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\centerline{\ttlfnt For 2-D lattice spin systems}
\centerline{\ttlfnt Weak Mixing Implies Strong Mixing}\vskip 1cm
\author{F. Martinelli\ddag , E.
Olivieri\dag and R.H. Schonmann\dag\dag}
\address{\ddag Dipartimento di Matematica,
III Universit\`a di Roma, Italy \hfill\break{\rm e-mail:
martin@mercurio.dm.unirm1.it}} \address{\dag Dipartimento di
Matematica, Universit\`a ``Tor
Vergata'' Roma, Italy \hfill\break{\rm e-mail:
olivieri@mat.utovrm.it}}
\address{\dag\dag Mathematics Department, UCLA. Los Angeles,
CA 90024, USA \hfill\break{\rm e-mail:
rhs@math.ucla.edu}}
\abstract{We prove that for
finite range discrete spin systems on the two dimensional lattice
$\Zt$, the (weak) mixing condition which follows, for instance, from the
Dobrushin-Shlosman uniqueness condition for the Gibbs
state implies a stronger mixing property of the Gibbs state,
similar to the Dobrushin-Shlosman complete analiticity condition,
but restricted to
all squares in the lattice, or, more generally, to all sets
multiple of a large enough square. The key observation
leading to the proof is that a
change in the boundary conditions cannot propagate either
in the bulk, because of the weak mixing condition, or along
the boundary because it is one dimensional. As a
consequence we obtain for ferromagnetic Ising-type systems
proofs that several
nice properties hold arbitrarily close to the critical
temperature; these properties include the existence of a convergent
cluster expansion and uniform boundedness of the logarithmic Sobolev
constant and rapid convergence to equilibrium of the associated
Glauber dynamics on nice subsets of $\Zt$, including the full lattice.}
\vskip 1cm
\noindent Work partially supported by grant SC1-CT91-0695
of the Commission of European Communities and by grant
DMS 91-00725 of the American NSF.
\pagina
\numsec=0\numfor=1
\tolerance=10000
\vskip 1cm
{\bf Section 0.}\par\noindent
\centerline{\bf Introduction.}
\bigskip
Let us consider a discrete, finite range lattice spin model in the
one phase region and let us analyze the problem of establishing
mixing properties of the corresponding Gibbs measure. As is well
known, a very powerful approach to the above question is to study the
{\it local specifications} of the Gibbs measure and to try to derive
the uniqueness and mixing properties (e.g. exponential clustering) of
the infinite volume state from suitable conditions on the local
specifications which express some sort of {\it weak} dependence on
the boundary conditions. We have in mind, in particular, the
Dobrushin [D] and Dobrushin-Shlosman [DS1], [DS2], [DS3]
conditions and we refer the interested reader to a recent paper by
two of the authors, [MO1], for a detailed critical review of these
conditions together with their implications both for the Gibbs
measure and for the associated Glauber-type dynamics.\par The above
conditions can be roughly divided into two classes that we call
{\it weak mixing} and {\it strong mixing}, respectively,
according to the following criterion
(see Section 1 for more details):
\medskip
\item{} A {\it weak mixing} condition implies that if in a finite
volume $\L$ we consider the Gibbs state with boundary condition $\t$,
then a local (e.g. in a single site $y\,\in \, \L^c$)
modification of the boundary condition $\t$
has an influence on the
corresponding Gibbs measure which decays exponentially fast inside
$\L$ with the {\it distance from the boundary} $\partial\L$.
\medskip
\item{} A {\it strong mixing} condition
implies, in the same setting as above, that the influence of the
perturbation decays in $\L$ exponentially fast with
the distance from the {\it support of the perturbation} (e.g. the
site $y$). \medskip
This distinction is very important since,
even if we are in the one phase region with a unique infinite
volume Gibbs state with exponentially decaying truncated
correlation functions, it may
happen that, if we consider the same Gibbs state in a finite volume
$\L$, a local perturbation of the boundary condition radically
modifies the Gibbs measure close to the boundary while leaving it
essentially unchanged in the bulk and this ``long range order
effect'' at the boundary persists even when $\L$ becomes arbitrarily
large. We will refer to this phenomenon as a
``boundary phase transition''. It is clear that if a ``boundary phase
transition'' takes place, then our Gibbs measure may satisfy a weak
mixing condition but not a strong one.\par A ``boundary phase
transition'' is apparently not such an exotic phenomenon since,
besides being proved for the so called Czech models [Sh]
(in dimension 3 and higher), it is also
expected to take place for the 3D ferromagnetic Ising model at low
temperatures and small enough magnetic field (depending on the temperature)
[DS4].\par
It is however very reasonable to conjecture that, for finite
range interactions, no ``boundary phase transition'' can take place in
2D for regular enough regions, e.g. squares, for
which the boundary is ``one dimensional''! In the present paper we prove this
conjecture (see Theorem 1.1). It is important to stress that the restriction
of the strong mixing condition to squares and other ``nice'' regions is not
a matter of our inability to extend further the result to arbitrary
regions. It is actually
known that, even in two dimensions and even for ferromagnetic systems,
under the same hypothesis of Theorem 1.1,
the strong mixing condition may not extend to arbitrarily shaped regions.
