} Thanks to Theorem 3.1 we need only to verify our
{\it weak mixing} condition (3.1). This in turn follows from part i)
of Theorem 2 of a recent paper by Higuchi [Hi].
\item{\bf b)} Also in this case we verify the
{\it weak mixing} condition and for this purpose
we use a result by
Martyrosian [M]. In order to state his result we need some notations.
A finite subset A of the cubic lattice is said to be connected if
for any two sites x and y in A there is a sequence of nearest
neighbor sites $x_o,\,x_1,\,..x_n$ in A connecting x to y i.e.
$x_o\,=\,x$ and $x_n\,=\,y$. The finite connected set A will be said
to be simply-connected if its complement is connected. Given a
connected set A, $\phi (A)$ will be the smallest simply-connected set
containing A. Then we have :\bigskip
{\bf Theorem 5.2}
(Martyrosian)\par {\it There exists a positive constant $\beta_o$ such
that for any $\beta \;\geq\;\beta_o$ and any $h\;>\;0$ there exists
a positive constant $C$ such that for every $L$ :
$$\mu_{Q_L}^{- ,\, \beta
,\,h}(\sigma\, ;\;\exists \hbox{ a connected set A with }\sigma
(x)\,=\,+1\quad\forall \,x\,\in\,A\; \hbox{and
}Q_{L\,-\,Clog(L)}\subset \phi (A) )$$ tends to one as $L\,\to
\,\infty$.}\bigskip {\bf Corollary 5.1}\par {\it There exists a positive
constant $\beta_o$ such that for any $\beta \;\geq\;\beta_o$ and
any $h\;>\;0$ there exist positive constants $C$ and $\epsilon$
such that for every $L$: $$\mu_{Q_L}^{+ ,\, \beta ,\,h}(\sigma
(0))\;-\; \mu_{Q_L}^{- ,\, \beta ,\,h}(\sigma (0))\;
\leq\;C\hbox{exp}(-\epsilon L)$$
}{\bf Proof}\par Using the method of Appendix 1, it is enough to show
that $$L^{d-1}[\mu_{Q_L}^{+ ,\, \beta ,\,h}(\sigma (0))\;-\;
\mu_{Q_L}^{- ,\, \beta ,\,h}(\sigma (0))]\Eq(5.1)$$
tends to zero as $L\,\to \,\infty$. For this purpose let us define
$$\eqalign{L_k\;&=\;[L/2\,+\,kClog(L)]\cr
\Lambda_k\,&=\,Q_{L_k}}\Eq(5.2)$$
for k=1...$K\,=\,[{L\over 2Clog(L)}]$ where $C$ is the constant
appearing in Theorem 5.2 and $[\cdot]$ denotes integer part. Let
also, for any k, $\Omega_k$ be the event that in the annulus
$\Lambda_k\setminus \Lambda_{k-1}$ there exists a connected set
$\Gamma_k$ such that: \item{i)}$\sigma_x\,=\,+1\quad \forall \; x\in
\Gamma_k$
\item{ii)}The set $Q_L\setminus\Gamma_k$ splits into two
disjoint connected sets A and B, with $0\in A$. \par
Let finally $\hat \Omega$ be the union of the events $\Omega_k$
. It is easy to see, using F.K.G., that: $$\mu_{Q_L}^{-
,\, \beta ,\,h}(\sigma (0)|\hat \Omega)\;\geq\; \mu_{Q_L}^{+
,\, \beta ,\,h}(\sigma (0))$$ so that
$$L^{d-1}[\mu_{Q_L}^{+ ,\, \beta ,\,h}(\sigma (0))\;-\;
\mu_{Q_L}^{- ,\, \beta ,\,h}(\sigma (0))]\,\leq\,
2L^{d-1}\mu_{Q_L}^{- ,\, \beta ,\,h}(\hat \Omega^c)\Eq(5.3)$$
