%% -------------------------------------------------------------------------
%%
%% Macro 1.1 - agg: 11-5-95
%%
%%
%% -------------------------------------------------------------------------
%%
%% Equazioni con nomi simbolici
%%
%% $$ x=1 \Eq(ciccio) $$
%% By \equ(ciccio) we get ...
%%
%% Dentro \eqalignno invece di \Eq si usa \eq.
%% Per far riferimento ad una formula definita nel futuro: \eqf
%% -------------------------------------------------------------------------
%%
%% Teoremi con nomi simbolici
%%
%% \nproclaim Proposition[peppe].
%% If bla bla, then blu blu.
%%
%% {\it Proof.} It is easy to check that ...
%%
%% Because of Proposition \thm[peppe], we know that ...
%%
%% Per far riferimento ad un teorema definito nel futuro: \thf
%% Per far riferimento a formule o teoremi definiti in altri file
%% di cui si dispone il .aux, includere lo statement
%% \include{file}
%% e usare \eqf o \thf
%%
%% Se e' presente il comando \BOZZA, viene stampato sul margine
%% sinistro il nome simbolico della formula (o del teorema).
%% -------------------------------------------------------------------------
%%
%% All'inizio di ogni sezione includere
%%
%% \expandafter\ifx\csname sezioniseparate\endcsname\relax%
%% \input macro \fi
%% \numsec=n
%% \numfor=1\numtheo=1\pgn=1
%%
%% dove n e' il numero della sezione
%% Le Appendici hanno numeri negativi (\numsec=-1, -2, ecc...)
%% -------------------------------------------------------------------------
%%
%% Fonti
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%% Vengono caricate le fonti msam, msbm, eufm. Se non sono
%% disponobili commentare lo statement \fnts=1
%% -------------------------------------------------------------------------
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%
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%-------------------------------------------------------------------
%
% ------- Per compatibilita'
%
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%
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%
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%
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%
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%
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%------------------------------ tilde
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%-------------------------------------------------------------------
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%@Ib
%--------------------------------- INIZIO
%
\expandafter\ifx\csname sezioniseparate\endcsname%--- non toccare
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%
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%
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%
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%
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%
% Address
%
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%
% Abstract
%
\def\abstract#1
{
\noindent{\bf Abstract.\ }#1\par}
%
\vskip 1cm
\centerline{\ttlfnt Relaxation of Disordered Magnets in the Griffiths' Regime}
\vskip 0.5truecm
\author{
F. Cesi $^{1}$,
C. Maes $^{2}$ and
F. Martinelli $^{3}$
}
%
\address{%
{\ninerm
$^1$
Dipartimento di Fisica,
Universit\`a \lq\lq La Sapienza", P.le A. Moro 2, 00185 Roma,
Italy \\
$\phantom{^1}$
e-mail: cesi@vaxrom.roma1.infn.it \\
$^2$
Instituut voor Theoretische Fysika, K.U. Leuven, Celestijnenlaan
200D B-3001 Leuven, Belgium \\
$\phantom{^2}$
and Onderzoeksleider N.F.W.O, Belgium \\
$\phantom{^2}$
e-mail: Christian.Maes@fys.kuleuven.ac.be \\
$^3$
Dipartimento di Energetica,
Universit\`a dell' Aquila, Italy \\
$\phantom{^3}$
e-mail: martin@mat.uniroma3.it
}}
%
\bigno
\abstract{
We study the relaxation to equilibrium of discrete spin systems
with random many--body (not necessarily ferromagnetic) interactions in the
Griffiths' regime. We prove that the speed of convergence to the unique
reversible Gibbs measure is almost surely faster than any stretched
exponential, at least if the probability distribution
of the interaction decays faster than exponential (e.g. Gaussian).
Furthermore, if the interaction is uniformly bounded,
the {\it average over the disorder\/} of the time--autocorrelation
function, goes to equilibrium as
$\exp[- k (\log t)^{d/(d-1)}]$ (in $d>1$), in agreement with
previous results obtained for the dilute
Ising model.
}
%
\vskip 1cm
\noindent
{\bf Key Words:}
Random spin systems, Glauber dynamics, relaxation time,
Griffiths singularities, logarithmic Sobolev inequalities
{\parindent=0pt
\footnote{}{\ninerm
Work partially supported by grant CHRX-CT93-0411 of the
Commission of European Communities\acapo
}
\footnote{}{\ninerm
Mathematics Subject Classification: 82B44, 82C22, 82C44, 60K35}
}
\vfill\eject
\endgroup
%--------------------------------- INIZIO
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \input xmac\fi
\numsec=1
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\pgn=1
\beginsection 1. Introduction
In the present paper we study the speed of convergence to equilibrium
of a single spin--flip stochastic dynamics with a reversible
Gibbs measure with {\it random\/} interactions in the so called
{\it Griffiths' phase.}
For simplicity (our models are introduced
in the next section) consider here a nearest--neighbor Ising model on the
$d$-dimensional
lattice with coupling coefficients $J = \{J_{xy}\}$. The $\{J_{xy}\}$,
are independent and identically distributed real--valued random
variables.
If the $\{J_{xy}\}$ are uniformly bounded, then at
all sufficiently high temperatures Dobrushin's uniqueness theory applies
and detailed information about
the unique Gibbs measure and the relaxation to equilibrium of
an associated
Glauber dynamics are available using the concept of
complete analyticity \ref[DS], \ref[SZ], \ref[MO1] and \ref[MO2].
This regime is usually referred to as the {\it
paramagnetic} phase.
There is then a range of temperatures, below the paramagnetic phase,
where, even if the Gibbs state is unique,
certain characteristics of the paramegnetic phase like
the analyticity of the free energy as a
function of the external field disappear.
This is the so called {\it Griffiths' regime\/} \ref[G]
(see also \ref[F] for additional
discussion on this and many other related topics).
This ``anomalous behavior'' is caused by the presence of arbitrarily
large clusters of bonds associated with ``strong'' couplings $J_{xy}$, which
can produce a long--range order inside the cluster.
Even above
the percolation threshold, \ie when one of such clusters
is infinite,
there may be a Griffiths phase, at temperatures between
a certain critical temperature $T_c(\hbox{disordered})$
and the critical temperature for the ``pure system''
(\ie the system with ``strong'' couplings everywhere on $\Z^d$).
What happens is that for almost all realizations of the disorder $J$
and for all sitex $x$
there is a finite length scale $l(J,x)$, such that correlations
between $\s(x)$ and $\s(y)$
start decaying exponentially at distances greater than $l(J,x)$.
In \ref[BD]
an ``elementary'' approach was given to the problem of uniqueness of the
equilibrium state of
disordered systems in the Griffiths regime (see also \ref[FI]).
In another paper
[D] Dobrushin prepared the mathematical background for the study
of (arbitrary order)
truncated correlation functions for spin glasses.
Bounds on $T_c(\hbox{disordered})$ have been obtained in \ref[ACCN] and
\ref[OPG].
More recent references
where, at least for the statics, the situation has
been considerably cleared up
are \ref[DKP], \ref[GM2], \ref[GM3] and \ref[Be].
In particular, under suitable conditions on the couplings distribution,
one proves that the
infinite volume Gibbs state is unique with
probability one and the static correlation functions decay exponentially
fast
uniformly in the size of the system and its boundary conditions.
Though the system is not
completely analytic there is still sufficient {\it local} analyticity to
ensure many high temperature properties.
The effect of the Griffiths' singularities on
the dynamical properties are much more serious since,
as we will see, the long time
behaviour of any associated Glauber dynamics is dominated by the
islands of strongly coupled spins produced by large statistical
fluctuations in the disorder
(see e.g. \ref[B1] and \ref[B2]).
Consider
the usual stochastic (or kinetic) Ising model associated to the model
discussed above. It is a stochastic spin flip dynamics for which the
(almost sure unique) Gibbs
state is a reversible measure. Let us denote by
$q(J,t)$ the absolute difference between the expectation at time $t$
of e.g. the spin at the
origin starting in some initial state, and its equilibrium value.
Up to now only little rigorous information was
available about the long time behaviour in the Griffiths'
regime of $q(J,t)$ or about its more physically
relevant disorder--average $q(t)$.
The first rigorous result was obtained in \ref[Z] where the absence of
a gap in the spectrum of the Markov generator was proven.
This result, in particular, rules out the possibility
that $q(J,t)$ decays exponentially fast in $t$ with probability one
with a rate independent of $J$.
In \ref[GM1], for quite general
models, an almost sure upper bound of the form
$$
q(J,t) \leq c(J) \exp[-\lambda (\log t)^\nu],
\qquad \nu > 1
$$ (faster than any polynomial) was
derived. This was further improved
for continuous spin systems in \ref[GZ1], \ref[GZ2] to almost sure upper bounds of the form
$$
q(J,t) \leq c(J) \exp[-\lambda t^\delta],
\qquad \delta < 1
$$ in
$d\leq2$
dimensions (stretched exponential), and
$$
q(J,t) \leq c(J)
\exp\bigl[ \, -\epsilon\exp[\, \theta(\log t)^{1/(d-1)^2}\, ] \, \bigr]
$$
(subexponential) for
dimensions $d\geq 3$ and {\it ferromagnetic\/} interactions.
We refer to the original papers for the precise
statement of the results and the models. We only observe that in
\ref[GZ1] the
probability distribution has
only exponential tails and that, in this case, the
stretched exponential bound
cannot be improved in general (see the remark after Theorem \thf[P20]).
One of the main results of the present
paper is the proof that the {\it almost sure}
decay is in fact faster than stretched
exponential at least for random interaction whose
probability distribution decays faster than exponential.
In particular, for bounded interactions,
we
show that
$$
q(J,t)\leq
c(J) \,
\exp\Bigl[ \,
- t \,
\exp\bigl[ \, - k \, ( \log t)^{1 - {1 \over d} } \,
(\log\log t)^{d-1} \, \bigr]
\, \Bigr]
$$
Our general assumption, called (H1), (see Section 3 for a precise
statement) is that in a cube of side length $L$, the correlation
between $\s(x)$ and $\s(y)$ start decaying exponentially fast
if, $|x-y|$ is greater than, say, $L/2$, with probability
(w.r.t the disorder) at least $1 - \exp( - c L)$.
We show that this assumption is indeed implied by the
assumptions used in \ref[GM2] and \ref[DKP]. In a related
paper \ref[CMM] we prove that, at least for the two dimensional
diluted Ising model, (H1) holds in a wider region
of the Griffiths' phase which extends above the percolation threshold.
We also analyze the average (over the
disorder or spatially) $q(t)$ of $q(J,t)$ and prove both upper and
lower bounds of the form
$$
\exp[-\lambda_1 (\log t)^{d/(d-1)}] \le
q(t) \le \exp[ -\lambda_2 (\log t)^{d/(d-1)}(\log\log t)^{-d}]
\Eq(aver)
$$
for suitable constants $\l_1,\, \l_2$.
This result agrees
with the predictions of \ref[B1], \ref[B2], \ref[DRS] and \ref[RSP]
for dilute Ising magnets.
Recent computer simulations \ref[J] suggest that behavior in \equ(aver)
is attained only when $q(t)$ is extremely small, while, at
intermediate times $q(t)$ is better fitted with a stretched
exponential.
We hope to come back to this problem in the next future.
Although we consider only discrete spin systems we don't
see any serious obstacle to extend our upper bounds to compact
continuous spins. The lower bound, on the other side, could be
very different if one considers, for instance, the Heisenberg
model (see our comment at the beginning of Section 4.3).
The basic questions treated in this paper can of course also be put in the
general context of interacting particle systems (with random interactions).
But there no reversibility is guaranteed and the methods of this paper or of
\ref[GZ1], \ref[GZ2] fail in that case. The only general results so
far are contained in \ref[GM1] but they are believed to be far from
optimal.
\bigno
The paper is organized as follows.
\noindent
Section 2 contains the definition of our models.\acapo
Section 3 contains the statements of the main results.\acapo
Section 4 contains several technical tools that are essential for
section 5. In particular we prove that, if the interactions are
bounded, then the relaxation time in an arbitrary set $V$ does not
grow faster than a exponential of the \lq\lq surface " $|V|^{d-1\over d}$.
\acapo
Section 5 represents the core of the paper.
We prove a rather sharp deterministic upper bound on the
logarithmic Sobolev constant for the finite volume Gibbs measure.\acapo
Section 6 is devoted to the proof of the main results. \acapo
Appendix 1 contains the proof of a key geometric bound.
\bigno
{\bf Acknowledgements.}
The authors are grateful to the Schr\"odinger Institute in Wien for
the kind hospitality and the opportunity to start this work.
A particular thank goes to M. \Zar\
for suggesting the main idea in the proof of the
geometric proposition contained in the Appendix 1.
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\beginsection 2. The model
{\it The lattice.}
We consider the $d$ dimensional lattice $\Z^d$ with {\it sites\/}
$x = \{x_1, \ldots, x_d \}$
and norm
$$
|x| = \max_{i \in \{1, \ldots, d\} } |x_i|
$$
The associated distance function is denoted by
$d(\cdot, \cdot)$.
By $Q_L$ we denote the cube of all $x=(x_1,\ldots, x_d) \in \Z^d$ such that
$x_i \in \{ 0, \ldots, L-1 \}$.
If $x\in \Z^d$, $Q_L(x)$ stands for $Q_L + x$.
We also let $B_L$ be the
ball of radius $L$ centered at the origin, \ie
$B_L = Q_{2L+1}( (-L, \ldots, -L))$.
A finite subset $\L$ of $\Z^d$ is said to be
a {\it multiple\/} of $Q_L$ if $\L$ is the union of a
finite number of cubes $Q_L(x_i)$ where $x_i \in L \Z^d$.
If $\L$ is a finite subset of $\Z^d$ we write $\L \ssset \Z^d$.
The cardinality of $\L$ is denoted by $|\L|$.
$\bF$ is the set of all nonempty finite subsets of $\Z^d$.
We define the exterior {\it n--boundary\/} as
$\dep_n \L = \{ x \in \L^c : \, d(x, \L)\le n \}$.
Given $r \in \Zp$ we say that a subset $V$ of $\Z^d$ is {\it r-connected\/}
if, for all $y,z \in V$ there exist $\{x_1, \ldots, x_n \} \sset V$
such that $x_1 = y$, $x_n =z$ and $|x_{i+1} - x_i| \le r$ for $i=2, \ldots, n$.
\medno
{\it The configuration space.}
Our {\it configuration space} is
$\O = S^{\Z^d}$, where $S=\pmu$, or
$\O_V = S^V$ for some $V\subset \Z^d$.
The single spin space $S$ is endowed with the discrete topology
and $\O$ with the corresponding product topology.
Given $\s\in \O$ and $\L \sset \Z^d$ we denote by $\s_\L$
the natural projection over $\O_\L$.
If $U$, $V$ are disjoint, $\s_U \h_V$ is the configuration on $U\cup V$ which
is equal to $\s$ on $U$ and $\h$ on $V$.
If $f$ is a function on $\O$, $\L_f$ denotes the smallest subset
of $\Z^d$ such that $f(\s)$ depends only on $\s_{\L_f}$.
$f$ is called {\it local} if $\L_f$ is finite.
$\cF_\L$ stands for the $\s-$algebra generated by the set of projections
$\{ \p_x \}$, $x\in\L$,
from $\O$ to $\pmu$,
where $\pi_x : \s \mapsto \s(x)$.
When $\L=\Z^d$ we set $\cF \equiv \cF_{\Z^d}$ and $\cF$ coincides
with the Borel $\s-$algebra on $\O$ with respect to the topology
introduced above.
By $\| f\|_\infty$ we mean the
supremum norm of $f$.
The {\it gradient} of a function $f$ is defined as
$$
(\nabla_x f)(\s) = f(\s^x) - f(\s)
$$
where $\s^x \in \O$ is the configuration obtained from $\s$, by flipping the
spin at the site $x$. If $\L \in \bF$ we
let
$$
| \nabla_\L f |^2 = \sum_{x\in \L} ( \nabla_x f)^2
$$
We also define
$$
\tnorm{f} = \sum_{x\in\Z^d} \ninf{ \nabla_x f }
$$
\medno
{\it The interaction and the Gibbs measures.}
We consider an abstract probability space $(\Th, \cB, \bP)$
and a set of real valued random variables
$J = \{ J_A \}$ with $A \in \bF$,
with the properties
\smallno
\item{(1)} $J_A$ and $J_B$ are independent if $A \ne B$
\item{(2)} $J_A$ and $J_{A+x}$ are identically distributed for all
$A \in \bF$ and all $x\in \Z^d$
\item{(3)} There exists $r>0$ such that with $\bP-$probability $1$,
$J_A = 0$ if $\diam A > r$. $r$ is called the {\it range}
of the interaction.
\smallno
The expectation with respect to $\bP$ is denoted by $\bE\,(\cdot)$.
For $x\in \Z^d$, we let
$$
\|J\|_x \equiv \sum_{A\ni x} |J_A|
$$
We write $\|J\|_V = \sup\{ \|J\|_x : \, x\in V\}$.
Given a {\it potential\/} or {\it interaction\/} $J$,
and $V \in \bF$ we define
the Hamiltonian $H^J_V : \O \mapsto \bR$ by
$$
H_V^J(\s) = - \sum_{A: \, A\cap V \ne \emp} J_A \prod_{x\in A} \s(x)
$$
For $\s, \t \in \O$ we also let
$H_V^{J,\t}(\s) = H_V^J (\s_V \t_{V^c} )$
and $\t$ is called the {\it boundary condition}.