Some simple counter-examples are presented in [MO1]. From the point of
view of the ``physics'' involved in the problem, it is not surprising
that one has to restrict the shape of the regions so that their
boundary be basically ``one-dimensional''. \par
%This is exactly the goal of this paper and it is achieved by
%showing that in 2D a {\it weak} mixing condition on all finite
%subsets of the lattice $\Z$ implies a {\it strong} mixing
%condition on at least all squares (see Theorem 1.1 in the next
%section).\par
The main result discussed above is relevant because:\medskip
\item{a)} A strong mixing condition on e.g. all squares implies
some nice consequences for the Gibbs measure
(see Section 3) like analyticity, the
existence of a convergent cluster expansion (see [O] and [OP]),
good behaviour under a decimation
transformation (see [MO2] and [MO3]), existence of a
finite logarithmic Sobolev constant (see [MO2] and [LY]), rapid
convergence to equilibrium of the associated Glauber dynamics (see
[MO1],[MO2]). \medskip
\item{b)} There are several cases in which the weak mixing condition
can be proven to hold. For instance, this is the case if the
Dobrushin-Shlosman uniqueness condition [DS1] is verified.
In Section 3 of this
paper we will show that for finite range Ising-like ferromagnets
weak mixing indeed holds in most of the one-phase region. In particular
we can show that it holds in the absence of an external field for
all temperatures which are larger than the critical one. As a
consequence, we obtain the results quoted in (a) arbitrarily
close to the critical temperature. For these models we believe
that weak mixing (and hence strong mixing, if the dimension is 2)
actually holds for every value of the temperature and external
field, except (of course) on the transition line and the critical
point. \medskip
The paper is organized as follows: in Section 1 we define the
models, the various mixing conditions and state the main result
(Theorem 1.1); in Section 2 we prove Theorem 1.1 and, finally, in
Section 3 we discuss some consequences of the main
theorem.\vskip 1cm
\numsec=1\numfor=1
\vskip 1cm
{\bf Section 1.}\par\noindent
\centerline{\bf General Definitions, Notation and Main Result.}
\bigskip
\noindent $\S \; 1$ The Model. \par We will consider
lattice spin systems in $d$ dimensions (later to be restricted to $d=2$)
with configuration space of a
single spin given by the finite set $S\, =\, \{1,\dots, N\},\ \
N\in{\bf N}$.
On the lattice $\Z$ we will use the norm
$$\vert \vert x\vert \vert\;=\;\sup_{i=1,...,d}\vert x_i\vert,$$
and measure distances accordingly.
The configuration space on a subset $\Lambda \subset
\Z$, namely $S^\Lambda $, will be denoted by $\Omega_\Lambda $
and a generic element of $\O_\L$ by $\sigma_\Lambda $. If
$\L$ coincides with the whole lattice $\Z$, then we simply write $\O$
and $\s$. By $\sigma_x \equiv\sigma(x)$ we denote the value of
the spin at the site $x\in\Lambda $ in the configuration
$\sigma$. Also, when $\Delta \subset \Lambda$, we denote by
$\sigma_{\Delta}$ the restriction of $\sigma_{\Lambda}$ to $\Delta$
(note that this is consistent with the previous use of the notation
$\sigma_{\Delta}$). The number $|X|$ denotes the cardinality of
$X\subset \subset \Z$ (we write $X \subset \subset \Z$ iff
$X $ is a {\it finite} subset of $ \Z$). \medskip Next we define the
potential or interaction $U$ as :
$$U \,=\, \{U_{X},\ X \subset \subset \Z\},$$
where, for every finite $X$,
$$U_{X}\ :\ \Omega \to \ {\bf R}$$
satisfies the condition $U_X(\sigma) = U_X(\eta)$ in case
$\sigma_x = \eta_x$ for all $x \in X$.
On the potential U we will always assume the following hypotheses:
\bigskip\indent
{\bf H1}.\ \ {\it Finite range:} $\exists r < \infty :
U_{X}\equiv 0$ if $\hbox{diam}(X) > r$.
\bigskip\indent
{\bf H2}.\ \ {\it Translation invariance:}
$\forall X \subset \subset \Z \quad\forall k\in \Z$ :
%\quad
%U_{X+k}(\sigma) \,=\, U_{X}(\eta) \ $ if $\ \sigma(y+k) = \eta(y)$ for
%all $y \in X$.}
$$U_{X+k}(\sigma) \,=\, U_{X}(\eta) \ \hbox{ if } \ \sigma(y+k) = \eta(y)
\hbox{ for all } y \in X.$$
\medskip
Given $\Lambda \subset \Z$ and
$\tau\in\Omega_{\Lambda ^c}
\;$ (where $\Lambda ^c \,=\, \Z\setminus\Lambda$), for every
$\sigma\in\Omega$
we denote by $(\sigma\tau)_{\Lambda}$ the configuration obtained from
$\sigma$ by changing it to $\tau$ outside $\Lambda $:
$$\eqalign{(\sigma \tau)_{\Lambda}(x) \,&=\,
\sigma(x)\quad \forall \; x\in\Lambda,
\cr (\sigma \tau)_{\Lambda}(x) \,&=\,