Thus we are left with the estimate of
$\mu_{Q_L}^{- ,\, \beta ,\,h}(\hat \Omega^c)$.\par
If $\chi_k$ denotes the characteristic function of the event
$\Omega_k^c$, we can write:
$$\mu_{Q_L}^{- ,\, \beta ,\,h}(\hat \Omega^c)\;\leq \;
\mu_{Q_L}^{- ,\, \beta ,\,h}(\prod_{k=2}^K\chi_k)
\mu_{\Lambda_1}^{- ,\, \beta ,\,h}(\chi_1)\Eq(5.4)$$
where we have used the D.L.R. equations and the fact that
$$\mu_{\Lambda_k}^{\tau ,\, \beta ,\,h}(\chi_{k})\,\leq\,
\mu_{\Lambda_k}^{- ,\, \beta ,\,h}(\chi_k)$$
for any $\tau$.\par
By Theorem 5.2 we have that :
$$\mu_{\Lambda_1}^{- ,\, \beta ,\,h}(\chi_1)\,\leq\,1/2\Eq(5.5)$$
provided that L is large enough. \par
Thus, if we iterate \equ(5.4) $K$-times, we get that
$$\mu_{Q_L}^{- ,\, \beta ,\,h}(\hat \Omega^c)\;\leq \;2^{-K}$$
which clearly proves the corollary since
$K\,\geq\,L/2Clog(L)\,-1$.\bigskip
\item{\bf c)} In
this case we verify that for any $h\;>\;0$ there exist two positive
constants $\beta_o(h)$ and $L_o(h)\,\geq \,\bar L$, where $\bar L$
is the numerical constant appearing in Theorem 4.3, such that if
$\beta \;\geq\;\beta_o$ then $SM(L_o,C,\gamma)$ mixing condition
holds. Let us fix $h\;>\;0$ and let us choose
$2L_o(h)\,=\,\lceil{A\over h}\rceil$. It is simple to verify
that if the constant $A$C is taken large enough (e.g. $A$ = 4 in d=2
and
$A$ = 6 for d=3 ) then 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
$\tau$. Thus if we estimate the variation distance appearing in our
$SM(L_o,C,\gamma)$ mixing condition by :
$$ 2\sup_{\tau}
\mu_\Lambda^{\tau ,\, \beta ,\,h}
(\;\exists\;x\,\in\,\Lambda_{L_o(h)}\quad ;\;\sigma (x)\,=\,-1
\,)\Eq(5.6)$$ then we can make the variation distance as small as we
like by taking $\beta$ large enough. Finally by taking the
constant $A$ large enough we can make the length scale $L_o(h)$
larger than the numerical constant $\bar L$ appearing in Theorem
4.3 . It is important to stress here that $\bar L$ does not depend
on the parameters $\beta$ and $h$ of the Hamiltonian.\par The theorem
is proved.
\numsec=1\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Appendix 1}\par\noindent
\centerline{\bf A simple proof of theorem 2.4}
\centerline{\bf in the attractive case}\bigskip
We give a simple proof of the "effectiveness" of the
$SM(L_o,C,\gamma)$ mixing condition in the case the interaction
$J(X)$ is ferromagnetic.\par
Let $\Lambda$ be an $L_o$-compatible set, let $\tau$ and
$\tau^{(y)}$ be
boundary configurations outside $\Lambda$, where $\tau^{(y)}
$ is
obtained from $\tau$ by flipping the spin at y
$\in \, \Lambda^c$ and
let x be a site
of $\Lambda$. Without
loss of generality we assume that $\tau^{(y)}\,\geq \, \tau$.