For each $V\in \bF$, $\t\in \O$ the
(finite volume) conditional Gibbs measure on $(\O, \cF)$, are given by
$$
\mu^{J,\t}_V(\s) =
\cases{ \bigl(Z^{J,\t}_V\bigr)^{-1}
\exp[ \,- H^{J,\t}_V(\s) \,] & if $\s(x) = \t(x)$
for all $x\in V^c$ \cr
\vphantom{\Bigl(}
0 & otherwise. \cr }
\Eq(finvolmea)
$$
where $Z^{J,\t}_V$ is the proper normalization factor called partition
function.
We will sometimes drop the superscript $J$ if that does not generate
confusion.
Given a measurable bounded function $f$ on $\O$,
$\mu_V f$ denotes the function $\s \mapsto \mu_V^\s(f)$.
Analogously, if $X\in\cF$, $\mu_V (X) \equiv \mu_V \id_X$, where
$\id_X$ is the characteristic function on $X$. $\mu(f,g)$ stands
for the covariance (with respect to $\mu$) of $f$ and $g$.
The set of measures \equ(finvolmea) satisfies the DLR compatibility
conditions
$$
\mu_\L( \mu_V (X) ) = \mu_\L (X) \qquad
\forall\, X \in \cF
\qquad
\forall\,
V\sset \L\ssset\Z^d
\Eq(DLR)
$$
A probability measure $\mu$ on $(\O, \cF)$ is called a {\it Gibbs measure\/}
for $J$ if
$$
\mu( \mu_V (X) ) = \mu (X) \qquad
\forall\, X \in \cF
\qquad
\forall\,
V\in \bF
\Eq(DLRi)
$$
Given any two measures $\mu$ and $\nu$ on $(\O, \cF)$, and given $V \in \bF$
such that for, each $X \in \cF_V$, $\nu(X) = 0$ implies $\mu(X) = 0$, we
define the $\cF_V-$measurable function
$$
{ d \mu \over d \nu} \Bigr|_V : \,
\h \mapsto
{\mu \{ \s\in\O : \, \s_V = \h_V \} \over
\nu \{ \s\in\O : \, \s_V = \h_V \}
}
\qquad
\h \in \O
\Eq(rd)
$$
where $0/0$ means $0$. We have, of course,
$$
\mu( f ) =
\nu( f \, { d \mu \over d \nu} \Bigr|_V )
\qquad
\forall f \in \cF_V
\Eq(rd2)
$$
\medno
{\it The dynamics.}
The stochastic dynamics we want to study is determined by the Markov generators
$L_V^J$, $V\sset \Z^d$, defined by
$$
(L_V^J f)(\s) = \sum_{x\in V} c_J(x,\s) (\nabla_x f)(\s)
\qquad
\s\in\O
\Eq(gnrt)
$$
The nonnegative real quantities
$c_J(x,\s )$
are the {\it transition rates\/} for the process.
\smallno
The general assumptions on the transition rates are
\item{(1)} {\it Finite range interactions.}
If $\s(y)=\s'(y)$ for all $y$ such that $d(x,y)\le r$, then
$c_J(x,\s) = c_J(x,\s')$
\item{(2)} {\it Detailed balance.} For all $\s\in\O$ and $x\in \Z^d$,
$$
\exp\bigl[ - H^J_\xx(\s) \bigr]
c_J(x,\s) =
\exp\bigl[ - H^J_\xx(\s^x) \bigr]
c_J(x,\s^x)
\Eq(dbal)
$$
\item{(3)} {\it Positivity and boundedness.}
There exist non--negative real numbers $c_m$, $\k_1$
$c_M$ and $\k_2$ such that
$$
c_m \nep{ - \k_1 \|J\|_x } \le
\inf_{x,\s} c_J(x,\s) \quad
\hbox{ and }\quad
\sup_{x,\s} c_J(x,\s) \le
c_M \nep{ \k_2 \|J\|_x }
\Eq(bounded)
$$
\smallno
Three cases one may want to keep in mind are
$$
\eqalignno{
&
c_J(x,\s) =
\min\bigl\{ \nep{ - (\nabla_{x} H_\xx)(\s) } \,,\, 1 \bigr\}
&\eq(MET) \cr
&
c_J(x,\s) = \mu^{J, \s}_\xx ( \s^x ) =
\left[ \,
1 +
\nep{ (\nabla_{x} H_\xx)(\s) }
\, \right]^{-1}
&\eq(HB) \cr
&
c_J(x,\s) =
\ov2 \,
\left[ \,
1 +
\nep{ - (\nabla_{x} H_\xx)(\s) }
\, \right]
&\eq(GRAD) \cr
}
$$
Notice that the first two examples, corresponding to the Metropolis and
heat--bath dynamics respectively, satisfy \equ(bounded) with $\k_2 = 0$.
When considering the infinite volume dynamics (see below) we will always assume
uniformly bounded transition rates ($\k_2=0$), even if this
assumption can be relaxed (see Theorem \teo[2.3] in \ref[GZ1]).
We denote by $L_V^{J,\t}$ the operator $L_V^J$ acting on
$L^2(\O,d\mu_V^{J,\t})$ (this amounts to choose $\t$ as the boundary
condition).
Assumptions (1), (2) and (3) guarantee that there exists a
unique Markov process whose generator is $L^{J,\t}_V$, and whose semigroup
we denote by $\{T^{J,\t}_V(t)\}_{t\ge 0}$.
$L^{J,\t}_V$ is a bounded operator on $L^2(\O,d\mu_V^{J,\t})$.
The process has
a unique invariant measure given by $\mu_V^{J,\t}$.
Moreover $\mu_V^{J,\t}$ is {\it reversible} with respect to the
process, \ie $L_V^{J,\t}$ is self--adjoint on $L^2(\O,d\mu_V^{J,\t})$.
\smallno
A fundamental quantity associated with the dynamics of
a reversible system is the gap of the generator, \ie
$$
\gap( L^{J,\t}_V ) =
\inf {\rm spec}\,
(- L^{J,\t}_V \restriction \identity^\perp )
$$
where $\identity^\perp$ is the subspace of
$L^2(\O, d\mu_V^{J,\t})$ orthogonal to the constant functions.
The gap can also be characterized as
$$
\gap( L^{J,\t}_V )
= \inf_{\st f\in L^2(\O, d\mu^{J,\t}_V)}
{\Dir^{\t}_V (f,f) \over \Var^{J,\t}_V(f) }
\Eq(gap)
$$
where $\Dir$ is the Dirichlet form associated with the generator $L$,
$$
\Dir^{J,\t}_V (f,f) =
{1\over 2} \sum_{\s\in \O_V} \sum_{x\in V}
\mu^{J,\t}_V(\s) \, c(x,\s)
\left[ (\nabla_x f)(\s) \right]^2
\Eq(var)
$$
and $\Var^{J,\t}_V$ is the variance relative to the probability
measure $\mu^{J,\t}_V$. When the transition rates are chosen
as in \equ(GRAD), it is easy to verify that
the Dirichlet form takes a particularly simple form
$$
\Dir^{J, \t}_V(f,f) = \ov2 \, \mu^{J, \t}_V( |\nabla_V f|^2 )
\Eq(DG)
$$
We define the {\it logarithmic Sobolev constant\/} $c_s(L^{J,\t}_V)$
associated with the generator $L^{J,\t}_V$ as the infimum over all
$c$ such that, for all positive functions $f$
$$
\mu^{J,\t}_V( f^2 \log f ) \le
c \, \Dir^{J,\t}_V(f,f) +
\mu^{J,\t}_V(f^2) \log \sqrt{ \mu^{J, \t}_V (f^2)}
\Eq(LSC)
$$
We define $c_s(L^J_V) = \sup_\t c_s(L^{J, \t}_V)$
and we denote by $c_s(\mu^{J, \t}_V)$ the logarithmic Sobolev constant
corresponding to the rates \equ(GRAD) and associated Dirichlet
form \equ(DG). Notice the following simple estimates relating $c_s(L^{J,\t}_V)$
to $c_s(\mu^{J, \t}_V)$:
$$
c_m \nep{ - \k_1 \|J\|_V } c_s(L^{J,\t}_V) \le c_s(\mu^{J, \t}_V) \le
c_M \nep{ \k_2 \|J\|_V } c_s(L^{J, \t}_V)
\Eq(LSC.1)
$$
\medno
{\it The infinite volume dynamics.}
Let $\mu$ be a Gibbs measure for $J$.
If the transition rates are bounded, \ie when $\k_2 = 0$, then
the infinite volume generator $L^J$ obtained by choosing $V= \Z^d$ in
\equ(gnrt) is well defined on the set of functions $f$
such that $\tnorm{f}$ is finite. The closure
of $L^J$ in $L^2(\O, d\mu)$ (or in $C(\O)$,
the metric space of all continuous functions on $\O$ with
the sup--distance)
is a Markov generator
(see, for instance Theorems \teo[3.9] in Chapter I and \teo[4.1]
in Chapter IV of \ref[L]),
which defines a Markov semigroup denoted by $T(t)$.
$L^J$ is self--adjoint on $L^2(\O, d\mu)$.
\medno
{\it The block dynamics.}
We will also consider a more general version of
heat--bath dynamics in which more than one spin can flip at once.
Let $\cD = \{ V_1, \ldots, V_n \}$ be an
arbitrary collection of finite sets $V_i \in \bF$ and let
$V = \cup_i V_i$. The generator of the Markov process corresponding
to $\cD$ is defined as
$$
L^J_\cD f = \sum_{i=1}^n ( \mu^J_{V_i} f - f )
$$
From the DLR condition \equ(DLR) it follows that $L^J_\cD$
is self--adjoint on $L^2(\O, d \mu_V^{J,\t})$.
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\beginsection 3. Main results
%@I
In this section we state our hypotheses and our main results on
\smallno
\item{(i)} the
growth of the logarithmic Sobolev constant in a cube of side
$L$ as a function of $L$,
\item{(ii)} the speed of relaxation to equilibrium for the infinite volume
dynamics for a set of potentials $J$ of measure one,
\item{(iii)} the speed of relaxation to equilibrium for the {\it averaged}
infinite volume dynamics.
\smallno
In order to state our hypotheses we need first the following definition.
Given $V\ssset \Z^d$, $n, \a >0$,
we say that the condition $SMT(V, n, \a)$ holds if for all
local functions $f$ and $g$ on $\O$
such that $d(\L_f, \L_g) \ge n$ we have
$$
\sup_{\t\in\O} |\mu_V^{J,\t}(f,g)| \le
|\L_f| |\L_g| \, \|f\|_\infty \|g\|_\infty
\exp[ - \a \, d( \L_f, \L_g ) ]
$$
\medno
Then our hypotheses on the random interactions $J = \{J_A\}_{A\in \bF}$ are:
\smallno
\item{(H1)}
There exist $L_0 \in \Zp$, $\a>0$, $\th>0$ such that for all
$L \ge L_0$
$$
\bP\{ \, SMT( Q_L, L/2 , \a) \, \} \ge
1 - \nep{ - \th L }
$$
\item{(H2)}
There exists $\d >0$, such that
$\bE\, ( \exp( \, \|J\|_x^{1+\d} \, ) ) \equiv G_\d < \infty$
\medno
Some of our result are given in the special case in which the $J_A$'s
are bounded, so we let
\medno
\item{(H3)}
There exists $J_0 >0$ such that with probability $1$,
we have $|J_A|\le J_0$ for all $A\in \bF$.
\bigno
{\it Remark.} The key hypothesis (H1) is
different from the
assumption that appears in the basic references on disordered systems in the
Griffiths phase (see e.g. \ref[GM3], \ref[DKP],
\ref[D], \ref[FI]), \ref[GZ1], \ref[GZ2]).
In these references, in fact, one sets a constraint
on either the inverse temperature
$\beta$ or the external fields (if present). Here we adopt a more
general hypothesis. In Section 3.2 we show that (H1) holds under the
assumption of \ref[GM3] or \ref[DKP], while
in \ref[CMM] we study the two dimensional diluted Ising model above the
percolation threshold.
As far as the second hypothesis is concerned, we observe that it is
definitely stronger than the assumption one needs in order to control the
equilibrium
(see e.g. \ref[DKP] or \ref[GM3]). This fact is, as we hope it will be
clear from the proofs of the various results, almost unavoidable when dealing
with dynamical problems if one wants to get sufficiently precise results.
Relaxation times are in fact much more sensitive
than correlation functions to
the occurrence of small regions of very large couplings
(see also the remark after Theorem \thf[P20]).
In any case we have
focused more on the bounded case (H3) since it appears to be the most
interesting one from the physical point of view.
\bigno
%@E
\beginsubsection 3.1. General theorems
%@Ig
\nproclaim Theorem [P20].
\item{(i)} Assume (H1) and (H2) and let $A_\d = (\log G_\d)^\ov{1+\d} \mmax 1$.
Then there exist $C(d,r,\a,\th)$,
and $L_1(d,r,\a,\th,L_0, A_\d)$ such that
for all $L\ge L_1$,
$$
\bP \bigl\{ \,
c_s( L_{Q_L}^{J} ) >
\exp \bigl[ \, C A_\d ( \log\log L)^{d-{\d\over 1+\d}}
(\log L)^{ 1- {\d \over d (1+\d)} }
\, \bigr]
\, \bigr\} <
L^{-1.5}
\Eq(P20.1)
$$
If, in addition, (H3) holds (bounded interactions) then
there exist $C_1$ and $C_2$ depending on $d$, $r$, $\a$ and $\th$ such that
for all $L \ge L_1$,
$$
\bP \bigl\{ \,
c_s( L_{Q_L}^J ) >
C_1 \exp \bigl[ \, C_2 J_0 \, ( \log\log L)^{d-1}
(\log L)^{ d - 1 \over d }
\, \bigr]
\, \bigr\} <
L^{-1.5}
\Eq(P20.2)
$$
\item{(ii)} Assume (H1) and (H3) and $d\ge 2$.
Then for any $\e\in (0,1]$ there exist positive constants $C_3$ and $L_2$
depending on
$d$, $r$, $\a$, $\th$, $J_0$ and $\e$ such that
for all $L \ge L_2$
$$
\bP \bigl\{ \,
c_s( L_{Q_L}^{J} ) > L^\e \, \bigr\} <
\exp \bigl[\, -C_3 \, ( \log\log L)^{-d} (\log L)^{d\over d-1}\bigr]
\Eq(P20.3)
$$
\noindent
{\it Remark 1.}
Using (i) together with the Borel--Cantelli lemma, it
follows that, with probability one,
$c_s( L_{Q_L}^{J} )$ does not grow faster than the exponential
appearing in \equ(P20.1). It is quite easy to see that, in this respect,
hypothesis (H2) is almost optimal. Let us in fact consider a ferromagnetic
model with
nearest neighbor couplings
$J_{xy} = J_{ \{x,y\}}$, with only exponential moments, e.g. such that
$\bP\{J_{xy} \ge n\} \le C\exp[-n]$, $n=0,1, \ldots$.
That would correspond to $\d=0$ in (H2).
Then, with large probability, one can find, in the
cube $Q_L$, a pair of nearest neighbor sites
$\{x,y\}$ such that, for some small $\e$, $J_{xy}= \e \log L$ and
$J_{z,z'}=0\;\forall z\in \{x,y\}, z'\in \{x,y\}^c$.
In this case the Gibbs
measure
$\mu_{Q_L(0)}^\t$ will factorize on the
product of the Gibbs measure on the pair
$\{x,y\}$ with free boundary conditions and of the Gibbs measure
$\mu_{Q_L(0)\setminus \{x,y\} }^\t$. In turn the logarithmic Sobolev
constant $c_s( L_{Q_L}^{J,\t} )$ will be at least as large as
the logarithmic Sobolev
constant for the pair $\{x,y\}$ with free boundary conditions and coupling
$J_{xy}= \e \log L$. A simple computation shows that this latter, for e.g. the
heat bath dynamics, is of the order of $\exp[J_{xy}] = L^\e$, i.e. much larger
than our bound in \equ(P20.1).
{\it Remark 2.} It is easy to check that, in the bounded case (H3),
the almost sure bound on the growth of $c_s(L_{Q_L}^{J,\t} )$ that follows
from
\equ(P20.2) is, apart from the $\log\log L$ factor, optimal.
To this purpose let us consider the simplest model, namely the
diluted Ising model without external field,
nearest neighbor interactions $J_{xy}$
taking only two values,
$0$ and
$\bar J\gg J_c$ with probability $1-p$
and $p\ll p_c$ respectively.
Here $J_c$
denotes the critical inverse temperature for the Ising
model
while $p_c$
is percolation thereshold for bond percolation in d--dimensions.
Since we are below $p_c$, it is not difficult to check that hypothesis (H1)
holds.
It is easy to see now that for almost all realizations $J$ there exists
$L_0(J)$ such that for all $L\ge L_0(J)$ there exists $x\equiv x(L,J)$, with
$|x|\le L/2$, such that {\it all} couplings
inside the cube
$Q_{l}(x)$, $l = (\e \log L)^{1/d}$, are equal to $\bar J$ and {\it all}
couplings connecting a point inside $Q_{l}(x)$ with one of its nearest
neighbors outside it are zero.
Let now $\L$ be the cube of side $2L$
centered at the origin. By construction $Q_{l}(x)\sset \L$. Since
the couplings across the boundary of $Q_{l}(x)$ are zero one
has
$$
c_s(L_{\L}^J) \ge c_s(L_{Q_{l}(x)}^{\bar J})\ge \gap( L_{Q_{l}(x)}^{\bar
J})^{-1}
$$
In turn, since $\bar J\gg J_c$, one has (see e.g
\ref[M]) that
$$
\gap( L_{Q_{l}(x)}^{\bar J})^{-1}\ge
\exp[k (\e \log L)^{d\over d-1}]
$$
for a suitable constant $k$.