\tau(x)\quad \forall \; x\in\Lambda ^c.} \Eq(1.1)$$
Given a set $\Lambda \subset \subset \Z$, a
{\it boundary
condition}, (b.c.), is a configuration
$\tau\in\Omega_{\Lambda ^c} $
and the {\it energy} associated to a
configuration $\sigma\in\Omega_\Lambda $ when
the boundary condition outside $\Lambda $ is $\tau\in\Omega_
{\Lambda ^c}$ is given by :
$$H^\tau_\Lambda (\sigma) \,=\, H_\Lambda (\sigma | \tau) \,=\,
\sum
_{X:X\cap\Lambda \ne\emptyset}U_{X}((\sigma\tau)_{\Lambda}). \Eq(1.2)$$
Because of the hypothesis H1,
$ H^{\tau}_{\Lambda}(\sigma)$ depends
only on $\tau_x$ for $x$ in
%$\partial^+_r\Lambda$ given by
$$\partial^+_r
\Lambda \,=\, \{x\not\in\Lambda : \hbox{dist} (x, \Lambda )\leq
r\}. \Eq(1.3)$$
Finally the {\it Gibbs measure} in
$\Lambda $ with b.c. $\tau\in\Omega_{\Lambda ^c}$ at inverse
temperature $\beta > 0$ is
the probability measure $\mu_\Lambda ^\tau$ on $\Omega_{\Lambda}$
given by
$$\mu_\Lambda ^\tau(\sigma) \,=\, {\exp(-\beta H^\tau_\Lambda
(\sigma))\over Z^\tau_\Lambda }, \Eq(1.4)$$
where the normalization
factor, called {\it partition function}, is given by
$$Z^\tau_\Lambda \,=\,\quad\sum_{\sigma\in\Omega_\Lambda }
\exp(-\beta H^\tau_\Lambda (\sigma)). \Eq(1.5)$$
If there exists a unique limiting
Gibbs measure for $\Lambda \to
\Z,$ independent of $\tau$, it will be denoted by $\mu$ .\bigskip
{\bf Remark.} In the next
sections it will be very convenient to consider
the Gibbs measure $\mu_\Lambda ^\tau(\sigma)$ directly as a measure
on the infinite volume configuration space $\Omega$ such that on
$\L^c$ we have a.s. $\s\;=\; \t$. Moreover we will frequently
write, for $\t$ and $\s$
in
$\O$, $\mu_\Lambda ^\tau(\sigma)$ instead of the more precise
but cumbersome $\mu_\Lambda ^{\tau_{\L^c}}(\sigma_\L)$.
\bigskip
\noindent $\S \; 2$.
Definition of Weak and Strong Mixing for the Gibbs
Measure.\medskip We first recall that the {\it variation
distance} between two probability measures $\mu_1, \mu_2$ on a
finite set $Y$ is defined as :
$$Var (\mu_1, \mu_2) \,=\, {1\over 2} \sum_{y\in Y}\vert \mu_1(y) -
\mu_2(y)
\vert \,=\,\sup_{X \subset Y}\vert \mu_1(X) -
\mu_2(X)\vert. \Eq(1.6)$$
More generally, given a metric $\varrho(\cdot, \cdot)$ on a finite
space $Y$
(a much more general framework can also be considered) the
{\it Kantorovich - Rubinstein - Ornstein - Vasserstein distance with
respect to $\varrho$} between two probability measures $\mu_1,
\mu_2$ on $Y,$
%that we denote by $ \hbox{KROV}_\varrho(\mu_1, \mu_2)$,
is defined as
$$\hbox{KROV}_\varrho(\mu_1, \mu_2) \,=\,
\inf_{\mu\in
K(\mu_1, \mu_2)}\sum_{y, y'\in Y} \varrho(y, y')\mu(y , y'), \Eq(1.
7)$$ \noindent
where $K(\mu_1, \mu_2)$ is the set of joint representations of
$\mu_1 ,\mu_2$, namely the set of measures on the cartesian product
$Y \times Y$ whose marginals are,
respectively, given by $\mu_1 , \mu_2$.
%Namely we have,
i.e., $\mu \in K(\mu_1,\mu_2)$ if
$\forall B \subset Y$
$$\eqalign{
&\mu(B\times Y) \,=\, \sum_{y\in B,\, y'\in Y}\mu(y , y')\,=\, \mu_1(B),
\cr &\mu(Y \times B) \,=\, \sum_{y\in Y,\,y'\in B} \mu(y ,
y') \,=\, \mu_2(B).}$$ \noindent
It is well known that for the particular case
$$\varrho(y , y') \,=\, 1\quad \hbox{iff }y\ne y'\;\hbox{ and }
0\; \hbox{ otherwise}, \Eq(1.8)$$
\noindent
$\hbox{KROV}_\varrho(\cdot,\cdot)$ coincides with the variation
distance $ Var(\cdot,\cdot)$.\bigskip
Given a measure $\mu_\Lambda $ on $\Omega_\Lambda $
we call
{\it relativization} of $\mu_\Lambda $ to $\Omega_\Delta$, with
$\Delta \subset\Lambda $, the measure $\mu_{\Lambda , \Delta}$
on $\Omega_\Delta$ given by
$$\mu_{\Lambda , \Delta}(\sigma_\Delta) \,=\,
%\sum_{\sigma_ {\Lambda \setminus\Delta}}\mu_\Lambda
%((\sigma_{\Lambda \setminus \Delta} \sigma_\Delta)_{\Delta}).
\sum_{\sigma_ {\Lambda \setminus\Delta}}\mu_\Lambda
(( \sigma_\Delta \sigma_{\Lambda \setminus \Delta} )_{\Delta}).
\Eq(1.9)$$
\noindent \bigskip
We are now in a position to define strong
mixing and weak mixing.\par
We say that the Gibbs measures $\mu_\Lambda^{\cdot} $ on
$\Omega_\Lambda $ satisfy the {\it strong mixing} condition with
constants $C, \gamma$ if for every subset $\Delta \subset \Lambda $
and every site $y \in \Lambda^{c}$
$$ \sup_{\tau,\tau^{\prime} \in \Omega _{\Lambda^c} :
\tau =_{y} \tau^{\prime}}
Var(\mu_{\Lambda , \Delta}^\tau\ , \ \mu_{\Lambda ,
\Delta}^{\tau^{\prime}})\leq C e^{-\gamma \hbox{dist}(\Delta, y)},
\Eq(1.10)$$
where\quad $\tau =_{y} \tau^{\prime}$ means that
$\tau_x = \tau^{\prime}_x$ for all $ x\ne y$.