Given $C$ and $\gamma$ we will
prove that there exists a constant $\bar L \,\geq \,R$
such that if $SM(L_o,C,\gamma)$ holds for
$ L_o\,\geq\,\bar L$,
then there exist positive constants $C_o$
$\gamma '$ such that :
$$\mu_{\Lambda}^{\tau^{(y)}}(\sigma_x\,=\,+1)\;-\;
\mu_{\Lambda}^{\tau}(\sigma_x\,=\,+1)\;\leq
\;C'\hbox{exp}(-\gamma '\hbox{dist}(x,y)\,)\Eqa(a1.1)$$
Clearly \equ(a1.1) proves the theorem. Let in fact
$\mu_{\Lambda}^{\tau^{(y)},\tau}(\sigma ,\eta )$ be a joint
representation of $\mu_{\Lambda}^{\tau^{(y)}}$ and
$\mu_{\Lambda}^{\tau}$ which is above
the diagonal. Then we have:
$$Var(\mu^{\tau}_{\Lambda , V}\,,\,
\mu^{\tau^{(y)}}_{\Lambda , V})\; \leq \;\sum_{x\in
V}\;
\mu_{\Lambda}^{\tau^{(y)},\tau}(\sigma_x\,\neq\,\eta_x)\Eqa(a1.2)$$
Using \equ(a1.1) and the fact that
$\mu_{\Lambda}^{\tau^{(y)},\tau}$ is
above the diagonal, the r.h.s. of \equ(a1.2) is bounded by
$C'$exp($-\gamma '\hbox{dist}(y,V)$) for some constant $C'$.\par
We now prove \equ(a1.1). Let, for any $\Lambda,\,\sigma\,\geq
\,\eta\, \in \Omega_{\Lambda ^c} $ $\mu_{\Lambda}^{\sigma,\eta}$
denote the joint representation of the
Gibbs states in $Q$ with boundary conditions $\sigma$
and $\eta$
which is above the diagonal. Let also, for any x in $\Lambda$,
$Q_{x}$ be a cube of side $2L_o\,+\,1$ such that x $\in \,
Q_{x}$ and dist(x,$\partial Q_x\setminus (\partial
\Lambda \cap \partial Q_{x} )\,)\;\geq \,{L_o\over
2} \,)$. Clearly such a cube always exists. Let
$\partial_rQ_{x}$ be the set of sites y in $\Lambda\setminus
Q_{x}$ with dist(y,$Q_{x})\,\leq\,r$ where r is the range of the
interaction. Then we can write:
$$\mu_{\Lambda}^{\tau^{(y)}}(\sigma_x\,=\,+1)\;-\;
\mu_{\Lambda}^{\tau}(\sigma_x\,=\,+1)\;=
\;\sum_{\sigma ,\sigma '}\mu_{\Lambda}^{\tau^{(y)},\tau}(\sigma
,\sigma
') \mu_{Q_x}^{\sigma,\sigma '}(\eta _x\,\neq \, \eta
'_x)\;\leq
$$ $$\sum_{\sigma ,\sigma '}\mu_{\Lambda}^{\tau^{(y)},\tau}(\sigma
,\sigma ')\chi (\exists \,z\,\in
\,\partial_rQ_x\,;\,\sigma_z\,\neq
\,\sigma_z'){ C\hbox{exp}(-\gamma L_o/2)}\Eqa(a1.3)$$
where we have used $SM(L_o,C,\gamma)$ in order to
estimate $\mu_{Q_x}^{\sigma,\sigma '}(\eta _x\,\neq
\, \eta '_x)$. It is at this point that attractivity becomes
important. Since
$\mu_{\Lambda}^{\tau^{(y)},\tau}$ is
above the diagonal the term
$\mu_{\Lambda}^{\tau^{(y)},\tau}(\sigma_z\,\neq
\, \sigma_z')$ is equal
to $\mu_{\Lambda}^{\tau^{(y)}}(\sigma_z\,=\,+1)\;-\;
\mu_{\Lambda}^{\tau}(\sigma_z\,=\,+1)$.
Thus, if we denote with $F(x)$ the l.h.s. of \equ(a1.3), we get :
$$F(x)\,\leq\, { C\hbox{exp}(-\gamma L_o/2)}\sum_{z\in
\partial_rQ_x} F(z)\Eqa(a1.4)$$
Iteration of \equ(a1.4) gives that
F(x) is bounded by the series :
$$F(x)\,\leq\,\sum_{n\,\geq\,[{L\over
L_o+R}]}(Cc(d,r)L_o^{d-1}\hbox{exp}(-\gamma L_o/2))^n\Eqa(a1.5)$$
where c(d,r) is a numerical constant. Clearly \equ(a1.5) gives
the desired exponential bound for F(x) provided that
$L_o$ is large enough depending on r and the dimension
d.\bigskip
\numsec=2\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Appendix 2}\bigskip\par\noindent