Actually one can
prove a similar lower bound on the logarithmic Sobolev constant even in the
more general case discussed in Theorem
\thf[Low-Bo] below.
%@E9
%@Ij
\nproclaim Theorem [UB].
Assume (H1) and (H2) and
uniformly bounded transition rates,
\ie $\k_2=0$ in \equ(bounded). Then
\item{(a)}
If $d \ge 1$
there exists a set $\bar \Th \sset \Th$
of full measure such that for each $J\in \bar \Th$ there exists a
unique infinite volume Gibbs measure $\mu^J$. Moreover
there exists a constant $k$ and,
for each $J\in \bar \Th$ and for any local function $f$
there exists
$0 < t_0(J,f) < \infty$ such that for all $t\ge t_0$
$$
\|T^J(t)f-\mu^J(f)\|_\infty\le
\exp\Bigl[ \, - t \,
\exp\bigl[ \,-k \, (\log t)^{1- {\d' \over d} } \,
(\log\log t)^{d- \d'}
\, \bigl]
\, \Bigr]
\Eq(UB1)
$$
where $\d' \in (0,1)$ is given by $\d' = \d (1 + \d)^{-1}$.
If in addition (H3) holds (bounded interactions), then for
all $t\ge t_0(J, f)$
$$
\|T^J(t)f-\mu^J(f)\|_\infty\le
\exp\Bigl[ \,
- t \,
\exp\bigl[ \, - k \, ( \log t)^{1 - {1 \over d} } \,
(\log\log t)^{d-1} \, \bigr]
\, \Bigr]
\Eq(UB2)
$$
\item{(b)}
Assume (H3) and $d\ge 2$. Then there exists a constant
$k$ and for any local function $f$ there exists $0 < t_0(J,f) < \infty$ such
that, if $t \ge t_0(f)$ then
$$
\bE \, \|T^J(t)f-\mu^J(f)\|_\infty \le
\exp\bigl[ \, -k \, (\log t)^{d \over d-1} \, (\log\log t )^{-d}
\, \bigr]
\Eq(UBb)
$$
\noindent
{\it Remark 1.} The ``constant'' $k$ as well as $t_0$
may depend on the
geometrical parameters $d$, $r$ and on the various parameters
appearing in our hypotheses, like $\a$, $\th$, $L_0$, $\d$, $J_0$.
\noindent
{\it Remark 2.} The almost sure speed of relaxation to
equilibrium is faster than any stretched exponential, at least under our
assumptions (H2) or (H3). It is possible to show, at least
for ferromagnetic systems, that if we assume an exponential tail for
the distribution of the couplings ($\d=0$),
then the almost sure bound cannot be
better than a stretched exponential (see also the Remark 1 after Theorem
\thf[P20])
In the general bounded case unfortunately we don't know any almost sure
lower bound on speed of relaxation of a local function $f$ that is slower than
exponential. We conjecture however, also on the basis of Remark 2 after
Theorem
\thf[P20], that for a truly interacting model (thus {\it not} for example the
dilute Ising model below the percolation threshold for which any local
function relaxes exponentially fast but with its own exponential rate) our
upper bound on the speed of relaxation is, apart from the
$\log\log L$ factor, optimal.
\acapo
The
bound (b) on the relaxation for the {\it averaged} dynamics in the bounded
case is, apart from the technical factor
$(\log\log t)^d$, optimal as the next Theorem \thf[Low-Bo] shows. We
don't give the analogous result in the unbounded case, \ie when only
(H2) holds,
since the computation of the new exponent of $\log t$ is quite
involved and, in our opinion, not particularly interesting from the physical
point of view.
%@E9
%@Il
\bigno
An interaction $J$ is said to be {\it nearest neighbor\/} (n.n.) if
$J_A = 0$ unless $A=\{x,y\}$
and the euclidean distance between $x$ and $y$ equals 1.
We also remind the reader that with $\pi_x$, for $x\in Z^d$,
we denote the projection
from $\O$ over $\pmu$,
given by $\pi_x : \s \mapsto \s(x)$.
\bigno
\nproclaim Theorem [Low-Bo].
Assume uniformly bounded transition rates,
\ie $\k_2=0$ in \equ(bounded),
nearest neighbor interactions $J$,
and suppose that
\item{(i)} for almost all $J \in \Th$,
there exists a unique Gibbs measure $\mu^J$
\smallno
Then for each $d\ge 2$ there exist $J_1(d) > 0$
such that if
\item{(ii)} $p_1 \equiv \bP\{J_{xy}= J_1\}>0$ and
$p_2 \equiv \bP\{|J_{xy}|\le 1/4 \}>0$
\smallno
then we have, for all large enough $t$,
$$
\bE \, \| T^J (t) \pi_0 - \mu^J( \pi_0) \|_\Ld \ge
\exp\bigl[ \, -k \, (\log t)^{d\over d-1} \, \bigr]
\Eq(LB)
$$
for some $k$ which depends on $d$, $p_1$ and $p_2$.
\medno
{\it Remark 1.}
If one assumes (H1) and (H2) then the uniqueness of the Gibbs measure
with $\bP-$probability one follows.
\smallno
{\it Remark 2.}
$\mu^J(\pi_0)$ is clearly equal to 0, by uniqueness of the Gibbs measure
and symmetry.
\smallno
{\it Remark 3.}
\equ(LB) is obiously also a lower bound for
$\bE \, \|T(t) \pi_0 -\mu( \pi_0 )\|_\infty$.
This lower bound is of the same order, apart from the
technical factor
$(\log\log t)^d$, as the upper bound given in (b) of Theorem \thf[UB].
Although a similar result was argued in \ref[DRS] for the diluted Ising
model, to our knowledge this is the first rigorous lower bound in a truly
interacting case.
\bigno
%@E9
%@Ip
\beginsubsection 3.2. Applications
In this Section we discuss the hypotheses in our Theorems from the point of
view of standard examples. Clearly the hypotheses (H2) and (H3) refer to the
nature of the disorder (the distribution of the interaction potential)
while the first hypothesis (H1) needs to be checked in a
given disordered equilibrium model.
General methods to verify (H1) can be
found in \ref[DKP] and \ref[GM3] or the
references therein.
Here we follow \ref[GM3] to discuss (H1) for the
important example of a random-field short-range spin glass with formal
Hamiltonian
$$
H=-\sum_{} J_{xy} \sigma(x) \sigma(y) - b\sum_x h_x \sigma(x) -
h\sum_x \sigma(x)
\Eq(1)
$$
determined by a realization of one-($h_x$) and two--body interactions
($J_{xy}$). To have in mind a specific example satisfying (H2) we could
e.g. take the $J_{xy}$
identically distributed independent Gaussian random variables
and let the $h_x$ be equal to $\pm1$ or $0$ each
with probability $1/3$.
In the notation of Section 2, $J_A = h_x$ if $A=\{x\}$ and
$J_A=J_{xy}$ if the set $A=\xy$ is a nearest neighbor pair $$ on the
lattice.
In \equ(1) $b$ and $h$ are just (constant) parameters.
To check (H1) we must consider the finite volume measure $\mu_V^\tau$
corresponding to \equ(1) with $V=Q_L$, and estimate truncated
correlation functions.
It is an immediate consequence of Corollary
2 in \ref[BM] as applied in the main Theorem of \ref[GM3]
that for all local functions $f$ and $g$
$$
|\mu_V^\tau(f,g)| \leq 2 |\Lambda_f| \, |\Lambda_g| \,
\|f\|_{\infty} \,
\|g\|_{\infty} \,
\max_{x \in \Lambda_f, \, y \in \Lambda_g} G(x,y)
\Eq(2)
$$
where $G(x,y)$ is the two-points connectivity function for independent site
percolation on $\Z^d$ with (random) densities $\{p_z, z\in \Z^d\}$
specified below. More precisely, $G(x,y)$ is the probability in the
independent site percolation process to find an open path from site $x$ to
site $y$; independently a site $z$ is open with probability $p_z$ and is
closed with probability $1-p_z$. The densities are an explicit function
of the interaction potential.
In the spin-glass example ($b=0$ in \equ(1))
the random densities are given by
$$
p_z^{SG}=
1/2 \, \Bigl[ \,
\tanh \Bigl[\, \sum_{} |J_{yz}| + h \, \Bigr] +
\tanh \Bigl[\, \sum_{} |J_{yz}| -h \, \Bigr] \,
\Bigr]
$$
while in the random-field case ($J_{xy}=J>0, h=0$ in \equ(1)) we get
$$
p_z^{RF}=
1/2\, \bigl[ \, \tanh(2dJ+bh_x) + \tanh(2dJ-bh_x) \, \bigr]
$$
It follows easily that $\{p_z^{SG}, z\in \Z^d\}$ is a one-dependent
stationary random field while the $\{p_z^{RF}\}$ are independent and
identically distributed.
All the above features are quite general and do not depend very much
on the model under investigation. As long as the interaction
is short range, we will find some independent percolation process with
random but
almost independent densities which allows a domination like \equ(2).
It is not difficult to see that if $\bE p_z$ is sufficiently small
(typically, below some percolation treshold),
then for all sites $x,y\in \Z^d$,
$$
\bE \, G(x,y) \le \nep{ - \a d(x,y)}
\Eq(3)
$$
for some $\a > 0$, with $\a \to \infty$ as $\bE p_z \to 0$.
For example, in order
to have $\a > 0$ for the random-field case it suffices that
$\bE p_z^{RF} < p_c(d)$ where $p_c(d)$ is the treshold (or critical) density
for Bernoulli site percolation on $\Z^d$; for the general model \equ(1) it
suffices that $\bE p_z < 1/(2d-1)^2$, see \ref[GM3].
The combination of the upper bounds in \equ(2) and \equ(3) with the Chebyshev
inequality yields (H1).
In fact the probability that $SMT(Q_L, L/2 , \a/2 )$ does not hold,
thanks to \equ(2),
is bounded by the probability that there exist $x$ an $y$ such that
$d(x,y)\ge L/2$ and $2 G(x,y) > \exp( -\a d(x,y)/2)$.
This latter probability is, in turn, not greater than
$$
\eqalign{
&
L^2 \, \sup_{x,y \in Q_L :\, d(x,y) \ge L/2 }
\bP \{ \, 2 G(x,y) > \exp( -\a d(x,y) /2 ) \, \} \le \cr
& \le
2\, L^2 \, \bE \, G(x,y) \, \exp( \a d(x,y) /2 ) \le
\nep{ -\a L /6 }
\cr }
$$
Hence, (H1) is verified.
Another (but very similar) approach to check (H1) can be found in \ref[DKP].
In particular their estimate (2.19) is almost the same as \equ(2)
above except that they are dominating via a bond percolation process.
Notice that checking (H1) as we have illustrated above requires a rather
strong ``high temperature'' or ``strong external field'' condition.
In certain
cases however (e.g. the dilute Ising ferromagnet) one can substantially
improve on this. We refer to \ref[CMM] for the details.
%@Ep
\fine
%--------------------------------- INIZIO
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \input xmac\fi
\numsec=4
\numfor=1
\numtheo=1
\pgn=1
\beginsection 4. Preliminaries
In this section we
collect several technical results to be used
in the next key section. Most of the results presented here, with the notable
exception of
Theorem \thf[F4] which seems to us completely new and of independent
interest,
are rather simple and some of them can actually be found in the literature.
We thought however useful, also for future purposes,
to put them
together in a sort of primitive tool--box for the subject.
\bigno
\beginsubsection 4.1. Mixing properties and bounds
on relative densities for Gibbs
measures
In this first part we give three equilibrium results on finite volume Gibbs
measures. The first one is what was called in \ref[MO1] \lq\lq effectiveness"
of property $SMT$. The second and the third one provide two
simple bounds on
the relative density between the projection over certain sets of
two different Gibbs measures, once
one assumes exponential decay of correlations.
%@I
\bigno
\nproclaim Proposition [MIX1].
Given $\a >0$ there exist positive numbers
$\bar l(d,r,\a)$, $\g_1(d,r)$, $\g_2(d,r)$,
$\hm(d,r,\a)$ such that, the following holds for all $l>\bar l$:
let $V \in \bF$ such that $V$ can be written as
union of (possibly overlapping) cubes of side $l$
and assume that
\smallno
\item{(i)} $SMT( Q_l(x), l/2 ,\a)$ holds if $Q_l(x) \sset V$
\item{(ii)} $\g_2 \, \|J\|_{V}\leq \a l$
\smallno
then $SMT( V, \g_1 l, \hm(d,r,\a) )$ holds.
\Pro\
It follows from the Lemma A2.1 of \ref[MO1] and Proposition 3.1,
Eq's (3.9), (3.11) of \ref[O] that any cube
$C_i$ for which $(i)$ and $(ii)$ hold also satisfies
condition ${\cal C}_{\bar l}$ of
\ref[OP] for some
$\bar l$ large enough depending
only on $\a,d,r$, provided that the constants $\g_1,\g_2$ are chosen
respectively large and small
enough depending only on the dimension $d$ and the range $r$. Then the result
follows from Propositions 2.5.1, 2.5.2, 2.5.3, 2.5.4 of \ref[OP].
We also refer the reader to Appendix A.1 of \ref[MO1] for a simple
proof in the attractive case.
\QED
\smallno
{\it Remark.}
In the sequel for any given $\a >0$ we will denote by $\hm(\a)$ the
constant $\hm(d,r, \a)$ given in the above proposition.
%@E
\bigno
%@Ia
\nproclaim Proposition [A0].
Let $V\sset \L \ssset \Z^d$, and let $x\in \L^c$ such that $d(x,V)> r$.
If $U \equiv \L\setm V$,
we have
$$
\sup_{\t \in \O} \,
\Bigl\| 1 - {d \mu_\L^{J,\t^x} \over
d \mu_\L^{J,\t} } \Bigr|_V \,
\Bigr\|_\infty
\le
\nep{ 14 \|J\|_x } \sum_{y\in V} \nep{ 2 \|J\|_y} \,
\sup_{\t\in\O} \,
\bigl| \,
\mu_{U}^\t
\bigl( \,
\nep{ - \nabla_x H_U } \, , \,
\nep{ - \nabla_y H_U }
\, \bigr)
\bigr|
$$
\Pro\
Let $\uno \in \O$ be the configuration with all spins equal to $+1$, and let
$$
W_{\L,V}(\s) =
\log Z^{\s}_U -
\log Z^{\s_{V^c} \uno_V}_U
$$
It is easy to show that
$$
\Bigl\| 1 - {d \mu_\L^{J,\t^x} \over
d \mu_\L^{J,\t} } \Bigr|_V \,
\Bigr\|_\infty
\le
\nep{ 2 \ninf{ \nabla_x W_{\L,V} } }
\qquad
\hbox{ for all $\t \in \O$ }
$$
which, using the trivial bound $\ninf{ \nabla_x W_{\L,V} } \le 4 \|J\|_x $,
gives
$$
\Bigl\| 1 - {d \mu_\L^{J,\t^x} \over
d \mu_\L^{J,\t} } \Bigr|_V \,
\Bigr\|_\infty
\le
\nep{ 8 \|J\|_x }
\ninf{ \nabla_x W_{\L,V} }
\qquad
\hbox{ for all $\t \in \O$ }
$$
By proceeding as in Lemma 3.1 of \ref[MO2] one can show that
$$
\ninf{ \nabla_x W_{\L,V} } \le
\nep{ 6 \|J\|_x } \sum_{y\in V} \nep{ 2 \|J\|_y} \,
\sup_{\t\in\O} \,
\bigl| \,
\mu_{U}^\t
\bigl( \,
\nep{ - \nabla_x H_U } \, , \,
\nep{ - \nabla_y H_U }
\, \bigr)
\bigr|
$$
which completes the proof.
\QED
%@E1
%@I11
\nproclaim Proposition [GP9].
For each $m>0$ there exists $C(d,r,m)$
such that the following holds.
Let $A \ssset \Z^d$, $A_0 \sset A$ and
$B_0 \sset \dep_r A$.
Let $\Ab = A \cup \dep_r A$ and assume that
\smallno
\item{(i)}
$m d_0 \equiv m d( A_0 , B_0 ) \ge
\max\{ \,
C \, , \, 100 \|J\|_{\Ab} \, ,
\, 10 \, ( \log | B_0 | + 1 ) \, \}$
\item{(ii)}
$SMT( A\setm A_0 , \, d_0 - 2 r, \, m)$ holds
\smallno
then for each pair of configurations $\s, \t \in \O$ which agree
on $\dep_r A \setm B_0$, we have
$$
\Bigl\| 1 -
{d\mu_A^\t \over
d\mu_A^\s
} \Bigr|_{A_0} \,
\Bigr\|_\infty \le
\nep{ - (m/4) d_0 }
\Eq(GP9.1)
$$
\Pro\
For each $\h\in \O_{A_0}$, consider the event
$
F_\h = \{ \s\in\O : \, \s_{A_0} = \h \}
$.