We denote this condition by
$ SM(\Lambda, C,\gamma)$.\medskip
We say that the Gibbs measures $\mu^{\cdot}_\Lambda $
satisfy the {\it weak
mixing } condition with constants $C , \gamma$ if, for every subset
$\Delta \subset \Lambda $,
$$\sup_{\tau, \tau'\in\Omega_{\Lambda ^c}}
\quad Var(\mu^\tau_{\Lambda , \Delta},
\mu^{\tau'}_{\Lambda , \Delta})\leq
C\sum_{x\in\Delta ,\,y\in
\partial^+_r\Lambda }\hbox{exp}(-\gamma\vert \vert x-y\vert \vert).
\Eq(1.11)$$
We denote this condition by $WM(\Lambda , C, \gamma)$.\par
Finally we say that $WM_\varrho(\Lambda , C,\gamma)$ holds if
$\forall\Delta \subset \Lambda $
$$\sup_{\tau, \tau'\in\Omega_{\Lambda ^c}}\ \
\hbox{KROV}_{\varrho}(\mu^\tau_{\Lambda ,\Delta},
\mu^{\tau'}_ {\Lambda , \Delta})\leq
C\sum_{x\in\Delta ,\,y\in
\partial^+_r\Lambda }\hbox{exp}(-\gamma \vert \vert x-y
\vert \vert).\Eq(1.11')$$
%C\sum_{x\in\partial^+_r\Lambda }\ \ \exp(-\gamma
%\hbox{dist}(x, \Delta))\Eq(1.11')$$
\bigskip
\noindent $\S \; 3$ The Main Result.\bigskip
The cube (we will also call it ``square'' if $d=2$)
of side $2L+1$ centered at the origin is the set
$$\L_L\;=\;\{\; x\in \Z \, ;\;\vert\vert
x\vert\vert\,\leq \, L\;\}.$$
%where $\vert\vert
%x\vert\vert\;=\;\sup_{i=1,...,d}\vert x_i\vert$.
Our main result reads as follows:
\bigskip {\bf Theorem 1.1} \par
{\it In 2 dimensions, if there
exist positive constants $C$ and $\g$ such that the Gibbs measures
$\mu_\L^{\cdot}$ satisfy the weak mixing condition $WM(\Lambda , C,
\gamma)$ for all $\L\subset \subset \Zt$, then there exist positive
constants $C'$ and $\g '$ such that they also
satisfy the strong mixing condition $SM(\Lambda_L ,
C', \gamma ')$ for every square $\L_L$.}\bigskip
{\bf Remark.} Actually, as we will see in the next section, the
conclusion of the above theorem remains true even if we assume the
weak mixing {\it not} for {\it all} finite subsets of $\Z$ but {\it
only} for all subsets of a square $\L_{L_o}$ provided that
$L_o$ is large enough (depending of course on the constants $C$ and
$\gamma$ and on the range of the interaction). Moreover it is
possible to show (see [MO1]) that, once $SM(\Lambda_L ,
C', \gamma ')$ holds for every square $\L_L$, then it also holds for
all sets ``multiple'' (i.e., union of disjoint translates) of a given,
large enough, square. \par
%It is however very important to notice that it is known, by means
%of explicit examples, that the above result becomes false if we
%try to replace in the above theorem {\it for all squares} with {\it
%for all finite subsets of $\Z$ } (see again [MO1]).
\vskip 1cm
\numsec=2\numfor=1
\vskip 1cm {\bf Section 2.}\par\noindent
\centerline{\bf Proof of Theorem 1.1} \bigskip
Theorem 1.1 actually follows from an apparently weaker result that
we state as a proposition. First we need to introduce a technical
definition; given $\Lambda \subset \subset \Z$, and $\tau, \tau^{\prime}
\in \Omega_{\Lambda}^c$ set
$$Q_{\Lambda, \t , \t '}\;=\;\{\; x\in \L \, ;\;\norm{x-y}\,\geq \,
L^{1/2}\;\forall \, y \hbox{ such that } \t (y)\,\neq \t ' (y)\;\}.$$
\bigskip
{\bf Proposition 2.1}\par
{\it In 2 dimensions, if there exist positive constants $C$ and
$\g$ such that the Gibbs measures $\mu_\L^{\cdot}$ satisfy
the weak mixing condition $WM(\Lambda , C, \gamma)$ for every
$\L \subset \subset \Zt$, then there exist positive constants $C_o$
and $\g_o$ such that for every square $\L_L$ and every site
$y \in \L_L^c$
$$ \sup_{\tau , \t^{\prime} \in \Omega _{\Lambda_L^c} :
\tau =_y \tau^{\prime}}
Var(\mu_{\Lambda_L , Q_{\L_L,\t,\t'} }^\tau\ , \
\mu_{\Lambda_L , Q_{\L_L,\t,\t'} }^{\t'}) \ \
%\mu_{\Lambda_L , Q}^{\tau^{(y)}}
\leq C_o e^{-\gamma_o L^{1/4}}.$$}
% where
%$$Q\;=\;\{\; x\in \L \, ;\;\norm{x-y}\,\geq \, L^{1/2}\;\}
\bigskip
From the above proposition
and from the triangle-inequality property of the variation
distance, it follows immediately that,
if $\t$ and $\t '$ are two arbitrary boundary
conditions outside the square $\L_L$,
then we have
$$Var(\mu_{\Lambda_L , Q_{\L_L,\t , \t '}}^\tau \ , \
\mu_{\Lambda_L , Q_{\L_L, \t ,\t '}}^{\tau '})\leq C_o4 r(L+2r)
e^{-\gamma_o L^{1/4}}, \Eq(2.1)$$ where $r$ is the range of the
interaction. We can use at this point the following result which was
proved in Section 4 of [MO1].