In this Appendix we want to prove Theorem 2.6. First we need some
definitions.\par
Let $Q_{L,3L} (0) $ be the box:
$$Q_{L,3L}(0)=\{x\in{\bf Z^d};|x_i|\leq {3L-1\over 2} , \; i=1\dots ,
d -1,\; |x_d|\leq{L-1\over 2}\}$$
Let $\bar \Gamma$ be the set of all subsets of $Q_{L,3L}(0)$ which
\item{i)} are
union of cubes $Q_L(x), x = Ly, y \in {\bf Z^d}$
\item{ii)} contain $Q_L(0)$ and
\item{iii)} are symmetric with respect to
all directions of the lattice \par Let us call "vertical" the $d$-th
direction of the lattice and "horizontal" the hyperplane orthogonal
to it. \par For any $\Lambda \in \bar \Gamma$ consider pairs of sites
$k,k'$ in $\partial ^+_r \Lambda $ "adjacent" to opposite horizontal
faces of $\Lambda $ in the sense that there exist
$x , x' \in \Lambda $,
$x\,=\,(x_1, ... ,x_d),x'\,=\,(x'_1, ... ,x'_d)$ with
$|x-k| \leq r,\; |x'-k'| \leq r, \; |x_d| =|x'_d|= {L-1\over 2}
, x_d =- x'_d$.\par
Let $$
\Delta_k = \{ x \in \Lambda : \hbox {dist} (x,k) \leq r\}$$
$$
\Delta_{k^{'}} = \{ x \in \Lambda : \hbox {dist} (x,k') \leq r\}$$
We also assume that if the horizontal distance between
$\Delta_k,\Delta_{k^{'}}$ is larger than one,
then there exists an $x\in \Lambda$ such that the cube
$Q_L(x)$ is such that $ \Lambda \supset Q_L(x)\supset \Delta_k ,\;
\Delta_{k^{'}} \subset \Lambda \setminus Q_L(x)$.\par
Following a
simple argument already used by Stroock and Zegarlinski (see
[SZ], proof of Eq.(3.4) )
we write, for $y \in \partial^+_r\Lambda$:
$$
\mu^{\tau}_{ \Lambda}(f) - \mu^{\tau^{(y)}}_{ \Lambda}(f) =
\mu^{\tau}_{ \Lambda}(f\psi^{(y)}_{\Lambda}) - \mu^{\tau}_{
\Lambda}(f) \mu^{\tau}_{ \Lambda}(\psi^{(y)}_{\Lambda})\Eqa(a2.6)$$
with $\psi^{(y)}_{\Lambda}$ such that:
$$
\mu^{\tau}_{ \Lambda}(\psi^{(y)}_{\Lambda}) = 1
$$
and
$$
\Vert \psi^{(y)}_{\Lambda} \Vert \leq \exp (4 \Vert U\Vert)\Eqa(a2.7)$$
Let us now state and
prove a Lemma.
\par \bigskip
{\bf Lemma A2.1 }\bigskip \noindent
{\it In the general case ( hypotheses H1,H2 satisfied)
suppose that $SM(Q_L,C,\gamma) $ holds for some $ C>0 , \gamma
>0,\, L > 8r,\; 2d r (L+r)^{d-1}\hbox{exp}(-\gamma L/8) < 1$. \par
\par\noindent
Then, for any $\Lambda \in \bar \Gamma, k,k'$ as above, given any
cylindrical function $f$, with support $S_f = \Delta_k,\;$ we have
: $$\sup _{\tau \in \Omega_{\Lambda^c}} \mu^{\tau}_{\Lambda}(f,g)
\leq \Vert f\Vert C' \exp (- \gamma ' L ) \Eqa(a2.1)$$
with
$$ C' = C \exp (4\Vert U\Vert) , \quad
4L \exp(-\gamma L/8) =\exp(- \gamma'L), $$
where $g = \psi^{(k')}_{\Lambda} $, and
$$
\Vert U \Vert = \sum _{X\subset \subset \Z, X \ni O}
| U_X| $$}
\bigskip
{\bf Proof}
\bigskip
For simplicity we shall only consider the case $d=2$ where
$\bar \Gamma$ contains only the square $Q_L(0)$ (for which (A2.1) is
true by hypothesis with $C' = C,\; \gamma' = \gamma$)
and the rectangle
$\Lambda \equiv Q_{L,3L}(0)$ with edges parallel to the $1,2$ (
horizontal and vertical) coordinate axes with length, respectively,
$L_1=3L,L_2=L$. The easy extension of the argument to the general,
d-dimensional, case is left to the reader.\par
Considering $\Lambda$ we distinguish two cases:
\item{1)} $\Delta_k ,\Delta_{k^{'} }$ have horizontal distance $\leq
L/2$ ; namely:
$$
\inf _{x\in \Delta_k,y \in \Delta_{k^{'}} } |x_1 - y_1| \leq L/2$$
\item {2)} $\Delta_k ,\Delta_{k^{'}} $ have horizontal distance $>
L/2$ .\bigskip
In the first case we observe that there exists $x \in \Lambda$
such that the square $Q_L(x)$, that for notation conveniece we call
$V$, is contained in $\Lambda$, contains both $\Delta_k
,\Delta_{k^{'}}$ and is such that dist$(\Delta_k,\partial V \cap
\Lambda), \;$ dist$(\Delta_{k^{'}},\partial V \cap \Lambda) \; \geq
L/8$ .\par
We then have, $\forall \; \tau \; \in \Omega
_{\Lambda^c}$ : $$
\mu^{\tau}_{\Lambda}(f g) = \sum_{\omega \in \Omega_{\Lambda
\setminus V}} \mu^{\tau}_{\Lambda,\Lambda \setminus V}(\omega)
\mu^{\tau, \omega}_{V}(fg)\Eqa(a2.2)$$
from which we get:
$$
\mu^{\tau}_{\Lambda}(f ,g)\equiv \mu^{\tau}_{\Lambda}(f g)-
\mu^{\tau}_{\Lambda}(f )\mu^{\tau}_{\Lambda}(g)=
\epsilon^{\tau}_{\Lambda}(f, g) +
\bar \epsilon^{\tau}_{\Lambda}(f, g)\Eqa(a2.3)$$
with
$$
\epsilon^{\tau}_{\Lambda}(f, g)=
\sum_{\omega \in \Omega_{\Lambda
\setminus V}}\mu^{\tau, \omega}_{ V}(f,g)
\mu^{\tau}_{\Lambda,\Lambda \setminus V}(\omega)\Eqa(a2.4)$$
$$
\bar \epsilon^{\tau}_{\Lambda}(f, g)=
\sum_{\omega,\omega' \in \Omega_{\Lambda
\setminus V}}
\mu^{\tau}_{\Lambda,\Lambda \setminus V}(\omega)
\mu^{\tau}_{\Lambda,\Lambda \setminus V}(\omega')
\mu^{\tau, \omega}_{ V}(f)[ \mu^{\tau, \omega}_{ V}(g)-
\mu^{\tau, \omega'}_{ V}(g)]
\Eqa(a2.5)$$
>From $SM(Q_L,C,\gamma) $, \equ(a2.4) we have immediately:
$$
\epsilon^{\tau}_{\Lambda}(f, g) \leq
\Vert f\Vert C
\hbox{exp} (- \gamma L/8 ) \Eqa(a2.5')$$
>From $SM(Q_L,C,\gamma) $, \equ(a2.5) and \equ(a2.6) we get
$$
\bar \epsilon^{\tau}_{\Lambda}(f, g) \leq
4L r C \Vert f\Vert\Vert g\Vert \ exp (-\gamma
L/4)\Eqa(a2.8)$$ Consider now the the second case (
horizontal distance of $\Delta_k ,\Delta_{k^{'}}\; > L/2 $ ).
By hypothesis there exists an $x\in \Lambda$ such that the
square $V
\equiv Q_L(x)$ is such that $ Q_{L,3L} \equiv \Lambda \supset
V\supset \Delta_{k} ,\;
\Delta_{k'} \subset \Lambda \setminus V$ (suppose, for instance,
that $\Delta_k$, between $\Delta_k ,\Delta_{k^{'}}$ is the set at
largest horizontal distance from the vertical edges of $\Lambda$).
\par
We then have:
$$
\mu^{\tau}_{\Lambda}(f ,g) =
\sum_{\omega,\omega' \in \Omega_{\Lambda
\setminus V}}
\mu^{\tau}_{\Lambda,\Lambda \setminus V}(\omega)
\mu^{\tau}_{\Lambda,\Lambda \setminus V}(\omega')
g(\omega)[ \mu^{\tau, \omega}_{ V}(f)-
\mu^{\tau, \omega'}_{ V}(f)]\Eqa(a2.9)$$
>From
\equ(a2.1),\equ(a2.6) and \equ(a2.9) we get
$$
\mu^{\tau}_{\Lambda}(f ,g) \leq
4L C \Vert f\Vert\Vert g\Vert\Vert \psi\Vert \hbox{exp} (-\gamma
L/8) \Eqa(a2.10)$$
From\equ(a2.3),\equ(a2.5'),\equ(a2.8) and \equ(a2.10) we get
the Lemma.\bigskip
>From Lemma A2.1 and Proposition 3.1, Eq.'s (3.9),
(3.11) of [O] we get that there exists $L\equiv L(C,\gamma )$ such
that Condition $C_L$ of [OP] (see eq. (1.8) there) holds. Then, from
Propositions 2.5.1, 2.5.2, 2.5.3, 2.5.4 of [OP] Theorem 2.6
follows.