Choose a pair of configurations $\s, \t$ which agree on
$\dep_r A\setm B_0$. Then
there exists a sequence of interpolating
configurations $\g_i \in \O$
for $i = 1, \ldots, n$ such that $ n \le |B_0|$, $\g_{i+1}$ differs
from $\g_i$ at exactly one site, $\g_1 = \s$ and $\g_n$ agrees with
$\t$ on $\dep_r A$. Thus,
for each $\h \in \O_{A_0}$, we can write
$$
\left| 1 -
{\mu_A^\t( F_\h) \over
\mu_A^\s( F_\h)
}
\right| =
\left| 1 -
\prod_{i = 2}^{n}
{\mu_A^{\g_i } ( F_\h) \over
\mu_A^{\g_{i-1}} ( F_\h)
}
\right|
\Eq(GP9.6)
$$
If we define
$$
a =
\sup_{\st\z\in\O, \, x\in B_0, \, \h \in \O_{A_0}}
\left| 1 -
{\mu_A^\z ( F_\h) \over
\mu_A^{\z^x} ( F_\h)
}
\right|
$$
then it is easy to check that, if
$a \le \ov{10}$ and
$a |B_0| \le 1$
then the RHS of \equ(GP9.6) cannot exceed $e a |B_0|$, so if we show that,
for instance, $a \le \nep{ - (m/2) d_0}$, the proposition follows.
\smallno
Let then, for $z\in \Z^d$,
$g_z = \exp( - \nabla_z H_{A\setm A_0} )$.
By Proposition \thm[A0], and the SMT property given in the hypotheses,
we find
$$
\eqalign{
&
a \le
\sup_{x\in B_0} \,
\nep{ 14 \|J\|_{B_0} }
\sum_{y\in A_0} \nep{ 2 \|J\|_y}
|\L_{g_x}|
|\L_{g_y}|
\, (\ninf{ g_x }
\ninf{ g_y } ) \,
\nep{ - m d(\L_{g_x}, \, \L_{g_y} ) } \le \cr
& \le
\, (2r+1)^{2d} \,
\nep{ 20 \|J\|_{\Ab} } \,
\sup_{x\in B_0} \,
\sum_{y\in A_0}
\nep{ - m \, | x - y | - 2 r }
\cr }
$$
In the second inequality we have used the fact that
$\L_{g_x}$ is contained in a ball of center $x$ and radius $r$,
and the fact that $\ninf{ g_x} \le \exp( 2 \|J\|_x )$.
Finally, using the hypothesis on $d_0$, we easily get
$$
a \le \nep{ - (m/2) d_0 }
\QED
$$
%@E11
\beginsubsection 4.2. Some results on the spectral gap of the block dynamics
Here we provide three lower bounds on the spectral gap of the block
dynamics with just two blocks.
\bigno
%@II
\nproclaim Proposition [GP1].
Let $V\ssset \Z^d$, and let $A,B$ be two (possibly intersecting)
subsets of $V$ such that $V = A \cup B$. Let $\cD = \{A, B \}$.
Assume that
$$
\sup_{\t \in \O} \,
\Bigl\| 1 -
{d\mu_A^\t \over
d\mu_V^\t
} \Bigr|_{\dep_r B} \,
\Bigr\|_\infty
\le \e < 1
\Eq(GP1.2)
$$
Then the gap for the block dynamics on $\cD$ satisfies
$$
\inf_{\t\in \O} \gap( L_\cD^\t ) \ge 1 - \sqrt{\e}
$$
\Pro\
The action of the semigroup $T_\cD(t)$ associated to the block dynamics
is given by
$$
T_\cD(t) f = \sum_{n=0}^\infty {t^n \over n!} ( L_\cD )^n f
$$
Using the explicit expression for $L_\cD$ and some elementary
combinatorics, it is not difficult to show that
$$
T_\cD(t) f =
\sum_{n=0}^\infty
{(2 t)^n \over n!}
\nep{-2t} \ov{2^n}
\sum_{X \in \{ A,B \}^n } \mu_{X_1} \cdots \mu_{X_n} (f)
\Eq(GP1.4)
$$
Since $(\mu_A)^2 = \mu_A$ (and similarly for $B$) the last summation
(over $X$)
in \equ(GP1.4) can be written as
$$
\sum_{k=0}^{n-1}
{n-1 \choose k}
( \hA_{k+1} + \hB_{k+1} ) f
\Eq(GP1.6)
$$
where
$$
\hA_k = ( \mu_A \circ \mu_B )^{ \inte{k/2} } \circ
\mu_A^{ k - 2 \inte{k/2} }
\qquad
\hB_k = ( \mu_B \circ \mu_A )^{ \inte{k/2} } \circ
\mu_B^{ k - 2 \inte{k/2} }
$$
If now $g$ is an arbitrary
bounded measurable function on $\O$, such that $\mu_V(g) = 0$, we get
$$
\ninf{ \mu_A \mu_B \mu_A g } \le
\ninf{ \mu_V \mu_B \mu_A g } +
\ninf{ \mu_V \mu_B \mu_A g - \mu_A \mu_B \mu_A g}
\Eq(GP1.8)
$$
By the DLR property \equ(DLR) the first term on the RHS of \equ(GP1.8)
is equal to $\mu_V(g) = 0$. Furthermore, since the interaction has
range $r$, the function $h \equiv \mu_B \mu_A g$ is
$\cF_{V^c \cup \dep_r B}$ measurable. This fact
together with hypothesis \equ(GP1.2) and the trivial
observation that $\mu_A$ and $\mu_V$ agree on $\cF_{V^c}$ implies
$$
\ninf{ \mu_A \mu_B \mu_A g } \le
\e \ninf{ \mu_B \mu_A g } \le
\e \ninf{ \mu_A g }
\Eq(GP1.10)
$$
Iterating this inequality we get, for each bounded measurable
$f$ with $\mu_V(f) = 0$,
$$
\ninf{ \hA_k f } \le (\sqrt{\e})^{ k - 3 } \ninf{f}
\qquad
\ninf{ \hB_k f } \le (\sqrt{\e})^{ k - 3 } \ninf{f}
\Eq(GP1.12)
$$
Thus, we get that the sup norm of \equ(GP1.6) is not greater than
$$
\ninf{f}
{2 \over \e^{3/2} }
( 1 + \sqrt{\e} )^{n-1}
$$
which, inserted back into \equ(GP1.4) yields
$$
\ninf{ T_\cD(t) f } \le
\ninf{f}
4 \e^{ -3/2 }
\nep{-2t}
\sum_{n=0}^\infty
{ t^n \over n!}
( 1 + \sqrt{\e} )^{n} =
\ninf{f}
4 \e^{ -3/2 }
\nep{- (1 - \sqrt{\e}) t}
\QED
$$
%@E
%@I2
\nproclaim Proposition [GP6].
Let $V$, $A$ and $B$ be as in Proposition \thf[GP1].
Let also $A_0 = A \cap \dep_s B$, with $s \ge r$,
$B_0 = B \cap \dep_r A$ and $\Ab = A \cup \dep_r A$.
For each $m>0$ there exists $C(d,r,m)$
such that if
\smallno
\item{(i)}
$m d_0 \equiv m d( A_0 , B_0 ) \ge
\max\{ \,
C \, , \, 100 \|J\|_{\Ab} \, , \, 10 \, ( \log | B_0 | + 1 ) \, \}$
\item{(ii)}
$SMT( A\setm A_0 , \, d_0 - 2 r, \, m)$ holds
\smallno
then
$$
\inf_{\t\in\O} \gap( L_{\{A,B\}}^\t ) \ge \ov2
$$
\Pro\
Thanks to Proposition \thm[GP1] it is sufficient to show that
$$
\sup_{\t\in \O}
\Bigl\| 1 -
{d\mu_A^\t \over
d\mu_V^\t
} \Bigr|_{A_0} \,
\Bigr\|_\infty
\le \ov4
\Eq(GP6.2)
$$
By the DLR property \equ(DLRi) we have
$$
\hbox{ LHS of \equ(GP6.2) } \le
\sup_{\t, \s\in \O \, : \,\t_{V^c} = \s_{V^c}}
\Bigl\| 1 -
{d\mu_A^\t \over
d\mu_A^\s
} \Bigr|_{A_0} \,
\Bigr\|_\infty
\Eq(GP6.4)
$$
At this point we can use Proposition \thf[GP9] and obtain the
result.
\QED
%@E2
%@I11
\bigno
\nproclaim Proposition [GP7].
Let $V,A,B$ be as in Proposition \thf[GP1]. Let
$$
N = |\dep_r A \cap B | \mmin |\dep_r B \cap A|
$$
Then there exists $k=k(d,r)$ such that
$$
\inf_{\t\in\O} \gap( L_{\{A,B\}}^{J,\t} ) \ge \exp[ \, - k \JV N \, ]
$$
\Pro\
We can assume $N = |\dep_r B \cap A|$. Consider a new interaction $J^0$
such that $A$ and $V\setm A$ are decoupled, \ie
$$
J^0_{X}=\cases{J_X & if $ X \cap \dep_r A \cap B = \emp$ \cr
0 & otherwise\cr}
$$
We have clearly $\|J- J^0\|_x =0$ unless $x$ is in a neighborhood of
radius $r$ of $\dep_r B \cap A$, hence
$$
\sum_{x\in V} \| J -J^0\|_x \le k_1 N \JV
$$
for some $k_1$ which depends on $d$ and $r$. This implies that
for all functions $f$ on $\O$,
$$
\eqalignno{
{\cal E}_{\{A,B\}}^{J, \t}(f,f)
&\geq
\exp(-4 k_1 \JV N ) \, {\cal E}_{\{A,B\}}^{J^0,\t}(f,f)
& \eq(GP7.1)
\cr
\Var_{V}^{J, \t}(f)
& \le
\exp( 4 k_1 \JV N ) \, \Var_{V}^{J^0,\t}(f)
& \eq(GP7.2) \cr }
$$
From \equ(GP7.1), \equ(GP7.2) and the variational
characterization of the gap \equ(gap), it follows that
$$
\gap( L_{\{A,B\}}^{J,\t} ) \ge
\exp[ - 8 k_1 \JV N ] \, \gap( L_{\{A,B\}}^{J^0,\t} )
\Eq(GP7.3)
$$
In order to estimate the gap for the block--dynamics with couplings $J^0$, we
just notice that
the hypotheses of Proposition \thf[GP1] are satisfied
with $\e = 0$, and thus $\gap( L_{\{A,B\}}^{J^0,\t} ) \ge 1 $.
\QED
%@E11
\beginsubsection 4.3. Some general results on the spectral gap and
the log--Sobolev constant
This is actually the most important part of this section since it contains
two key results. The first one, Proposition \thf[F1], has been
essentially
proved in \ref[MO2] and, at least in the case of bounded interaction,
it roughly says the following. If in
a given cube $Q_L$ of side $L$ truncated
correlations decay exponentially fast on all length scales
larger
than $l_1$, with $l_1\ll L$, then the logarithmic Sobolev constant in that
cube is not larger than the largest among the logarithmic Sobolev constants of
all cubes of side $l_1$ inside $Q_L$. In order to appreciate this result
one should consider that, if hypotheses (H1) and (H3) hold, then with
probability one, truncated correlations in a cube of side $L$ centered at the
origin decay
exponentially fast on all length scales larger than $l_1\approx \log L$.
Thus in this case we would have a logarithmic contraction of the
starting length scale, namely from $L$ to $\log L$. This result, together with
a very rough estimate of the logarithmic Sobolev constant for a cube (see
Proposition \thf[F3] and Theorem \thf[F4] below), allows us to conclude
immediately that, with probability one, $c_s(L_{B_L}^\t)$ cannot grow faster
than $\exp[C (\log L)^{d-1}]$. Notice that that in two dimensions this
bound is just a power law in the side $L$.
The second important result is a very general lower bound on the spectral gap
of Glauber dynamics (or upper bound on the logarithmic Sobolev constant)
in an {\it
arbitrary} set $V\ssset \Z^d$. It says that the spectral gap is always larger
than a
negative exponential of $|V|^{d-1\over d}$. Notice that if $V$ is cube then
$|V|^{d-1\over d}$ is simply its surface. In this case the bound is certainly
optimal, at least in our general setting, since it is known that for
several models
of lattice {\it discrete spins} in the phase coexistence region, the
activation energy between different stable phases is proportional to the {\it
surface} of the region in consideration (see \ref[M] and \ref[CGMS] for more
precise statements for the Ising model).
Apparently the situation for {\it continuous} spins systems can be very
different. For Heisenberg models, in fact, it is believed on the basis of
spin--wave
theory (see \ref[B1], \ref[B2]) that, at least for cubic regions, the spectral
gap does not go to zero faster than the inverse of the volume. It is a
challenging problem to actually prove it!
%@Iq
\bigno
\nproclaim Definition [D1].
A cube $C = Q_l(x)$ is said to be $\a-$regular if,
letting $n = \inte{ l/ (2 \g_1) }$
\smallno
\item{(i)} $SMT( Q_n(y), n/2 ,\a)$ holds for all $y \in Q_l(x)$
\item{(ii)} $(\g_2\mmax 100) \, \|J\|_{\bar C} \le \hm(\a) l$
\smallno
where $\bar C = C \cup \dep_r C$ and the constants $\g_1$,
$\g_2$ and $\hm(\a)$ are those appearing in Proposition \thm[MIX1].
\smallno
We immediately observe the following
\nproclaim Proposition [REG].
Assume (H1) and (H2). Then there exist $L'_0 \in \Z_+$, $\th' >0$
(depending on $\a$, $\th$, $\g_1$, $\g_2$ and $G_0$)
such that for all $L \ge L'_0$,
$$
\bP\{ \, \hbox{$Q_L$ is $\a-$regular} \, \} \ge
1 - \nep{ - \th' L }
$$
Furthermore, if $V$ is a union of $\a-$regular
cubes of side length $l$, then $SMT(V, l/2, \hm(\a))$ holds.
\Pro\
The probability that $Q_L$ is not $\a-$regular is bounded by
(we use the exponential Chebyshev inequality)
$$
\eqalign{
&
\bP\{ \hbox{ (i) does not hold } \} +
\bP\{ \hbox{ (ii) does not hold } \} \le \cr
& \le
L^d \, \nep{ - \th / (3 \g_1) } +
(L+2r)^d \, G_0 \, \nep{ - \hm L /( \g_2 \mmax 100 ) } \le
\nep{ - \th' L }
\cr }
$$
if $L$ is greater than some $L'_0$. The second statement follows from
Proposition \thf[MIX1].
\QED
%@E
%@I
\nproclaim Proposition [F1]. Let $l_1 \in \Zp$ and let
$\L\ssset \Z^d$ be a multiple of $Q_{l_1}$, \ie $\L = \cup_{i=1}^n B_i$
where $B_i = Q_{l_1}(x_i)$ for some $x_i\in l_1\Z^d$. Let,
for any
$I\sset \{1,\ldots,n\}$, $\L_I = \cup_{i\in I} B_i$.
Let also
$\cA$ be the set of all $I\sset \{1,\ldots, n\}$ such that
$\diam(\L_I) \le 3 l_1$. Assume that each
$B_i$ is $\a-$regular for $\a>0$.
Then there exist two positive constants $\bar l_1$ and
$k$ depending on $\a, d, r$
such that if $m \equiv \hm(\a)$, and
\smallno
\item{(i)}
$ l_1 \ge \bar l_1$
\item{(ii)}
$\inf_{I\in \cA} \inf_{\t\in\O} \gap ( L_{\L_I}^\t) \ge
\exp[ - m l_1 / 2 ]
$
\smallno
then
$$
\sup_{\t\in \O} c_s( \mu_{\L}^\t ) \le
k \sup_{I\in \cA} \sup_{\t\in\O} c_s( \mu_{\L_I}^\t )
$$
\Pro\
Since each $B_i$ is $\a-$regular, using
Proposition \thm[MIX1], we
get that for any $I\sset \{1,\ldots, n\}$, $SMT(\L_I, l_1/2, \, m)$
holds. This
fact, together with hypothesis $(ii)$,
allows us to apply Theorem 2.1 of \ref[MO2] and to conclude that
there exists $k(d,r,\a)$ such that
$$
\sup_{\t\in\O}c_s( \mu_{\L}^\t ) \leq
{k\over 4} \sup_{i \in \{1, \ldots, n\} }
\sup_{I\sset \{1\dots n\} \atop I\ni i }
\sup_{\t\in\O}
\g_i(\mu_{\L_I}^\t) \Eq(F1.1)
$$
provided that $l_1$ was taken large enough.
Here, for any $I\sset \{1,\ldots, n\}$ containing $i$,
$\g_i(\mu_{\L_I}^\t)$ is the smallest constant $\g$ such that the
logarithmic Sobolev inequality
$$
\mu_{\L_I}^\t (f^2 \log f) \leq
\g \mu_{\L_I}^\t (|\nabla_{\L_I}f|^2) +
\mu_{\L_I}^\t(f^2)\log \sqrt{ \mu_{\L_I}^\t (f^2) }
$$
holds for all positive functions $f$ that depend only on the spins in
$B_i$.
It is clear from the above definition that
$\g_i(\mu_{\L_I}^\t)\le c_s( \mu_{\L_I}^\t)$.
Assume that the supremum in the RHS
of \equ(F1.1) is attained over a
set $I\notin {\cal A}$ (otherwise the proof of the proposition
would be finished).
Given $i, I$ such that $i\in I\notin {\cal A}$, let
$I_0$ be the largest subset of $I$ such
that $d(B_i,B_j) < l_1$ for all $j\in I_0$.
By construction $i \in I_0 \in {\cal A}$ .
We claim that
$$
\g_i(\mu_{\L_I}^\t) \le
4 \sup_{\t\in \O }c_s( \mu_{\L_{I_0}}^\t )
\Eq(F1.2)
$$
provided that $l_1$ is large enough depending on $\a, d, r$. Such a bound
clearly completes the proof.