We quote the result in our special 2D setting.\bigskip
{\bf Theorem 2.1}\par
{\it In 2 dimensions, there exists a constant $\bar L$ such that if
$$ \sup_{\tau , \t^{\prime} \in \Omega _{\Lambda_{L_o}^c}}
Var(\mu_{\Lambda_{L_o} , Q_{\L_{L_o},\t , \t '}}^\tau \ , \
\mu_{\Lambda_{L_o} , Q_{\L_{L_o},\t ,\t '}}^{\tau '})\leq \,L_o^{-d-2}
\Eq(2.1bis)$$ for some $L_o\,>\,\bar L$, then there exist
positive constants
$C$ and $\gamma$ such that the strong mixing condition
$SM(\Lambda_L , C, \gamma)$
%the strong mixing condition (1.10)
holds for all squares $\L_L$, $L\,\geq \,0$.}\bigskip
From \equ(2.1) (where $L$ is arbitrary)
and the above theorem we immediately get
Theorem 1.1 .\bigskip {\bf Remark.} Actually in order to be able to
use the results of [MO1] we only need to prove the result of the
proposition for $L$ so large that the r.h.s. of \equ(2.1) is smaller
than $K\over L^{4}$ where $K$ is a suitable constant. As one can
easily check in the proof given below, this requires as input the
weak mixing condition only for all subsets of a large enough
square.\bigskip Thus we are left with the proof of Proposition 2.1.
Suppose that $L$, $y \in \L_L^c$, $\t$ and $\t'$ are fixed and such that
$\t =_y \t'$, and abbreviate $Q=Q_{\L_L,\t,\t'} =
\{x\in \L \, ;\;\norm{x-y}\,\geq \, L^{1/2}\;\}.$
The key idea is to show that it is possible
to construct, by succesive ``improvements'' over an increasing sequence
of length scales, a coupling,
$$\nu\; \in \; K(\mu_{\L_L , Q}^\t\,,\, \mu_{\L_L , Q}^{\t'}),$$
of $\mu_{\L_L , Q}^\t$ and $\mu_{\L_L , Q}^{\t'}$ such that
% $$\sup_{\tau,\tau^{(y)} \in \Omega _{\Lambda_L^c}}
$$\nu (\s \, \neq \, \eta)\;\leq \,C_o e^{-\gamma_o
L^{1/4}},\Eq(2.2)$$ for suitable positive
constants $C_o$ and $\g_o$, which do not depend on $L$, $y$, $\t$
and $\tau'$. Clearly
\equ(2.2) proves the proposition, because of the remark concerning
(1.8). \par
Each ``improvement'' will be
carried out via the so called ``surgery'' technique (used for instance
in [DS1])
that we will now recall (here it is very convenient to adopt the
convention pointed out in the remark which followed (1.5)).\medskip The
ingredients of a surgery are:\bigskip \item{i)} A finite set $\L\,
\subset \subset \Z$
%(or ${\bf Z^d}$ in general)
and a subset $\Gamma
\subset \L$. \item{ii)} A coupling $\nu\; \in \; K(\mu_{\L
}^{\t_1}\,,\,\mu_{\L , }^{\t_2})$ for some $\t_1$ and $\t_2$.
\item{iii)} A transition kernel $T(\eta ,\xi\,;\,\bar \eta ,\bar \xi
)$ (where $\eta ,\xi ,\bar \eta ,\bar \xi$ run over $\O$) which has
to satisfy the following property:
%\itemitem{a)} For each pair $\eta \, ,\, \xi$ : $T(\eta ,\xi\,;\,
For each pair $\eta \, ,\, \xi$ $$T(\eta ,\xi\,;\,
,\ )\;\in \;K(\mu_{\G}^{\eta}\,,\,\mu_{\G}^{\xi}).$$
%\itemitem{b)} $T(\eta ,\xi\,;\,\bar \eta ,\bar \xi )\;=\;0 \hbox{
In particular, $T(\eta ,\xi\,;\,\bar \eta ,\bar \xi )\;=\;0 \hbox{
if } \eta_{\G^{c}}\,\neq \,\bar \eta_{\G^{c}} \hbox{ or }
\xi_{\G^{c}}\,\neq \,\bar \xi_{\G^{c}}.$\bigskip
\noindent
The result of a surgery is a new coupling $\bar \nu \, \in \,
K(\mu_{\L }^{\t_1}\,,\,\mu_{\L }^{\t_2})$ defined by:
$$\bar \nu (\bar \eta ,\bar \xi)\;\equiv \;
\sum_{ \eta , \xi}\nu (\eta ,\xi )
T(\eta ,\xi\,;\,\bar \eta ,\bar \xi ).\Eq(2.3)$$
One can check that the measure
$\bar \nu$ defined above is indeed a new coupling
of $\mu_{\L }^{\t_1}$ and $\mu_{\L }^{\t_2}$
with the following simple type of computation:
$$\eqalign{\sum_{\bar \xi}\bar \nu (\bar \eta ,\bar \xi )\;&=\;
\sum_{ \eta , \xi}\nu (\eta ,\xi )\sum_{\bar \xi}
T(\eta ,\xi\,;\,\bar \eta ,\bar \xi )\,=\,
\cr \sum_{ \eta , \xi}\nu (\eta ,\xi )\mu_\G^\eta(\bar \eta )\;&=\;
\sum_{ \eta }\mu_\L^{\t_1}(\eta )\mu_\G^\eta(\bar \eta )\;=\;
\mu_\L^{\t_1}(\bar \eta ),}\Eq(2.4)$$
where the second equality follows from property (iii) above, while the
third and fourth equalities follow from the definition of coupling
and the DLR equations, respectively.\par
%In a similar way it also follows that if $A$ is an event in
%$\O_{\G^c}\times \O_{\G^c}$ then $\bar \nu (A)\;=\;\nu (A)$. \par
The simplest example is the so called product surgery
defined by:
$$T(\eta ,\xi\,;\,\bar \eta ,\bar \xi )\;=\;\mu_\G^\eta(\bar \eta )
\mu_\G^\xi(\bar \xi ).\Eq(2.5)$$
Let us now go back to the proof of Proposition 2.1.