\tolerance=10000
\numsec=3\numfor=1
\vskip 1cm
{\bf Appendix 3}\par\noindent
\bigskip
In this final Appendix we prove Theorem 2.5'. Our goal is to show that theorem 2.6 can be
viewed as a corollary of Theorem 2.2 .\par
Let $\L$ be a subset of $\Z$, let $\t$ be a boundary configuration
outside $\L$ i.e. $\t\in \O_{\L^c}$, let
$y\,\in \,\partial_r^+\L$ and let $\D\,\subset\,\L$. We want to
estimate
$$Var(\mu^\t_{\L,\D}\,,\,\mu^{\t^{(y)}}_{\L,\D})\Eqa(a3.1)$$
by supposing true $K(\L_o,\d)$ for some finite set $\L_o$ and
$\d\,<\,1$. \par
For this purpose let for any $x\,\in \,\D$
$$l_x\;=\;\hbox{dist}(x,y)$$ and let
$$B\;=\;\bigcup_{x\in \D}\{z\in \Z
;\;\hbox{dist}(z,x)\,<\,l_x\,\}\,\cup\,\L\Eqa(a3.2)$$ Then by
construction $\hbox{dist}(\D,\partial_r^+B)\, \geq\,\inf_{x\in
\D}l_x\;=\;\hbox{dist}(\D,y)$ and $y\,\in\,\partial_r^+B$.\par The
idea at this point is to estimate \equ(a3.1) by applying theorem 2.2
to a suitable " Gibbs " measure $\nu^\t_B$
on $\O_B$, whose specifications satisfy,
thanks to $K(\L_o,\d)$, the condition $DSU(\L_o,\d)$ with
$\d\,<\,1$. In order to define the new measure $\nu^\t_B$, let us
denote by $\xi$ the restriction of the configuration $\t$ to the set
$B\setminus \L$; by abuse of notation, the restriction of $\t$
to $\Z\setminus B$ will also be called $\tau$. If for every
configuration $\s \,\in\,\O_B$ we denote by $\s_{A}$ its restriction
to $A\subset B$,
then the measure $\nu^\t_B$ is given by: $$\eqalign{\nu^\t_B
(\s)\;&=\;0\quad\quad \hbox{if } \s_{B\setminus \L}\,\neq\,\xi\cr
\nu^\t_B (\s)\;&=\;\mu^{\t\xi}_{\L}(\s_\L)\quad \hbox{if }
\s_{B\setminus \L}\,=\,\xi}\Eqa(a3.3)$$
where $\t\xi$ has been defined n (1.1).\par
Thus, by construction,
\equ(a3.1) can be written as :
$$Var(\nu^\t_{B,\D}\,,\,\nu^{\t^{(y)}}_{B,\D})\Eqa(a3.3bis)$$ It is
easy to check that $\nu^\t_B$ is "Gibbsian" in the sense that it
satisfies the DLR equations for the following local
{\it specifications} $q^\zeta_V$ :
$$q^\zeta_V(\sigma_V)\,=\,\mu_{V\cap\L}^{\zeta}(\s_{V\cap \L}){\bf
1}_{(\s_{V\cap (B\setminus\L)}= \x_{V\cap
(B\setminus\L)})}\Eqa(a3.4)$$ where $\zeta\in \O_{V^c}$ and in general,
for any set A $$\eqalign{{\bf 1}_{(\s_A=\eta_A)}\;&=\;0\quad
\hbox{if } \s_A\neq\eta_A\cr {\bf 1}_{(\s_A=\eta_A)}\;&=\;1\quad
\hbox{if } \s_A\,=\,\eta_A}$$ We next show that $K(\L_o,\d)$, with
$\d\,<\,1$, implies that the specifications $q^\zeta_V$
satisfy $DSU(\L_o,\d)$ with $\d\,<\,1$ {\it uniformly} in the
location of the cube $\L_o$ inside the set B.\par Thus, let us
choose $x\in B$ in such a way that $\L_o\,+\,x\;\subset \,B$ and let
us compute : $$\sup_{\zeta ,\zeta
^{(y')}}KROV_{\rho}(q^\zeta_{\L_o+x},\,q^{\zeta^{(y')}}_{\L_o+x})
\Eqa(a3.6)$$
where $\rho$ is given by (1.8) and $y'\in \partial_r^+(\L_o+x)$. We