In order to prove \equ(F1.2) it is enough to estimate the relative
density between the projection over $\cF_{B_i}$ of the two Gibbs measures
$\mu_{\L_{I_0}}^\t$ and
$\mu_{\L_I}^\t$ uniformly in the boundary condition $\t$.
More precisely, let
$$
g^\t_{max} \equiv
\|{ d \mu_{\L_I}^{\t} \over d \mu_{\L_{I_0}}^\t }
\Bigr|_{B_i}\|_\infty\, ; \qquad
g^\t_{min} \equiv
\min_{\s\in \O_{B_i}}
{ d \mu_{\L_I}^{\t} \over d \mu_{\L_{I_0}}^\t } \Bigr|_{B_i}(\s)
$$
Then, using
Exercise 6.1.27 of \ref[DeSt], \equ(DLR) and the bound
$\g_i(\mu_{\L_{I_0}}^\t)\le c_s(\mu_{\L_{I_0}}^\t)$, we get
$$
\g_i(\mu_{\L_I}^\t) \le
\sup_{\t\in\O} \, {g^\t_{max} \over g^\t_{min}}
\g_i(\mu_{\L_{I_0}}^\t)
\le
\sup_{\t\in\O} \,
{g^\t_{max} \over g^\t_{min}}
c_s(\mu_{\L_{I_0}}^\t)
\Eq(F1.3)
$$
We then use the DLR equations and write
$$
g^\t_{max}
\le
\sup_{\t, \t' \in \O \, : \,
\t_{\L_{I}^c}=\t'_{\L_{I}^c}} \,
\Bigl\|
{ d \mu_{\L_{I_0}}^{\t} \over
d \mu_{\L_{I_0}}^{\t'}
} \Bigr|_{B_i}
\Bigr\|_\infty
\Eq(F1.5)
$$
$$
g^\t_{min}
\ge
\inf_{\t, \t' \in \O \, : \,
\t_{\L_{I}^c}=\t'_{\L_{I}^c}} \,
\min_{\s\in \O_{B_i}}
{ d \mu_{\L_{I_0}}^{\t} \over
d \mu_{\L_{I_0}}^{\t'}
} \Bigr|_{B_i}(\s)
\Eq(F1.5.1)
$$
Thanks to Proposition \thm[GP9] applied to the
sets $A\equiv \L_{I_0}$, $A_0 \equiv B_i$ and
$B_0 \equiv \dep_r \L_{I_0} \cap \L_I $, we know that the
RHS of \equ(F1.5) is less than $2$ while the RHS of \equ(F1.5.1) is greater
than $1/2$, provided that $l_1$ is taken large enough depending only on $\a, d,
r$. In this way we have proven \equ(F1.2), and, by consequence, the proposition.
\QED
%@E
%@Ia
\nproclaim Proposition [F3].
For each $\L\ssset\Z^d$ we have
$$
c_s( L^{J, \t}_\L) \le
\Bigl[ \,4 + 4 \sum_{x\in \L} \|J\|_x + 2 |\L| \log 2 \,\Bigr] \,
( \gap(L_\L^{J,\t} ) )^{-1}
$$
\noindent
The proposition follows from \equ(gap), Proposition \thf[LS1] below,
and from a trivial estimate on $\inf_\s \mu_\L^{J,\t}(\s)$.
%@E
%@I
\nproclaim Proposition [LS1].
Let $\O$ be a finite set, let $\mu$ be a probability
measure on $(\O, 2^\O)$ and assume
$$
\mu_0 \equiv \inf_{x\in\O} \mu(x) >0
$$
Then, for each positive function $f$ on $\O$, we have
$$
\mu( f^2 \log f) \le ( 4 + 2 \log \mu_0^{-1} ) \Var (f) +
\mu(f^2) \log \sqrt{ \mu(f^2)}
$$
\Pro\
We can assume $\mu(f^2) =1$.
If we let $f = \mu (f) (1 + g)$, we find $\mu (g) = 0 $ and
$\mu(g^2) = \Var (f) / \mu (f)^2$. Let $A$ be the set of all $x\in \O$
such that $|g(x) |<1$. We can then write
$$
\mu ( f^2 \log f) = \mu (f^2 \log f \, \id_A ) +
\mu (f^2 \log f \, \id_{A^c} )
\Eq(LS1.1)
$$
Let's denote by $X_1$ respectively $X_2$ the first and the second term in
the RHS of \equ(LS1.1). Using the inequalities $\log(1+g)\le g$ and
$\log \mu (f) \le \log \mu (f^2) \le 0$, we get
$$
\eqalign{
X_1 & \le
\mu (f)^2 \mu[ (g + 2g^2 + g^3) \id_A ] \cr
& \le
\mu (f)^2 \bigl[ \,
3 \mu(g^2) + \mu( g \id_A )
\, \bigr] =
3 \Var (f)+ \mu (f)^2 \mu( g \id_A )
\cr }
\Eq(LS1.2)
$$
To take care of the last term we remember that $\mu g = 0$, so
$\mu( g \id_A) = - \mu( g \id_{A^c} )$ which implies, using the
Schwarz and then the Chebyshev inequalities
$$
| \mu( g \id_A) | \le \mu( |g| \id_{A^c} ) \le
( \mu(g^2) \mu( \id_{A^c} ) )^{1/2} \le \mu( g^2)
$$
Thus we get $X_1 \le 4 \Var f$. As for $X_2$, we write
$$
X_2 \le
( \, \sup_{x\in \O} \, \log f(x) \, ) \,
\mu( f^2 \id_{A^c} ) \le
\log \ninf{f} \,
\mu( f^2 \id_{A^c} )
\Eq(LS1.5)
$$
Finally we observe that $\ninf{f}$ is bounded by
$( \mu(f^2) / \mu_0 )^{1/2} = \mu_0^{-(1/2)}$
while
$$
\mu( f^2 \id_{A^c} )=
\mu(f)^2 \mu( (1 + 2g + g^2 ) \id_{A^c} ) \le
4 \mu (f)^2 \mu(g^2) = 4 \Var (f)
$$
This concludes the proof.
\QED
%@E
%@I1
\bigno
\nproclaim Theorem [F4].
There exist $k(d,r, \k_1)$, such that,
for each $\L \ssset \Z^d$ and for each $\t \in\O$, we have
($c_m$ was defined in \equ(bounded))
$$
\gap( L_\L^\t ) \ge
c_m \exp\bigl[ \, - k \, \JL\, |\L|^\dd \, \bigr]
\Eq(F4.1)
$$
\Pro\
For each non--negative integer $n$,
let
$$
(K_n) =
\hbox{ the inequality \equ(F4.1) holds for all
$\L\in\bF$ such that $|\L| \le (3/2)^n$ }
$$
We want to show, that $(K_n)$ holds for all $n\in \Zp$, by proving
that there exists $n_0(d,r) \in \Zp$ such that
$(K_{n_0})$ holds, and such that, for all $n \ge n_0$,
$(K_n)$ implies $(K_{n+1})$.
\smallno
Assume then that $K_{n-1}$ holds, and take any $\L$
such that $(3/2)^{n-1} < |\L| \le (3/2)^n$. Let $v = |\L|$.
By Proposition \thf[GEO0],
it is possible to write $\L$ as the disjoint union of two
subsets $X$ and $Y$, such that
\smallno
\item{(a)}
$v/2 - k_1 v^\dd \le |X| \le v/2$
\item{(b)}
$\d_r(X,Y) \le k_1 v^\dd$
\smallno
where $k_1$ depends only on $d$ and $r$. There exists then $n_0(d,r)$
such that if $n>n_0$ (and thus $v> (3/2)^{n_0-1}$), then
$|Y| \le (2/3) |\L|$. So we can apply the inductive hypothesis
to both $X$ and $Y$. Furthermore a simple calculation
(see Proposition \teo[A1.1] in \ref[CM]) shows that
$$
\inf_{\t\in\O}
\gap( L^{J, \t}_\L ) \ge
\inf_{\t\in\O} \,
\inf_{ W \in \{X,Y\} }
\gap( L^{J, \t}_W ) \,
\inf_{\t\in\O}
\gap( L^{J, \t}_{ \{X, Y\} } )
\Eq(F4.10)
$$
where, as usual, the last term refers to the block dynamics.
By Proposition \thf[GP7] we know that
$$
\inf_{\t\in\O}
\gap( L^{J, \t}_{ \{X, Y\} } ) \ge
\nep{ -k_2 \|J\|_\L v^\dd }
$$
for some $k_2(d,r)$. Together with the inductive hypothesis on $X$
and $Y$, this gives
$$
\inf_{\t\in\O}
\gap( L^{J, \t}_\L ) \ge
c_m \exp\bigl[ \,
- k \JL |Y|^\dd
-k_2 \JL v^\dd
\, \bigr]
\Eq(F4.12)
$$
Since $|Y| \le (2/3) v$, we have
$$
\inf_{\t\in\O}
\gap( L^{J, \t}_\L ) \ge
c_m \exp\bigl[ \,
- k \JL v^\dd
\, \bigr]
\qquad
\hbox{if }
k \ge { k_2 \over 1 - (2/3)^\dd }
$$
In this way we have shown that $(K_n)$ implies $(K_{n+1})$ for
all $n \ge n_0(d,r)$.
\smallno
All is left is to prove $(K_{n_0})$. For this purpose we observe
that
$$
\nep{ - 2 \| J - J'\|_\L |\L| } \le
{ \mu^{J, \t}_\L(\s) \over
\mu^{J', \t}_\L(\s) }
\le
\nep{ 2 \| J- J'\|_\L |\L| }
\qquad
\hbox{ for all }
\t, \s, \L, J, J'
\Eq(F4.14)
$$
Choose now any $\L$ with volume not exceeding $(3/2)^{n_0}$ and
let $\tL_\L$ be the generator of the heat--bath dynamics
with $J=0$, \ie
$$
\tL_\L = \sum_{x\in\L} \tL_\xx =
\sum_{x\in \L}
( \mu^{J=0}_\xx - \id )
$$
Since all $\tL_\xx$ commute, it follows that
$\gap( \tL_\L ) = \gap( \tL_\xx ) = 1$
(the last equality can be checked via an explicit calculation).
From \equ(bounded), \equ(gap) and \equ(F4.14),
it now follows that
$$
\gap( L^{J, \t}_\L ) \ge
\gap( \tL_\L ) \,
\nep{ -6 \JL |\L| } \,
c_m \nep{ -\k_1 \JL } \ge
c_m \exp\bigl[ \,
- (6+\k_1) \JL (3/2)^{n_0}
\, \bigr]
$$
which implies $(K_{n_0})$ (and then \equ(F4.1)), if we take
$k \ge (6+\k_1) (3/2)^{n_0}$.
\QED
%@E1
\fine
%--------------------------------- INIZIO
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \input xmac\fi
\numsec=5
\numfor=1
\numtheo=1
\pgn=1
\beginsection 5. The deterministic problem
%@I12
This section is the core of the paper. We give a deterministic
upper bound
on the logarithmic Sobolev constant $c_s(\mu_\L^{J,\t})$ in the cube
$\L\equiv Q_L$.
In order not to obscure the discussion of our ideas with less
relevant details due to unbounded interactions, we present the main
steps of our strategy in the bounded case.
For this purpose,
consider first the so--called two dimensional
diluted Ising model with nearest
neighbor interactions $J_{xy}$ which are either zero or $\bar J$, and
call ``regular'' any site
$x\in \L$ such that $J_{xy}=0$ for all neighboring sites
$y$. Let us also consider the set $W$ of all non--regular
sites and its connected components (in the obvious sense)
$\{W_i\}_{i=1}^n$
inside $\L$. Fix a volume scale $v$
and assume that $\sup_i|W_i| \le v$.
Then we claim that in this case
$$
c_s(\mu_\L^{J,\t})\le C_1 L^d\exp[C_2 v^{d-1\over d}]
\Eq(idea.1)
$$
for suitable constants
$C_1$ and $C_2$ independent of $L$ and $v$.
The proof follows immediately from Proposition \thf[F3]
if we can prove the key
inequality
$$
\gap(L_\L^{J,\t}) \ge C'_1 \exp[ - C'_2 v^{d-1\over d}]
\Eq(idea.2)
$$
for another pair of constants $C'_1, C'_2$.
The above inequality follows from Theorem \thf[F4]
once we observe that, and this is the key feature of the diluted model,
the connected components of $W$ are non--interacting since
they are separated from
each other by a \lq\lq safety belt" of completely decoupled sites.
Therefore the
spectral gap of $L^{J,\t}_\L$ is not smaller than the smallest among the
spectral gaps of $L^{J,\t}_{W_i}$.
Using now Theorem \thf[F4]
and the assumption
$\sup_i|W_i| \le v$ we get the required bound \equ(idea.1).
It is very important to observe that, thanks to some of the results of section
4.2, the above conclusion remains true, modulo some irrelevant constant
factors, even if the value $J_{xy}=0$ is replaced by a very small
number $J_{min}$, provided that $|J_{min}| |W|\ll 1$.
This remark
suggests how to transpose to a truly interacting model the previous
ideas.
In a certain sense our original model behaves after a suitable
``coarse-graining'' quite closely to this diluted model.
Let us in fact make a coarse--grained description of the model
on a new scale $l_0 \ll L$, by replacing sites with disjoint
cubes $C_i$ of side $l_0$ and declare ``regular" those cubes $C_i$
in which truncated correlations decay exponentially fast with rate
$\a>0$. In this way, if $B$ is a collection of \lq\lq non--regular"
cubes $C_i$ surrounded by a safety--belt of regular cubes, then
the effective interaction of $B$ with any other region outside the
safety--belt will be not larger than $|B|\exp(-\a l_0)$. Thus, if $l_0$
is chosen so large that the effective interaction among the connected
components of the set $W_{l_0}$ of non--regular cubes $C_i$ is much
smaller than one, e.g. if $|W_{l_0}|\exp(-\a l_0) \ll 1$, then our system,
on scale $l_0$, will behave like a diluted Ising model. In particular
we will be able to apply the results of Section 4.2 and, as a
consequence, we will get the bound $\equ(idea.2)$ on the spectral gap,
with $v$ equal to the volume of the largest connected component of the set
$W_{l_0}$. We refer the reader to Proposition \thf[F2] below for a
precise formulation of this result in the more general case of
unbounded $J$. Once we have \equ(idea.2) then we also get \equ(idea.1)
simply by applying Proposition \thf[F3].
Although the above reasoning looks quite appealing
from a physical point
of view, it is still unsatisfactory for the following reason. In a
typical configuration of $J$, the volume of the set $W_{l_0}$ is roughly
$p(l_0) L^d$, where $p(l_0)$ is the probability that a cube $C_i$
is not regular. Using our basic assumption (H1), $p(l_0)\approx
\exp(-\th l_0)$ so that the minimal scale $l_0$ satisfying
$|W_{l_0}|\exp(-\a l_0) \ll 1$ becomes of order $\log L$. This
unfortunately is a too large scale:
since $v$ is at least $l_0^d$, the corresponding bounds
\equ(idea.2) or \equ(idea.1)
on the spectral gap or on the logarithmic
Sobolev constant, become at least of the order of a power of $L$.
In order to overcome this difficulty, we appeal to Proposition
\thf[F1]. More precisely we introduce an intermediate length scale
$l_1\ll L$ and we assume that the $J$ in $\L$ are such that the
hypotheses of Proposition \thf[F1] apply for $l_1$. If this is the
case, then Proposition \thf[F1] basically allows us to replace the
initial cube $\L=Q_L$ with a smaller cube $Q_{l_1}(x)$, for a
suitable $x\in \L$.
Once we have reduced the initial scale $L$ to the
new scale $l_1$, we make the coarse--grained analysis on scale
$l_0\ll l_1$ on the new cube $Q_{l_1}(x)$ and proceed as explained
before.
The advantage of the above two--scale analysis is twofold. First
of all the shortest scale $l_0$ is now at most of the order of
$\log l_1$ instead of $\log L$. Secondly the prefactor $L^d$ in
\equ(idea.1) is replaced by $l_1^d$. If one considers that in a
typical configuration the intermediate scale $l_1$ can be taken already
of the order of $\log L$ (see the comments before Proposition
\thf[F1]), we see that the smallest scale becomes
$l_0\approx \log\log L$ with an enormous
gain in precision. We conclude this short heuristic discussion by observing
that it is precisely
the coarse--grained
analysis on scale $\log\log L$ that is responsible for the various
$\log\log L$ factor in Theorem \thf[P20].
\smallno
We are now ready for a precise formulation of our results.
%@E12
%@Id
\nproclaim Definition [D2].
Let $l \in \Zp$, $\a>0$ and
let $\L$ be a mutilple of $Q_l$ and write
$\L = \cup_{i=1}^n Q_l(x_i)$ for some $n\in\Zp$ and $x_i \in l \Z^d$.
Let $K$ be the set of all $i\in \{1,\ldots,n\}$ such that
$Q_l(x_i)$ is not $\a-$regular. Then we let
$$
\eqalign{
W(\L, l, \a) &=
\bigl\{ \, x\in \L : \,
d \bigl(\, x, \cup_{i\in K} Q_l(x_i) \, \bigr)
\le 2 l \,
\bigr\}
\cr
v(\L, l, \a) &=
\hbox{the cardinality of the largest
$r-$connected component of $W(\L, l, \a)$ }
\cr}
$$
\medno
Given $\l \ge 0$, we also define a cutoff interaction
$J^{(\l)}$ as
$$
J^{(\l)}_A = (\sgn J_A) \, ( |J_A| \mmin \l )
\Eq(cutoff)
$$
%@E3
%@Ia
\bigno
\nproclaim Proposition [F2].