%Given the
%constants $C$ and $\gamma$,
Recall that $y \in \L_L^c$ is fixed.
Given an integer $l_o$ to be chosen
conveniently later on {\it independently} of $L$ (in (a) below),
we define for each integer
$i=1,2,\dots,[{L^{1/2}\over L^{1/4}}]$,
$$\eqalign{
\G_i\;&\equiv \;\{x\in \L_L\,;\;\norm{x-y}\,>\,(i-1)L^{1/4}\,\}, \cr
%\Eq(2.7)
D_i\;&\equiv \;\{ x\in \L_L\,;\;iL^{1/4}-r\,<\,\norm{x-y}\,
\leq \,iL^{1/4}\,\}, \cr
A_i\;&\equiv \;\{x \in D_i\, ;\; \min_{z\notin \L_L}
\norm{x-z}\,\leq\,l_o\,\}, \cr
B_i\;&\equiv \;D_i\setminus A_i.} $$%\Eq(2.6)$$
%{\bf Remark.} The following minor nuisance, unfortunately, will
%appear in the argument below. For some value of the index $i
%\,\leq \, L^{1/4}$ it may happen that the set $A_{i}$ touches
%only one face of the square $\L_L$ and that the set $B_i$ is at a
%distance (measured in the norm $\norm{\,}$) from one of the other faces
%less than $L^{1/4}$. As one can easily convince himself, if $L$ is
%large enough there can be at most one such value of $i$ and it
%will be denoted by $i_o$. This particular value of $i$ (when it exists)
%will have to be treated differently from the others below. \bigskip
It is easily seen that for large $L$ there can be at most one value of
$i$ for which the set $D_i$ is at a distance less than $L^{1/4}/2$ from
one of the four vertices of the square $\L_L$. If such a value of $i$
exists, we will denote it by $i_o$, and below we will have to
treat it differently than the other values of $i$.
\par The following two geometric facts can be easily checked:
%\item{b)} If $L$ is large enough, then for each $i \neq i_o$,
%$$ \vert A_i \vert = 2 l_o r. $$
\item{a)} Let $C$ and $\gamma$ be the positive constants which
appear in the statement of the proposition. We can choose $l_o$ so
large (depending only on $C$, $\gamma$ and $r$) that for all
large $L$ and each $i \neq i_o$,
$$ C\sum_{u\in B_i ,\,v\in \partial^+_r\Gamma_i}
\hbox{exp}(-\gamma\vert \vert u-v\vert \vert) \ \leq \ 1/2.$$
\item{b)} If $L$ is large enough, then for each $i \neq i_o$,
$$ \vert A_i \vert = 2 l_o r. $$
\medskip \par
The key idea in the proof is to iterate
the procedure described below for $i=1,2,\dots,[{L^{1/2}\over
L^{1/4}}]$.
We describe now the $i-th$ step of the iteration.
Suppose that from steps $1,\dots,i-1$ we have been
able to costruct a joint representation of $\mu_{\L_L}^{\t}$ ,
$\mu_{\L_L}^{\t'}$ that we denote by $\nu_{i-1}$. At the i-th step
we obtain a new coupling $\nu_i\,\in \, K(\mu_{\L_L}^{\t_1} ,
\mu_{\L_L}^{\t_2})$ as follows: \par \medskip
\noindent I) If $i\,=\,i_o$ we do nothing,
namely we set
$$\nu_{i_o}\;=\;\nu_{i_o-1}.$$ \medskip
\noindent II) If, on the contrary, $i\,\neq \,i_o$, then we modify
$\nu_{i-1}$ via the following surgery applied to
$\nu_{i-1}$ on the set $\G_i$.
%$$\G_i\;\equiv \;\{x\in \L_L\,;\;\norm{x-y}\,>\,(i-1)L^{1/4}\,\}\Eq(2.7)$$
The transition kernel of the surgery, $T_i$, is defined as follows:
%\item{\quad II.i)} If $\eta_{D_{i-1}}\,=\,\xi_{D_{i-1}}$, then
II.i) If $\eta_{D_{i-1}}\,=\,\xi_{D_{i-1}}$, then
keep in mind that the range of the interaction is $r$ and define
$$\eqalign{T_i(\eta ,\xi\,;\,\bar \eta , \bar \xi
)\,&=\,\mu_{\G_i}^\eta(\bar \eta ) \;\;
\hbox{ if }\; \;\bar\eta_{\G_{i}}\,=\,\bar\xi_{\G_{i}}\;\hbox{and }
\bar\xi_{\G^c_{i}}\,=\,\xi_{\G^c_{i}}\,,\,\bar\eta_{\G^c_{i}}\,=\,
\eta_{\G^c_{i}},\cr
T_i(\eta ,\xi\,;\,\bar \eta , \bar \xi )\,&=\,0 \quad\hbox{
otherwise}.}\Eq(2.8)$$
%\item{\quad II.ii)}
II.ii) If $\eta_{D_{i-1}}\,\neq \,\xi_{D_{i-1}}$, then
we observe that, since $i\neq i_o$, we can use
(a) above and the weak mixing property on the
set $\G_i$ to deduce that there exists $\tilde \nu_i\;\in \;
K(\mu_{\G_i}^{\eta} , \mu_{\G_i}^{\xi})$ such that :
$$\tilde \nu_i((\tilde \eta , \tilde \xi) \,;\;\tilde \eta_{B_i} \,=\,
\tilde \xi_{B_i})\;\geq\; 1/2, \Eq(2.9)$$
for all $L$ large enough
(independently of $i$) provided that $l_o$ was chosen properly.