have to distinguish three different cases: \item{i)}
$\L_o+x\,\subset \,B\setminus \L$ ; in this case \equ(a3.6) is zero by
construction. \item{ii)} $\L_o+x\,\subset \,\L$; in this case
\equ(a3.6) is equal to:
$$\sup_{\zeta ,\zeta^{y'}}KROV_{\rho}
(\mu^\zeta_{\L_o+x},\,\mu^{\zeta^{(y')}}_{\L_o+x})
\Eqa(a3.7)$$
which, because of $K(\L_o,\d)$, is bounded from above by $\a_{y'}$
with $$\sum_{y' \in \partial_r^+(\L_o+x)}\a_{y'}\,\leq \, \d\vert
\L_o\vert \Eqa(a3.8)$$
\item{iii)} $\L_o+x$ intersects
both $\L$ and $B\setminus \L$; in this case let $V\,=\,(\L_o+x)\cap
\L$. Then \equ(a3.6) becomes equal to :
$$\sup_{\zeta ,\zeta
^{y'}}KROV_{\rho}(\mu^{\zeta\xi}_{V},\,\mu^{\zeta^{(y')}\xi}_{V})
\Eqa(a3.9)$$
where $\zeta\xi$ is the configuration in $V^c$ which coincides
with $\zeta$ outside $\L_o+x$ and with $\xi$ in
$(\L_o+x)\cap(B\setminus \L)$. Again because of
$K(\L_o,\d)$, \equ(a3.9) is bounded from above by $\a_{y'}$ with
$$\sum_{y' \in \partial_r^+(\L_o+x)}\a_{y'}\,\leq \, \d\vert
\L_o\vert \Eqa(a3.10)$$ \par We stress that it is precisely in the
third case iii) that one uses the full strength of $K(\L_o,\d)$
since the set V can be an arbitrary subset of $\L_o$.\par At this
stage we can apply theorem 2.2 to the measure $\nu^\t_B$ and
estimate \equ(a3.3bis) from above by:
$$Var(\nu^\t_{B,\D}\,,\,\nu^{\t^{(y)}}_{B,\D})\,\leq\,C\sum_{x\in \D
,z\in \partial_r^+B}\hbox{exp}(-\gamma '\vert x\,-\,z\vert)\,\leq
\, C'\hbox{exp}(-\gamma ''\hbox{dist}(\D ,y)) \Eqa(a3.11)$$ for a
suitable ,positive constant $\gamma ''$. \bigskip {\bf Remark} We notice that Theorem 2.2 has been
stated only for translation invariant Gibbs measures and certainly
the specifications in \equ(a3.4) do not satisfy this requirement.
However, as one can easily check in the original proof in [DS1],
translation invariance becomes irrelevant provided that one is able
to verify $DSU(\L_o,\d)$ with $\d\,<\,1$ {\it uniformly} in the
location of the cube $\L_o$ inside the set B.\bigskip
The theorem is proved.
\def\refj#1#2#3#4#5#6#7{\parindent 2.2em
\item{[{\bf #1}]}{\rm #2,} {\it #3\/} {\rm #4} {\bf #5} {\rm #6}
{(\rm #7)}}
\tolerance=10000
\vskip 1cm
{\bf Acknowledgments}\bigskip\noindent
We are in debt with several colleagues for many useful discussions.
We want to thank, in particular, G.B.Giacomin for actively taking
part to discussions in the very early stage of the work;
R.L.Dobrushin, A.v.Enter . 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 with Aizenman we received a preprint by
Higuchi where this and many other new interesting results were
proved for the Ising model. We would also like to thank H.T.Yau for
informing us about his work prior to publication. 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.
\pagina
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\end
ENDBODY