Choose the transition rates $c_J$ as in \equ(GRAD).
Then, for each $\a>0$ there exists $\bar l(d,r,\a)$
such that the following holds for all $l_0 \ge \bar l$:
let $V$ be a multiple of $Q_{l_0}$,
$v = v(V, l_0, \a)$ (see Definition \thf[D2]), and
let $m = \hm(d,r,\a)$.
Let also $\l, \g \ge 0$,
and assume that
\smallno
\item{(i)}
$ m l_0 \ge 10 \, ( 1 + \log |W( V, l_0, \a)|) $
\item{(ii)}
For each $r-$connected subset $X$ of $V$ with $|X| \le v$, we have
$\sum_{x\in X} \| J - \Jl \|_x \le \g$
\smallno
Then,
$$
\inf_{\t\in\O}
\gap( L_V^\t ) \ge
|V|^{-\o}
\exp\bigl[ \, - ( \, 8 \g + k \l v^\dd + k' m l_0^d \, )
\, \bigr]
\Eq(F2.0)
$$
where $\o$ can be taken equal to $d \log 4 / \log(3/2)$,
$k=k(d,r)$ is the quantity defined in Theorem \thf[F4] and $k' = 9^{d-1} k$.
\smallno
{\it Remark.} The reader who does not want to bother with the extra
complications due to the unboundedness of the interaction may just
consider the bounded case and take $\l$ equal to $\sup_x \|J\|_x$
and $\g=0$.
\smallno
\Pro\
Write
$V = \bigcup_{i=1}^s C_i$, where
$C_i = Q_{l_0}(y_i)$
for some
$y_i \in l_0 \Z^d$.
Let $B = W(V, l_0, \a)$ and let $A$ be the union of all
those ($\a$--regular) cubes $C_i$ such that $d(C_i, C_j) > l_0$ for all
$C_j$ which are not $\a-$regular.
Let also $A_0 = A \cap \dep_{l_0} B$ and $B_0 = B \cap \dep_r A$.
By Proposition \teo[A1.1] in \ref[CM], we have
$$
\inf_{\t\in\O}
\gap( L^{J, \t}_V ) \ge
\ov2 \,
\Bigl[ \,
\inf_{\t\in\O} \,
\inf_{ D \in \{A,B\} }
\gap( L^{J, \t}_D ) \,
\, \Bigr]
\inf_{\t\in\O}
\gap( L^{J, \t}_{ \{A,B\} } )
\Eq(F2.2)
$$
The proof of the Proposition can be organized in the following steps
\smallno
\textindent{(a)}
We can use Proposition \thf[GP6] to show that the gap of the block
dynamics generator $L^{J, \t}_{ \{A,B\} }$ is at least $1/2$.
In order to show that \thf[GP6] does indeed apply to our case,
we first notice that $d(A_0, B_0) \ge l_0$,
which, together with the fact that all cubes in $A$ are
$\a-$regular and the trivial inequality $|B_0| \le |W(V,l_0, \a)|$,
implies the hypothesis (i) of \thf[GP6]. Then we observe that
$A \setm A_0$ can be expressed as a union of
$\a-$regular cubes $C_i$. So, by Proposition \thm[REG],
the property $SMT( A\setm A_0, l_0/2, m)$ holds.
\textindent{(b)}
Since the set $A$ is a union of $\a-$regular cubes,
using the ideas in \ref[MO1] one can prove that
$\gap( L_A^{J,\t} )$ is bounded from below by a quantity
which does not depend on the size of $A$. In
the Appendix 2, we give a simple proof of the much weaker result
$$
\gap( L_A^{J,\t} ) \ge
8 |A|^{-\o} \,
\exp( -k' m l_0^d )
$$
Such an inequality, even if far from optimal, is sufficient anyway for
our purposes.
\textindent{(c)}
For what concerns the gap of $L_B^{J,\t}$,
we write $B$ as the disjoint union of its $r-$connected
components $\tB_1, \ldots, \tB_n$. Since $L_{\tB_i}^{J,\t}$
commutes with $L_{\tB_j}^{J,\t}$ for all $i\ne j$, it follows
that
$$
\gap( L_B^{J,\t} ) =
\inf_{i \in \{1, \ldots, n\} }
\gap( L_{\tB_i}^{J,\t} )
$$
\textindent{(d)}
Now we get rid of those couplings which are too
strong, by introducing, on each $\tB_i$, a cutoff interaction
$\{\Jl_X\}_{X \in \bF}$ (see \equ(cutoff)).
By \equ(gap) and hypothesis (ii), we obtain
$$
\gap( L_{\tB_i}^{J,\t} ) \ge
\nep{ - 8 \g } \,
\gap( L_{\tB_i}^{\Jl,\t} )
$$
\smallno
From \equ(F2.2), (a), (b), (c) and (d), together with Theorem \thf[F4]
(for the dynamics \equ(GRAD) we can take $c_m=1/2$ and $\k_1 = 0$
in \equ(bounded)) and the fact that trivially $\|\Jl\|_{\tB_i} \le \l$,
we get
$$
\inf_{\t\in\O}
\gap( L_V^{J,\t} ) \ge
\ov4 \,
\min\bigl\{ \ov2 \,
\inf_i
\nep{ - ( 8 \g + k \, \l \, |\tB_i|^\dd )
}
\, , \,
8 |A|^{-\o} \,
\exp( -k' m l_0^d )
\bigr\}
$$
In order to obtain \equ(F2.0) we now observe that
by definition of $v$, we have $|\tB_i| \le v$, and that the minimum
of the two quantities in braces is greater than their product
if $l_0$ is such that both terms are less than 1.
\QED
%@E7
%@I3
\nproclaim Theorem [F0.7].
If the transition rates are given by \equ(GRAD), then
for each $\a>0$ there exist $\bar l$, $C_1$ and $C_2$
depending on $d$, $r$ and $\a$ such that the following
holds for all positive integers $l_0 \ge \bar l$:
let $l_1$ be a multiple of $l_0$ and let $\L$ be a
multiple of $Q_{l_1}$ so that we can write
$$
\txt
\L = \bigcup_{i=1}^n B_i = \bigcup_{i=1}^s C_i
\Eq(multip)
$$
where
$B_i = Q_{l_1}(x_i)$
and
$C_i = Q_{l_0}(y_i)$
for some $x_i \in l_1 \Z^d$ and $y_i \in l_0 \Z^d$
Let $v = v(\L, l_0, \a)$ (see Definition \thf[D2]), and
let $m = \hm(d,r,\a)$.
Let also $\l, \g \ge 0$,
and assume that
\smallno
\item{(i)}
For each $i\in \{1,\dots n\}$ the cube $B_i$ is $\a-$regular
\item{(ii)}
$8 \g + k \l v^\dd \le m l_1 /4$, where $k(d,r)$ is the quantity
defined in Theorem \thf[F4]
\item{(iii)}
$30 d \log l_1 \le m l_0 \le (l_1)^{1/(2d)}$
\item{(iv)}
For each $r-$connected $V\sset\L$ with $|V| \le v$, we have
$
\sum_{x\in V} \| J - J^{(\l)} \|_x \le \g
$
\smallno
Then we have
$$
\sup_{\t\in\O}
c_s(\mu_\L^{J,\t}) \le
C_1 \exp \bigl[ \,
8\g + C_2 \, (\l v^\dd + l_0^d )
\, \bigr]
$$
\Pro\
Let $V \sset \L$ be a union of cubes $C_i$ such that
$\diam(V) \le 3 l_1$. The hypotheses (iii) and (iv) tell us that
Proposition \thf[F2] can be applied to $V$. Therefore
we have
$$
\gap( L_V^{J, \t} ) \ge
|V|^{-\o} \,
\exp\bigl[ \, - ( \, 8 \g + k \l v^\dd + k' m l_0^d \, )
\, \bigr] \ge
\nep{ - m l_1 /2 }
\Eq(F0.2)
$$
where,
$\o = d \log 4 / \log(3/2)$, and
in the second inequality, we have used hypotheses (ii) and (iii).
Thanks to \equ(F0.2) we can now apply Proposition \thf[F1], which,
combined with Proposition \thf[F3] and again with \equ(F0.2) implies that
$$
\eqalign{
\sup_{\t\in\O}
c_s(L_\L^{J,\t})
& \le
C' \,
\bigl[ \, 4 + 4 (3l_1)^d (\JL + 2 \log 2) \, \bigr] \,
(3l_1)^{\o d} \,
\exp\bigl[ \, 8 \g + k \l v^\dd + k' m l_0^d
\, \bigr] \le \cr
& \le
C_1 \exp \bigl[ \,
8\g + C_2 (\l v^\dd + l_0^d )
\, \bigr]
\cr }
$$
for some $C'$, $C_1$, $C_2$ depending on $d$, $r$ and $\a$.
\QED
%@E3
\fine
%--------------------------------- INIZIO
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \input xmac\fi
\numsec=6
\numfor=1
\numtheo=1
\pgn=1
\beginsection 6. Proof of the main results
\beginsubsection 6.1. The upper bounds
In this section we finally prove our main results. Before doing that we need
a simple probabilistic estimate on independent
random variables.
%@I
\bigno
\nproclaim Lemma [P6].
Let $\{X_i\}_{i=1}^n$ be real independent random variables such that
$$
\bE\, ( \exp( \, X_i^{1+\d} \, ) ) \le G_\d < \infty
$$
for some $\d>0$, for all $i$. Then, for all $\l, \g>0$
$$
\bP
\Bigl\{
\, \sum_{i=1}^n (X_i-\l)_+ \ge \g \,
\Bigr\} \le
\exp \bigl[ \, -\l^\d \g + n G_\d \nep{ -\l^{1+\d}} \,
\bigr]
$$
\Pro\
By the Chebyshev inequality, and using $\log(1+x) \le x$, we obtain
$$
\eqalign{
&
\bP \bigl\{ \, \sum_{i=1}^n (X_i-\l)_+ \ge \g \, \bigr\} \le
\nep{ - \a \g} \bigl[ \bE \nep{ \a (X_i - \l)_+ } \bigr]^n \le \cr
& \le
\nep{ - \a \g} \left[ 1 + \nep{ - \a \l}
\bE\, \bigl( \nep{ \a X_i } \id\{ X_i \ge \l \} \bigr) \right]^n \le
\exp\left[ -\a \g + n \nep{ - \a \l}
\bE\, \bigl( \nep{ \a X_i } \id\{ X_i \ge \l \} \bigr) \right]
\cr }
$$
Now take $\a = \l^\d$ and notice that
$$
\bE\, \bigl( \nep{ \l^d X_i } \id\{ X_i \ge \l \} \bigr) \le
\bE\, \bigl( \nep{ (X_i)^{1+\d} } \id\{ X_i \ge \l \} \bigr) \le
\bE\, \nep{ (X_i)^{1+\d} } \le G_\d
\QED
$$
%@E
%@I
\bigno
\nproclaim Proposition [P1].
Assume (H2). Then there exists $k = k(d,r) >0$ such that if
$v > \log L$ and if we let
$$
A_\d = (\log G_\d)^{1\over 1+\d}\mmax 1\qquad
\l = v^{1\over d(1+\d)}A_\d \qquad
\g = k v^{1 - {\d\over d(1+\d)} }
$$
then, for all $L \in \Zp$ (see \equ(cutoff))
$$
\bP
\Bigl\{ \,
\exists V\sset Q_L :
\hbox{$V$ is $r-$connected }, \ |V|\le v, \
\sum_{x\in V} \| J - J^{(\l)} \|_x \ge \g \,
\Bigr\}
\le L^{-3d}
\Eq(P1.0)
$$
\Pro\
For each $V\in \bF$ we have
$$
\bP
\Bigl\{ \,
\sum_{x\in V} \| J - J^{(\l)} \|_x \ge \g \,
\Bigr\} \le
\bP
\Bigl\{ \,
\sum_{ A : \, A \cap V \ne \emp } ( J_A - \l)_+ \ge \g/k_1 \,
\Bigr\}
\Eq(P1.2)
$$
where $k_1$ can be taken equal to $\sup \{ |A| : \, \diam A \le r \}$.
Using lemma \thm[P6] and the fact that the number of sets
$A \in \bF$ with a diameter not greater than $r$ which intersect $V$
can be bounded by $|V| k_2(d,r)$ we obtain
$$
\bP
\Bigl\{ \,
\sum_{x\in V} \| J - J^{(\l)} \|_x \ge \g \,
\Bigr\} \le
\exp\Bigl[
- \l^\d {\g\over k_1} + k_2 |V| G_\d \nep{ - \l^{1+\d} }
\Bigr]
\Eq(P1.4)
$$
Furthermore, since the number of $r-$connected $V\sset Q_L$
such that $|V|\le v$
is not greater than
$L^d \exp( k_3 v )$ for some $k_3(d,r)$, if $v \ge \log L$
and $\l$ is chosen as in the hypothesis, we get
$$
\hbox{RHS of \equ(P1.0)} \le
\exp\Bigl[
v ( d + k_3 + k_2 ) - \l^\d {\g\over k_1}
\Bigr]
$$
If now $k \ge k_1 ( 4d + k_2 + k_3)$, we find
$$
\hbox{RHS of \equ(P1.0)} \le
\nep{ - 3 d v } \le L^{- 3d }
\QED
$$
%@E
%@Ia
\bigno
{\it Proof of Theorem \thm[P20].}
We give the proof in the special case of $L$ which is a power of $2$,
which is enough to prove Theorem \thf[UB].
A proof which works for all $L$ requires a modification of Theorem
\thf[F0.7] where one considers more general coverings of $\L$
with cubes and cuboids with slightly different
sidelengths. This generalization is straightforward.
\medno
{\it Part (i).}
By combining hypothesis (H2) with the exponential
Chebyshev inequality, one gets
$$
\bP \bigl\{ \,
\|J\|_{Q_L}\ge
3 \, (\log L)^{1-{\d\over d(1+\d)}}
\bigr\}
\le L^d G_\d
\exp\bigl[- 3 (\log L)^{1+\d-{\d\over d}}\bigr]
\ll L^{- 2}
\Eq(P20-2)
$$
for all $L$ large enough. Therefore, using \equ(LSC.1),
it is enough to prove
\equ(P20.2) with
$c_s(L_{Q_L}^{J,\t})$ replaced by
$c_s(\mu_{Q_L}^{J,\t})$ and $L^{-1.5}$ replaced by $3 L^{-2}$.
For this purpose we are going to use the key deterministic estimate of
$c_s(\mu_{Q_L}^{J,\t})$ given in Theorem \thm[F0.7].
The idea is to prove that with probability greater
than $1- 3 L^{-2}$, it is possible to choose the four parameters in
Theorem \thm[F0.7], $l_0$, $l_1$, $\l$ and $\g$ in such a way that the
deterministic upper bound on $c_s(\mu_{Q_L}^{J,\t})$ given in
that proposition is not greater than
$$
\exp \bigl[ \, C A_\d ( \log\log L)^{d-{\d\over (1+\d)}}
(\log L)^{ 1- {\d \over d (1+\d)} }
\, \bigr]
$$
More precisely we define $l_0$ and $l_1$ as those powers of $2$
(they are uniquely defined) such that
$$
{60 d\over m}\log\log L \le l_0 < {120 d\over m}\log\log L \qquad
{3 d\over \th'}\log L \le l_1 < {6 d\over \th'} \log L
\Eq(P20-a)
$$
where $m \equiv \hm(\a)$ (see Proposition \thm[MIX1])
and $\th'$ is given in Proposition \thf[REG].
We then take
$$
v_* = l_0^d \log L
\quad
\l = v_*^{1\over d(1+\d)}A_\d \quad
\hbox{and} \quad
\g = k v_*^{1 - {\d\over d(1+\d)}}
\Eq(P20-3)
$$
and $k(d,r)$ is given in Proposition \thf[P1].
Since $l_0$ divides $l_1$ and $l_1$ divides $L$,
we can write $Q_L$ as in \equ(multip).
We now
observe that, if $L$ is large enough,
the hypotheses (i) -- (iv) of Theorem \thf[F0.7]
are satisfied for all $J \in \tilde\Th \equiv \cap_{i=1}^3 \Th_i$, where
\smallno
\item{} $\Th_1 = \{J: \, \hbox{each $B_i$ is $\a-$regular} \} $
\item{} $\Th_2 = \{J: \, v(Q_L,l_0,\a) \le v_* \}$
\item{} $\Th_3 = \{J: \,
\hbox{for each $r-$connected $V\sset\L$ with
$|V| \le v_* ,\; \sum_{x\in V} \| J - J^{(\l)} \|_x \le \g $ } \}$
\smallno
and $v(Q_L,l_0, \a)$ has been defined in \thm[D2].
Notice that, for all $J\in \Th_1$ the bound
$\|J\|_{Q_L}\le k'\log L$ holds for some constant $k'$,
because of the definition of $\a-$regular cubes
and of our choice of $l_1$.
By Theorem \thm[F0.7], \equ(P20-a) and \equ(P20-3), for any
$J\in\tilde\Th$, we have
$$
c_s(\mu_{Q_L}^{J,\t}) \le
\exp \bigl[ \, C A_\d ( \log\log L)^{d-{\d\over 1+\d}} (\log L)^{ 1- {\d
\over d(1+\d)} }
\, \bigr]
$$
for a suitable constant $C$ independent of $J$.
In order to prove the theorem it is therefore sufficient to
estimate from above
$\bP\bigl( \tilde \Th^c \bigr)$.