%is taken large enough depending only on $r$, $C$ and $\gamma$.
In this case we define
$T_i(\eta ,\xi\,;\,\bar \eta , \bar \xi)$ as :
$$T_i(\eta ,\xi\,;\,\bar \eta , \bar \xi )\;=\;
\sum_{\tilde \eta , \tilde \xi }\tilde \nu_i(\tilde \eta , \tilde
\xi )\mu_{A_i}^{\tilde \eta}(\bar \eta )\mu_{A_i}^{\tilde
\xi}(\bar \xi ).\Eq(2.10)$$
(In \equ(2.10) we are taking the composition of two surgeries,
the second of which is the product surgery on $A_i$.)
\medskip \par \noindent
It is easy to check that indeed $T_i(\eta ,\xi\,;\,\bar \eta , \bar
\xi )$, defined by (II.i) and (II.ii),
has the right properties to be the kernel of a surgery; moreover,
thanks to \equ(2.8), \equ(2.9), \equ(2.10) and (b) above,
the following crucial estimate holds:
$$\inf_{\eta ,\xi }\sum_{\bar \eta ,\bar \xi \,
;\;\bar \eta_{D_i}\,=\,\bar \xi_{D_i}} T_i(\eta ,\xi\,;\,\bar \eta ,
\bar \xi )\;\geq \;{1\over 2}a^{2\vert A_i\vert}\, = \,{a^{4rl_o}\over
2}\,\equiv\, \d \;>\;0,\Eq(2.11)$$
where the constant $a>0$ depends only on the range and the norm of the
interaction $U$.\bigskip
{\bf Remark.} Notice that the constant $\d$ is
independent of the side $L$ and of $i$ just because we are working in two
dimensions; it is only in two dimensions in fact that the
cardinality of the sets $A_i$, can be bounded above
uniformly in $L$ and $i$ (by $2rl_o$)!
%$\vert A_i\vert \,=\,rl_o$ is
\bigskip Finally we set, in case (II), according to
\equ(2.3), $$\nu_i (\bar \eta ,\bar \xi)\;\equiv \; \sum_{ \eta ,
\xi}\nu_{i-1} (\eta ,\xi ) T_i(\eta ,\xi\,;\,\bar \eta ,\bar \xi
).\Eq(2.12)$$ Let now $i_Q$ be the largest $i$ such that $Q\,\subset
\, \G_i$.
% and let $$\nu\;=\;\nu_{i_Q}\Eq(2.12bis)$$
Then $\nu_{i_Q}\,\in
\,K(\mu_{\L_L }^{\t}\,,\,\mu_{\L_L , }^{\t'})$ and
$$\nu_{i_Q}(\,(\eta \,,\xi)\,;\;\eta_Q\,\neq \,\xi_Q\,)\,\leq\,
\prod_{i=2,\dots,i_Q\,,\, i\neq i_o}\nu_i(\,(\eta \,,\xi)\,;\;\eta_{D_i}\,\neq
\,\xi_{D_i}\,)\;\leq\;(1\,-\,\d )^{i_Q-3}.\Eq(2.13)$$
Clearly \equ(2.13) implies \equ(2.2) and hence proves Proposition 2.1
and therefore also Theorem 1.1 .\eop \bigskip
{\bf Remark.} We notice that in the above proof we used the
weak mixing condition only for the sets $\G_i$ %defined in \equ(2.7)
which, in turn, can have only two particular geometric shapes.
Actually we can weaken our hypotheses even further if we use the
results of Appendix 2 in [MO1]. There it was shown that, in order
to prove strong mixing, it is sufficient to establish, in a
large enough square (or hypercube in d-dimensions), a bound like
\equ(2.1) on the truncated correlation functions of two
local observables $f$ and $g$ with support of diameter equal to
the range $r$ of the interaction and located on two opposite
faces of the square. Using this geometry it is easy to show that one
could modify slightly the geometric construction used in the proof,
in order to require the weak mixing condition {\it only} on rectangles.