\smallno
From Proposition \thf[REG], it follows
$$
\bP\bigl( \Th_1^c \bigr ) \le
L^d \nep{ - \th' l_1 }
\le L^{-2d}
\Eq(P20-4)
$$
for all $L$ large enough.
Let $p(l)$ be the probability that a cube
$Q_{l}$ is not $\a-$regular.
Then $p(l)$ goes to zero as $l\to\infty$,
and a standard estimate for $2-$dependent
site percolation implies
$$
\bP\bigl( \Th_2^c \bigr) \le L^{d} \,
\bigl( k_1 \, p(l_0) \bigl)^{k_2 v_* l_0^{-d}} \le
L^{-3d}
\Eq(P20-5)
$$
for $L$ large enough, where $k_1$ and $k_2$
are two suitable geometrical constants.
Finally, by
Proposition
\thm[P1], we have
$\bP\bigl( \Th_3^c \bigr) \le L^{-3d}$, and, by consequence
$\bP\bigl( \tilde\Th^c \bigr) \le 3 L^{-2}$.
This completes the proof in the general unbounded case.
The bounded case can be treated in the same way, by choosing
$\l= J_0$ and $\g = 0$.
\medno
{\it Proof of part (ii).}
The proof is the same as in part (i), with a
different choice of the three basic parameters $l_0$, $l_1$ and $v_*$.
More precisely we define $l_0$ and $l_1$ as those powers of $2$
(they are uniquely defined) such that (let again $m = \hm(\a)$)
$$
\txt
{60 d^2\over (d-1) m}\log\log L \le l_0 < {120 d^2\over (d-1) m}\log\log L
\qquad
(\log L)^{d\over d-1} \le l_1 < 2 ( \log L)^{d\over d-1}
$$
Given $\e\in(0,1)$ we then let
$$
\txt
\l = J_0 \quad
\g = 0 \qquad
\hbox{and} \quad
v_* = \left( {\e \log L \over 2J_0 C_2 } \right)^{d\over d-1}
\Eq(P20-7)
$$
where $C_2$ appears in Theorem \thm[F0.7].
Write $Q_L$ as in
\equ(multip) and define the events
$\tilde \Th$, $\Th_i$ as in the proof of part (i).
Thanks to Theorem \thm[F0.7], we get
$$
c_s(\mu_{Q_L}^{J,\t}) \le
C_1 L^{\e\over 2} \exp(C_2 l_0^d) \le L^\e
\qquad \forall J\in \tilde \Th
\Eq(P20-8)
$$
for all $L$ sufficiently large.
In order to prove \equ(P20.3)
it is therefore sufficient to bound
from above $\bP \bigl( \tilde \Th^c \bigr)$. As before,
we find
$$
\bP\bigl( \Th_1^c \bigr) \le
L^d \nep{ - \th' l_1 } =
L^d \exp\bigl[- \th' (\log L)^{d\over d-1}\bigr]
\Eq(P20-9)
$$
and
$$
\bP\bigl( \Th_2^c\bigr ) \le
L^{d} \, ( k_1 p(l_0) )^{k_2 v_* l_0^{-d}} \le
\exp\bigl[ \, -C_3 \, (\log\log L)^{-d} (\log L)^{d\over d-1}\bigr]
\Eq(P20-10)
$$
for a suitable constant $C_3$ and all $L$ large enough.
Clearly \equ(P20-9) and \equ(P20-10) complete the proof of (ii).
\QED
%@E9
\bigno
%@Ij
\noindent
{\it Proof of Theorem \thm[UB].}
The proof of the almost sure bounds (part (a))
is a simple consequence of Theorem \thm[P20].
We prove only \equ(UB1) since the case of bounded interactions
\equ(UB2) is identical.
Let $\bar \Th$ be the set of interactions $J$ such that for each
$J\in \bar \Th$
there exists $L_1(J)$ such that for all $L\ge L_1(J)$
($C$ is given in Proposition \thf[P20])
\smallno
\item{(i)}
$
c_s( L_{B_L}^{J} ) <
\exp \bigl[ \, C A_\d \,
(\log L)^{ 1 - {\d'\over d } } \,
( \log\log L)^{ d - \d'}
\, \bigr]
$
\item{(ii)} $ SMT(B_L,\g_1 (2L+1), \a)$ holds
\item{(iii)} $\|J\|_{\bar B_L} \le \log L$, where
$\bar B_L = B_L \cup \dep_r B_L$
\smallno
Using Theorem \thm[P20], (H1), (H2) and the Borel--Cantelli lemma,
one can check that
$\bP(\bar \Th) =1$.
Moreover, thanks to (ii) and (iii),
for all $J\in \bar \Th$ there exists a unique infinite volume Gibbs
measure that in the sequel will be denoted by $\mu^J$.
Let, in fact, $f$ be any local function on $\O$,
and take $L$ large enough such that $B_L \supset \L_f$.
Then, given two arbitrary boundary conditions $\t$ and $\h$,
and using a telescopic interpolation between them, we get
$$
\eqalign{
&
\sup_{\t, \h \in \O}
\bigl| \, \mu_{B_L}^{J,\t}(f) - \mu_{B_L}^{J,\h}(f) \,\bigr|
\le
|\dep_r B_L|
\sup_{x\in \dep_r B_L} \,
\ninf{ \nabla_x [\mu_{B_L}^{J}(f)] } =
\cr
&
= |\dep_r B_L|\sup_{x\in \dep_r B_L}
\sup_{\t\in \O}
\left| \, {\mu_{B_L}^{J,\t}(h_x, f) \over \mu_{B_L}^{J,\t}(h_x)} \, \right|
}
\Eq(UB.0)
$$
where $h_x\equiv \exp[- \nabla_x H_{B_L} ]$.
\smallno
Notice that, because of (iii) in
the definition of $\bar \Th$, we have $\|h_x\|_\infty\le \exp(2 \log L) =L^2$.
Therefore, if $L$ is larger than $L_1(J)$ and if
$d(\L_f, (B_L)^c) > \g_1 (2L+1) +r$,
we can use $SMT(B_L,\g_1 (2L+1), \a)$, and write
$$
\sup_{\t, \h \in \O}
\bigl| \, \mu_{B_L}^{J,\t}(f) - \mu_{B_L}^{J,\h}(f) \,\bigr|
\le
k L^{d+4} |\L_f| \ninf{f} \nep{ -\a d(\L_{h_x},\L_f) }
\Eq(mumu)
$$
for a suitable constant $k$, and the uniqueness follows.
In order to prove
inequalities \equ(UB1) and \equ(UB2) we first need to recall a
standard result on the ``finite speed of information
propagation'' for Glauber dynamics with bounded rates.
\bigno
\nproclaim Lemma [approx].
Assume the transition rates uniformly bounded, \ie
$\k_2=0$ in \equ(bounded).
Then there exists a constant $k_{0}$ depending
on $d$, $r$ and $c_M$ and for any local function $f$,
there is $A(f)$ such that for all $V \ssset \Z^d$, $t \ge 0$
with $d(V^c,\L_f) \ge k_0 t$, we have
$$
\sup_{\t\in \O}
\|T(t)f-T_{V}^{\t}(t)f\|_\infty \leq A(f) \, \nep{-2t}
$$
\Pro\
One can see for instance Lemma \teo[1.7] in \ref[HS],
or Lemma \teo[1] in \ref[S] which makes use of the explicit
``graphical construction'' of the process.
\QED
\medno
Let now $L_t = \inte{k_1 t}$ for some $k_1 > k_0$
($k_0$ is given in Lemma \thm[approx]) and, for simplicity, let
$\L_t = B_{L_t}$.
Choose an arbitrary boundary condition $\t$. Then we have
$$
\eqalign{
\|T^J(t)f-\mu^J(f)\|_\infty
& \le
\|T^{J,\t}_{\L_t}(t)f-\mu^{J,\t}_{\L_t}(f)\|_\infty + \cr
& +
\|T^J (t)f-T^{J,\t}_{\L_t}(t)f\|_\infty +
|\mu^{J,\t}_{\L_t}(f)-\mu^J(f) |
\cr }
\Eq(UB.1)
$$
Let us examine separately the three terms appearing in the RHS of \equ(UB.1).
The first one, using
(i) and (iii) above, together with
hypercontractivity (see the proof of Theorem 4.1 in
\ref[GZ1]), can be bounded from above by
$$
\eqalignno{
\|T^{J,\t}_{\L_t}(t)f-\mu^{J,\t}_{\L_t}(f)\|_\infty
& \le
2 \, \tnorm{f} \,
\exp\bigl[-{t\over 2}c_s(L_{\L_t}^{J,\t} )^{-1}\bigr] \le
& \eq(UB.2) \cr
&
\le 2 \, \tnorm{f} \,
\exp\Bigl[-{t\over 2}
\exp\bigl[\, -C A_\d \,
(\log L_t)^{ 1 - {\d' \over d } } \,
(\log\log L_t)^{ d - \d' } \,
\bigr]
\, \Bigr]
& \cr }
$$
for any sufficiently large $t$.
\smallno
The second term in \equ(UB.1),
thanks to Lemma \thf[approx] is not greater than
$A(f) \nep{ -2t}$. The last term is bounded by the RHS of \equ(mumu)
which, if $k_1 > 3 \a^{-1}$, is bounded by $A'(f) \nep{- 2 t}$
for large $t$.
This concludes the proof \equ(UB1).
\medno
{\it Proof of part (b).}
Define $L_t$ as in part (a)
and, for any $\e\in (0,1)$,
let
$\Th(t,\e)$ be the set of interactions $J$ such that
\smallno
\item{(i)} $c_s( L_{\L_t}^{J} ) \le L_t^\e $
\item{(ii)} $SMT(\L_t,(\log L_t)^{d\over d-1}, \hm(\a))$.
\smallno
We can write, for any $\t\in \O$,
$$
\eqalign{
\bE \, \|T^J(t)f-\mu^J(f)\|_\infty
& \le
\tnorm{f} \,
\bP \bigl( \Th(t,\e)^c \bigr) +
\sup_{J \in \Th(t,\e)}
\|T^{J,\t}_{\L_t}(t)f-\mu^{J,\t}_{\L_t}(f)\|_\infty
+ \cr
& +
\|T^J(t)f-T^{J,\t}_{\L_t}(t)f\|_\infty +
\sup_{J \in \Th(t,\e)}
|\mu^{J,\t}_{\L_t}(f)-\mu^J(f)|
\cr }
\Eq(UB.4)
$$
We denote by $X_1$, $X_2$, $X_3$ and $X_4$ the four terms on the RHS
of \equ(UB.4).
For the last two terms we can proceed as in part (a) and we get
$$
X_3 + X_4 \le
( A(f) + A'(f) ) \, \nep{ -2 t }
\Eq(UB.5)
$$
Furthermore, we have
$$
\eqalign{
\bP \bigl( \Th(t,\e)^c \bigr)
&\le \bP \bigl\{ \, c_s( L_{\L_t}^{J} )
\ge L^\e \, \bigr\} +
\cr
&+ \bP \bigl\{ \, SMT(\L_t,(\log L)^{d\over d-1}, \hm(\a))
\hbox{ does not hold } \bigr\}
\cr }
\Eq(UB.6)
$$
Of the above two terms the first
one is estimated via (ii) of Theorem \thm[P20], which implies
$$
\bP \bigl\{ \, c_s( L_{\L_t}^{J} ) \ge L^\e \, \bigr\}
\le
\exp \bigl[\, -C_3 \, ( \log\log L_t)^{-d} (\log L_t)^{d\over d-1}\bigr]
\Eq(UB.7)
$$
provided that $t$ is large enough.
The second term in the RHS of \equ(UB.6)
can be bounded from above, using Proposition \thm[MIX1], by
the probability that there exists a cube $Q_l(x)$ in $\L_t$,
with $l=\ceil{(\log L)^{d\over d-1}}$,
which is not $\a-$regular. Using
Proposition \thf[REG] such a probability is bounded from above
by
$$
L_t^d \exp[ -\th' \, (\log L_t)^{d\over d-1}]
\Eq(UB.8)
$$
provided that $t$ is so large that $L_t\ge L_0$.
In this way we have obtained
$$
X_1 \le
\tnorm{f} \,
\Bigl[ \,
\exp [\, -C_3 \, ( \log\log L_t)^{-d} (\log L_t)^{d\over d-1}]
+ L_t^d\exp[-\th (\log L_t)^{d\over d-1}] \, \Bigr]
\Eq(UB.9)
$$
As for $X_2$,
we use
hypercontractivity (see again the proof of Theorem 4.1 in
\ref[GZ1]) and the fact that now
$c_s( L_{\L_t}^{J} ) \le L_t^\e$, and we get
$$
X_2 \le
2 \, \tnorm{f} \, \exp\bigl[ \,- k' \, t^{1-\e} \, \bigr]
\Eq(UB.10)
$$
for any $t$ sufficiently large.
From \equ(UB.5), \equ(UB.9) and \equ(UB.10) we get that
for large
$t$ the dominant term in \equ(UB.4) is the first one and, by consequence
\equ(UBb) follows.
\QED
%@E9
%@Il
\beginsubsection 6.2. Proof of the lower bound, Theorem \thf[Low-Bo]
The main idea behind the proof is not new (see \ref[DRS])
and it can be summarized as follows.
If {\it all\/} the couplings $J_{xy}$
in the cube $B_L$
are above the critical value for the standard Ising
model,
then the spin at the origin reaches the equilibrium after a
time $t$ which is at least relaxation time of the cube $B_L$.
Since the
relaxation time for the stochastic Ising model in a cube $B_L$, at low
temperature and zero external field, grows like the exponential of the {\it
surface} $L^{d-1}$,
it follows that if
$L\approx (\log t)^{1\over d-1}$,
then at time $t$ the spin at the origin has not yet equilibrated.
To complete the argument one has to observe that, under our
assumptions, the probability that
the $J_{xy}$'s in $B_L$
are all equal and large is not smaller than an exponential of the volume
$L^d$.
\medno
Let us now provide the details.
Given
$J_1>0$ and a positive integer
$L$, let
$\L=B_L$ and let
$\Thb$ be the set of all interactions $J\in \Th$ such that hypothesis (i)
holds and
\smallno
\item{(a)} $J_{xy} = J_1$ for all $\xy$ such that $\xy \sset \L$
\item{(b)} $|J_{xy}| \le \ov4$ for all $\xy$ which intersect both $\L$
and $\L^c$ (the boundary edges)
\smallno
If we denote with
$m_{\L}= |\L|^{-1} \sum_{x\in \L} \s(x)$
the normalized magnetization in $\L$,
we can write (remember that $\mu^J(\pi_0) = 0$)
$$
\bE \, \| T^J(t) \pi_0 \|_\Ld \ge
\bE \, \|T^J(t) \, m_\L \|_\Ld \ge
\bP( \Thb ) \,
\inf_{J\in \Thb}
\|T^J(t) \, m_{\L} \|_\Ld
\Eq(low.1)
$$
Choose $J \in \Thb$ and let
$$
\txt
F_\L = \{ \s\in\O : \, m_\L(\s) > \ov2 \}
$$
Then we have
$$
\|T^J(t) \, m_{\L} \|_\Ld \ge
\sqrt{\mu(F_\L)} \;
\|T^J(t) \, m_{\L} \|_\LdF
$$
and
$$
\|T^J(t) \, m_{\L} \|_\LdF
\ge
\|T^J(t) \, m_{\L} \|_\LuF \ge
\mu( T^J(t) \, m_{\L} \tc F_\L )
\Eq(l2l1)
$$
For $\s \in \O$, let $\{\h^\s_t\}_{t\ge 0}$ be the process
associated with $T^J(t)$
with initial condition $\h^\s_0 = \s$, and let
$\{\h^\mu_t\}_{t\ge 0}$ be the stationary process
(the one with initial distribution $\mu^J$).
Consider the events
$$
G_{\L,t}^\s \equiv \{ \, \exists s \in [0,t] : \,
| m_\L( \h^\s_s) - 1/2 | \le 1/(100) \, \}
\qquad
\s \in \O \cup \{\mu\}
$$
For each $\s\in F_\L$, if $|\L| > 100$, we have
$$
m_\L(\h^\s_t) \ge
\ov2 \id_{ (G_{\L,t}^\s)^c } -
\id_{ G_{\L,t}^\s } =
\ov2 -
{3\over 2} \id_{ G_{\L,t}^\s }
$$
which implies
$$
\mu( T^J(t) \, m_{\L} \tc F_\L ) \ge
\ov2 - {3\over 2} \int_\O \mu( d \s \tc F_\L) \prob( G_{\L,t}^\s ) \ge
\ov2 - {3\over 2} \mu(F_\L)^{-1} \prob( G_{\L,t}^\mu )
\Eq(3/2)
$$
If $t_1, t_2, \ldots$ are the (random) times at which the stationary process
$\h^\mu_t$
is updated inside $\L$ and $n_t$ is the number of updates up to time $t$,
we have, for all $j \in \Zp$,
$$
P( G_{\L,t}^\mu )
\le
j \mu^J \{ |m_{\L}(\s )-1/2| \le 1/(100) \} +
\prob \{ n_t > j \}
\Eq(low.d)
$$
which, taking $j = k |\L| t$ with $k = 2 c_M$, can be bounded by
(remember that we have $\k_2 = 0$)
$$
k \, |\L|\, t \; \mu^J
\{ |m_{\L}(\s )-1/2|\le 1/(100) \} + \nep{-k' |\L| t}
\Eq(low.3)
$$
for a suitable positive constant $k'$.
\smallno
The idea is now to
\smallno
\textindent{(1)} replace
$\mu^{J}$ with
$\mu_{\L}^{J_1,\emptyset}$, \ie with the Ising Gibbs measure in $\L$ with
coupling $J_1$ and free boundary conditions.
Thanks to the properties (a) and (b) of $\Thb$ and the DLR condition,
the price to pay can be estimated as
$$
\nep{- |\dep\L| / 2}
\mu_{\L}^{J_1,\emptyset} (X) \le
\mu^{J} (X) \le
\nep{ |\dep\L| / 2}
\mu_{\L}^{J_1,\emptyset} (X)
\qquad
\forall X \in \cF_\L
\Eq(low.4)
$$
\textindent{(2)}
use the following key result of \ref[P]
for the large deviations of the
magnetization for the $d-$dimensional Ising model in $\L$
without external field and with free boundary conditions.
\nproclaim Theorem [Low-Bo.1].
There exists $J_1>0$ such that
$$
\eqalign{
\mu_{\L}^{J_1,\emptyset}
\{ m_{\L}(\s )\ge 1/2 \} &\ge {1\over 3}\cr
\mu_{\L}^{J_1,\emptyset}
\{ |m_{\L}(\s )-1/2|\le 1/(100) \}
&\le \nep{-2|\dep \L|}
\cr}
$$
\Pro\
See \ref[P]
\medno
{\it Remark.} The results of \ref[P] are stated for the standard
Ising model,
namely when the couplings $J_{xy}$ are all {\it equal\/} and large
enough. We expect the same result to hold also when the $J_{xy}$'s
are not all equal, but just large enough.
\medno
Choose now $J_1$ as in Lemma \thf[Low-Bo.1], and take
$L=L_t$ as the smallest integer for which
$|\dep \L|\geq 2\log t$.
In this way we find
$$
\mu( T^J(t) \, m_{\L} \tc F_\L ) \ge
{1\over 2} -
{9 \over 2} \, k \, |\L| \, t \, \nep{-|\dep \L|} -
{9\over 2} \nep{-k'|\L|t + |\dep \L|/2 }\ge {1\over 3}
\Eq(low.5)
$$
for all $t$ large enough.
From \equ(low.1) \dots \equ(low.5) it follows
$$
\bE \, \| T^J(t) \pi_0 \|_\Ld \ge
{1\over 3} \nep{ - |\dep\L|/2 } \, \bP( \Thb )
\ge \exp\bigl[\, -k'' \, (\log t)^{d\over d-1} \, \bigr]
$$
for a suitable positive constant $k''$.
\QED
%--------------------------------- INIZIO
\expandafter\ifx\csname sezioniseparate\endcsname\relax%
\input macro \input xmac\fi
\numsec=-1
\numfor=1
\numtheo=1
\pgn=1
\beginsection A1. Appendix 1
%@In
Given two subsets $A$, $B$ of $\Z^d$ we let
$$
\d_r(A,B) = (\dep_r A \cap B ) \cup (\dep_r B \cap A)
$$
\nproclaim Proposition [GEO0].
For each $d, r \in \Zp$,
there exists $k(d,r)$ such that for each $V\ssset \Z^d$,
and for each $v \in [0, |V| ]$ there exists $X_v \sset V$ such that,
if we let
$S \equiv \ceil{ 2 |V|^{d-1\over d} }$
and $Y_v \equiv V \setm X_v$, we have
\smallno
\item{(a)}
$v - k S \le |X_v| \le v$
\item{(b)}
$\d_r(X_v,Y_v) \le k S$
\item{(c)}
$X_0 = \emp$, $X_{|V|} = V$ and $X_v \sset X_w$ if $v < w$.
\Pro\
Given $V \ssset \Z^d$, we define the {\it $i-$width\/} of $V$ as
the smallest $k \in \Zp$ such that there exists $n\in \Z$
with the property that, for all $x\in V$, we have
$x_i \in \{n , \ldots, n+k-1 \}$ where $x_i$ is the $i^{th}$ coordinate
of $x$. We start with the following result
\nproclaim Lemma [GEO5].
Let $V$ be a finite subset of $\Z^d$, let $i\in \{1, \ldots, d\}$
and let $a \in [0, |V|]$. Let $L, S$ be two positive numbers such that
$ L S > |V|$.
Then there exist $k = k(d) > 0$ and two
disjoint subsets of $V$, $W_1$ and $W_2$ such that
\smallno
\item{(a)} $|W_1| \le a$
\item{(b)} The $i-$width of $W_2$ is less than or equal to $L$
\item{(c)} $|\d_1( W_1, V\setm W_1)| \le k S$ and
$|\d_1( W_2, V\setm W_2)| \le k S$
\item{(d)}
If $W_2 = \emp$ then $|W_1| \ge a - S$, while, if $W_2 \ne \emp$,
then $|W_1 + W_2| > a $.
\Pro\
If $a= |V|$, we take $W_1 = V$ and $W_2 = \emp$ and the Lemma follows.
Assume now that $a < |V|$.
For $j\in \Z$, let
$$
V^{(i)}_j = \{ x\in V : \, x_i = j \}
\Eq(GEO5.1)
$$
and
$$
m = \inf \{ j \in \Z : \, | \bigcup_{k\le j} V^{(i)}_k | > a \}
$$
Since $a < |V|$, $m$ is always finite. If $|V^{(i)}_m| \le S$
we set
$$
W_1 = \bigcup_{j S $. Define
$$
m_1 = \sup\{ j < m : \, |V_j^{(i)}| \le S \} \qquad
m_2 = \inf\{ j > m : \, |V_j^{(i)}| \le S \}
$$
Let then
$$
W_1 = \bigcup_{j \le m_1} V_j^{(i)} \qquad
W_2 = \bigcup_{j = m_1 +1}^{m_2 -1} V_j^{(i)}
$$
The statements (a) -- (d) are easily verified.
\QED
\medno
We now prove Proposition \thm[GEO0] when $r=1$ and then we will show
that this is enough to treat the case of $r$ arbitrary.
We let
$$
f_a^{(i)}(V) = W_1 \qquad g_a^{(i)}(V) = W_2
$$
where $W_1$ and $W_2$ are the
subsets of $V$ found in the previous lemma for given values of $a$ and $i$,
with $L$ and $S$ chosen as
$$
S \equiv \ceil{ 2 |V|^{d-1\over d} } \qquad
L \equiv \ceil{ |V|^{1\over d} }
$$
Then we define
$$
A_1 = f^{(1)}_v (V) \qquad D_1 = g^{(1)}_v(V)
$$
and recursively
$$
D_i = g^{(i)}_{v - |A_{i-1}|}(D_{i-1}) \qquad
E_i = f^{(i)}_{v - |A_{i-1}|}(D_{i-1}) \qquad
A_i = A_{i-1} \cup E_i
\Eq(GEO0.1)
$$
We let then $k = \min \{ i : \, D_i = \emp \}$ and we claim that
Proposition \thm[GEO0] holds with $X_v = A_k$.
We observe that necessarily $k \le d$. In fact, previous lemma together
with the fact that $D_i \sset D_{i-1}$, imply that the $i-$width
of $D_{d-1}$ is no greater than $L$ for all $i \in \{1, \ldots, d-1 \}$.
By consequence, all the slices of $D_{d-1}$ in the direction
perpendicular to the $d$ direction (see \equ(GEO5.1)) have a cardinality
not greater than $L^{d-1}$ which is less than $S$.
Thus, by the proof of the lemma (see \equ(GEO5.2)),
it is clear that $D_d = \emp$.
Then we check that, for all $i \in \{ 1, \ldots, k-1 \}$
we have $0 \le v - |A_{i}| \le |D_{i}|$, so that
definitions \equ(GEO0.1) make sense for $i\le k$.
This can be done by induction.
From statement (a) of previous lemma, we get
$$
|A_i| \le |A_{i-1}| + |E_i| \le |A_{i-1}| + v - |A_{i-1}| = v
$$
On the other hand, since $D_i \ne \emp$, statement (d) of
Lemma \thm[GEO5] implies
$$
|A_i| + |D_i| = |A_{i-1}| + |E_i| + |D_i| > |A_{i-1}| + v - |A_{i-1}|
= v
$$
Since $D_k = \emp$, Lemma \thm[GEO5] implies $|E_k| > v - |A_{k-1}| - S$,
thus, by consequence we have $|A_k| > v -S$. Together with
the statement $|A_k| \le v$ , this gives
part (a) of the Proposition.
In order to prove part (b), we notice that, thanks to Lemma \thm[GEO5],
we have
$$
|\d_1(E_i, D_{i-1}\setm E_i)| \le k S \qquad
|\d_1(D_i, D_{i-1}\setm D_i)| \le k S
\Eq(GEO0.2)
$$
We then claim that
$$
\eqalignno{
& \d_1( A_i, V\setm A_i) \sset \d_1(A_{i-1}, V\setm A_{i-1}) \cup
\d_1(E_i, D_{i-1} \setm E_i) \cup \d_1( D_{i-1}, V\setm D_{i-1})
& \eq(GEO0.4) \cr
& \d_1(D_i, V\setm D_i) \sset \d_1(D_i, D_{i-1}\setm D_i) \cup
\d_1( D_{i-1}, V\setm D_{i-1} )
& \eq(GEO0.5) \cr }
$$
Iterating \equ(GEO0.5) and using \equ(GEO0.2)
we find $|\d_1(D_i, V\setm D_i)| \le k i S$,
which, inserted into \equ(GEO0.4), together with \equ(GEO0.2),
gives $|\d_1( A_i, V\setm A_i)| \le k d^2 S$ which completes the proof
of the Proposition. To obtain \equ(GEO0.4), we write
$$
\d_1(A_i, V\setm A_i) =
\d_1( A_{i-1}, V\setm A_i ) \cup
\d_1( E_i, V\setm A_i ) \sset
\d_1( A_{i-1}, V\setm A_{i-1} ) \cup
\d_1( E_i, V\setm A_i )
$$
The last term can be written as
$$
\eqalign{
&
\d_1( E_i, V\setm A_i ) =
\d_1( E_i, D_{i-1} \setm A_i ) \cup
\d_1( E_i, V \setm (A_i \cup D_{i-1}) ) \sset \cr
& \sset
\d_1( E_i, D_{i-1} \setm E_i ) \cup
\d_1( D_{i-1}, V \setm D_{i-1} ) \cr }
$$
which proves \equ(GEO0.4). Furthermore, to get \equ(GEO0.5), we observe that
$$
\d_1( D_i, V\setm D_i ) =
\d_1( D_i, D_{i-1} \setm D_i ) \cup
\d_1( D_i, V \setm D_{i-1} ) \sset
\d_1( D_i, D_{i-1} \setm D_i ) \cup
\d_1( D_{i-1}, V \setm D_{i-1} )
$$
This proves (b). Property (c) follows from the construction.
\smallno
Finally we want to show that Proposition \thm[GEO0] with $\d_1$
(\ie for $r=1$)
implies that the same result holds for $\d_r$ but
with a different constant $k$.
Choose then $v \in [0, |V|]$, let $s= 2 r$ and consider the mapping
$$
\Z^d \ni x = (x_1,\ldots, x_d) \mapsto \p(x) \equiv
( \ceil{x_1/s}, \ldots, \ceil{x_d/s} ) \in \Z^d
$$
Applying the Proposition \thm[GEO0] (with $r=1$)
to the set $\p V$, we get
that, for each $u \in [0, |\p V|]$,
$\p V$ is the disjoint union of two subsets $\p V= X'_u \cup Y'_{u}$
such that properties (a), (b) (with $r=1$) and (c) hold.
Let then
$$
w = \sup \{\, u \in [\,0, |\p V|\,] : \, | (\p^{-1} X'_u) \cap V | \le v \,\}
$$
We claim that, if we define
$$
X_v = (\p^{-1} X'_w) \cap V
$$
then (a), (b) and (c) are satisfied.
\smallno
By definition of $v$ we have $|X_v| \le v$.
Now, let $\bw = (w+1) \mmin |\p V|$ and let $\D = X'_\bw \setm X'_w$.
Using (a) we get
$$
|\D| = |X'_\bw| - |X'_w| \le \bw- ( w - k S ) \le (k+1) S
\Eq(GEO0.8)
$$
Moreover, by considering both cases $\bw = w+1$ and $\bw = |\p V|$,
it easy to verify that $| (\p^{-1} X'_\bw) \cap V | \ge v$.
Thus we obtain
$$
|X_v| = |(\p^{-1} X'_\bw) \cap V| - | (\p^{-1} \D) \cap V | \ge
v - s^d |D| \ge v - (k+1) s^d S
$$
which proves (a).
\smallno
To prove (b) all we need is to observe that
$$
\d_r(X_v, Y_v) \sset \d_r( \p^{-1} X'_w, \p^{-1} Y'_w)
\sset \p^{-1} \d_1( X'_w, Y'_w)
$$
which implies
$$
| \d_r(X_v, Y_v) | \le s^d k S
$$
The proof of (c) is straightforward.
\QED
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\beginsection A2. Appendix 2
\nproclaim Proposition [XX].
Take the transition rates as in \equ(GRAD).
Let $l\in \Zp$ and let $V$ be a multiple of $Q_l$, \ie
$$
\txt
V = \bigcup_{i=1}^s Q_l(x_i)
$$
with $x_i \in l \Z^d$. Assume that each $Q_l(x_i)$ for $i=1, \ldots, s$
is $\a-$regular for some $\a>0$. Then, if $l$ is larger than
some $\bar l(d,r,\a)$, we have
$$
\gap( L_V^{J,\t} ) \ge
8 |V|^{-\o} \,
\exp( -k' m l^d )
$$
where $\o = d \log4 / \log (3/2)$,
$k' = 9^{d-1} k$ and $k$ is the constant given in Theorem
\thf[F4].
\Pro\
We can assume that $V$ is $r-$connected, since, otherwise
one could just consider the $r-$connected components of $V$.
Since $V$ is a multiple of $Q_l$, this implies, if $l>r$, that
$V$ is actually connected (\ie $1-$connected).
Choose $l\in \Zp$ and let, for $n=0, 1, 2, \ldots$
$$
R_n =
\bigl(\, [ 0, a_{n+1}) \times
[ 0, a_{n+2}) \times \cdots \times [0, a_{n+d}) \, \bigr)
\cap \Z^d
$$
where $a_n = 6 l b^n$ and $b = (3/2)^{1/d}$.
Let $C^*_n$ be the set of all volumes $V \sset \Z^d$ such that
\item{(1)}
$V$ is a multiple of $Q_l$
\item{(2)}
$V\sset R_n$ modulo translations and permutations of the coordinates.
\smallno
Let also $\Th_r(V,l,\a)$ be the set of all interactions $J$ such that
each $Q_l(x_i) \sset V$ with $x_i \in l \Z^d$ is $\a-$regular.
Define
$$
g_n =
\inf_{\t\in \O} \,
\inf_{V \in C^*_n} \,
\inf_{J \in \Th_r(V,l,\a) }
\gap( L_V^{J,\t} )
$$
Thanks to the $\a-$regularity we know that
$$
\|J\|_x \le 100^{-1} m l \equiv J_0
\qquad
\forall x\in V
\Eq(XX.1)
$$
where, as usual, we have set $m = \hm(\a)$ (see Proposition \thf[MIX1]).
\smallno
We will show that for all $n \ge 1$ we have
$$
g_n \ge \ov4 g_{n-1}
\Eq(XX.2)
$$
which implies
$g_n \ge 4^{-n} g_0$.
Using Theorem \thf[F4] to estimate $g_0$ we get
$$
g_n \ge 4^{-n} \ov2 \exp( -k J_0 |R_0|^\dd )
\Eq(XX.4)
$$
Once we have \equ(XX.4) the proposition easily follows from
\equ(XX.1) and from the following observations:
\smallno
\item{(1)} $|R_0| \le ( 6 l b^d)^d = (9l)^d$
\item{(2)} Since $V$ is connected, if $V \in C^*_n \setm C^*_{n-1}$
then $|V| \ge a_n = 6 l (3/2)^{n/d}$, which implies
$$
4^{-n} \le (6l)^\o \, |V|^{-\o}
$$
\medno
So we are left with the proof of \equ(XX.2). For this purpose we want
to use \equ(F2.2) and Proposition \thf[GP6]. So we let
$$
\eqalign{
&
p_1 =
\sup \{ s \in l \Z : s \le a_n \} \cr
&
p_2 =
\inf \{ s \in l \Z : s \ge a_{n+d} - a_n \} \cr
}
$$
and
$$
A = \{ x = (x_1, \ldots, x_d) \in V : x_d < p_1 \}
\qquad
B = \{ x = (x_1, \ldots, x_d) \in V : x_d \ge p_2 \}
$$
Both $A$ and $B$ are in $C^*_{n-1}$,
so $\gap(L_A^{J,\t}) \ge g_{n-1}$ and the same holds
for $B$. Moreover, if we let $A_0 = A \cap \dep_l B$,
$B_0 = B \cap \dep_r A$, we get
$$
d(A_0, B_0) = p_1 -p_2 + 1 \ge 2a_n - a_{n+d} - 2l \ge l \, (3/2)^{n/d}
$$
One can then check that, thanks also to \equ(XX.1) and
Proposition \thf[MIX1], the
hypotheses of Proposition \thf[GP6] are satisfied, so that
the gap for the block dynamics on $\{A,B\}$ is at least $1/2$.
Combining this fact with formula \equ(F2.2) we get \equ(XX.2)
\QED
\fine
%@E
%-------------------------------------------------------------------------
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