\pagina
\numsec=3\numfor=1
{\bf Section 3.}\par\noindent
\centerline{\bf Some Consequences and Applications}
\bigskip
In this final section we first prove the equivalence, in 2D,
of the validity of
weak mixing with several other statements that involve either the
Gibbs states or a Glauber dynamics reversible with respect
to them; afterwards we check that weak mixing actually holds
true in most part of the one phase region of Ising-like
ferromagnets. In order to state our result we first have to
give some definitions. \medskip A Glauber dynamics is defined by
means of its {\it generator} $L$ which is formally given by:
$$L f
(\sigma) \,=\, \sum_{x,a} c_x (\sigma, a) \bigl ( f
(\sigma^{x,a})-f (\sigma) \bigr ).\Eq(3.1)$$ where $\sigma^{x,a}$ is
the configuration obtained from $\sigma$ by setting the spin at $x$
equal to the value $a\,\in \, S$ and the non-negative quantities
$c_x (\sigma, a) $ are called ``jump rates''.\par We will also
consider the dynamics associated to the above described
jump rates
in a {\it finite volume} $\Lambda$ with boundary conditions
$\tau$
outside $\Lambda.$ By this we mean the dynamics on
$\Omega_\Lambda$
generated by $L^\tau_\Lambda$, defined by
$$L_{\L}^\t f
(\sigma) \,=\, \sum_{x \in \L,a} c_x^{\t,\L} (\sigma, a) \bigl ( f
(\sigma^{x,a})-f (\sigma) \bigr ),$$ with
$$c^{\tau, \Lambda}_x (\sigma, a) \equiv c_x ((\sigma \tau)_\L, a).$$
%where, given $\tau \in \Omega_{\Lambda ^c}\hbox{ and } \; \sigma \in
%\Omega_\Lambda \; ,\,\sigma \tau$
%is the configuration which agrees with $\s$ in $\L$ and with $\t$
%outside $\L$. \par
The general hypotheses on the jump rates, that we shall always
assume, are the following ones.
\item{{\bf H1.}}\ {\it Finite range $r$}. This means that
there exists a finite $r$ such that for each $x \in \Z$,
if the configurations $\sigma$ and $\eta$ satisfy
$\eta (y) \,=\, \sigma (y) \
\hbox{ whenever } \ ||y - x|| \leq r $, then for every $a \in S$
$$c_x (\sigma, a) \,=\, c_x (\eta , a).$$
\item{{\bf H2.}}\ {\it Translation invariance}. That is, if for some
$x \in \Z$ we have for every $y \in \Z$ that $\eta (y) \,=\,
\sigma (y+x)$, then we also have for every $y \in \Z$ and every $a \in S$
$$\;c_{y+x} (\sigma, a) \,=\, c_x (\eta, a).$$
\item{{\bf H3.}}\ {\it Positivity and boundedness}. There exist two
positive constants $k_1$, $K_2$ such that
$$0\,<\, k_1\,\leq \, \inf_{\sigma, x, a} \ c_x (\sigma, a )
\,\leq\,\sup_{\sigma, x, a} \ c_x (\sigma, a )\,\leq \,k_2. $$
\item{{\bf H4.}}\ {\it Reversibility with respect to
the Gibbs measures}. For each $x \in \Z$ and $a \in S$
%(in finite or infinite volume):
%$$\hbox{exp}(\sum_{ X \ni x}U_X(\prod_{x\in X}(\s )_x))c_x(\sigma
%,a)\;=\;
% \hbox{exp}(\sum_{X \ni x}U_X((\prod_{x\in X}(\s^{x,a} )_x)))
%c_x(\sigma^{x,a} ,\sigma_x)\quad \forall\,x\in \L\Eq(3.2)$$
%A similar equation holds in finite volume
%$\Lambda$ with boundary conditions $\tau$, provided that we
%replace in \equ(3.2) $\sigma$ with the configuration $ \sigma
%\tau$.
$$\hbox{exp}(-\beta \sum_{ X \ni x}U_X(\s))c_x(\sigma
,a)\;=\; \hbox{exp}(-\beta \sum_{X \ni x}U_X(\s^{x,a} ))
c_x(\sigma^{x,a} ,\sigma_x).
\Eq(3.2)$$
\par
Under these conditions above
$L$ ({\it resp.} $L^\tau_\Lambda$) generates
a unique positive selfadjoint contraction semigroup on the space
$L^2(\O ,d\mu )$ ({\it resp.}
$L^2(\O_\L ,d\mu_\L^\t )$) that
will
be denoted by $P_t$ ({\it resp.} $P_t^{\Lambda ,\tau}$) (see $[L]$).\par
It is immediate to check that, in finite volume,
reversibility implies that
the unique invariant measure of the dynamics coincides with the Gibbs
measure $\mu^\tau_\Lambda$.
%This important fact holds also in infinite volume
%provided that there exists a unique Gibbs measure in the
%thermodynamic limit.
\par \medskip
Given a family (finite or infinite) $\Gamma$ of subsets of the
lattice $\Z$ we say that $ECU(\G,m)$ holds if for each cylindrical
(i.e., depending on finitely many spins) $f:\O \,\to \, R$ there
exists a finite constant $C_f$ such that: $$\sup_{\t, \L\in
\Gamma}\parallel P_t^{\t,\L}f\,
-\,\mu_\L^\t(f)\parallel_\infty\,\leq \, C_f
\hbox{exp}(-mt).$$\medskip
Below we will use $\nu (f)$ to denote the average of the
continuous function $f: \Omega \to {\bf R}$
with respect to the measure $\nu$.
(This is a common abuse of notation, since $\nu(\sigma)$ still denotes
the $\nu$-probability assigned to the configuration $\sigma$.)
Using this notation we define $\hbox{gap}(\L , \t )$ as:
$$\hbox{gap}(\L , \t )\;\equiv\;
\inf_{f : \,\mu_\L^\t(f)\,=\,0 \ ,\ \mu_\L^\t(f^2)\,=\,1} \
\ {1\over 2}\sum_{x,a,\sigma \in \Omega_{\Lambda}}\mu_\L^\t(\s ) c_x
(\sigma, a) \bigl ( f (\sigma^{x,a})-f (\sigma) \bigr )^2.$$
\medskip
We define the
\LSC $c_s(\nu)$ for an arbitrary measure $\nu$ on $\O_\L$ as the
smallest number $c$ such that for every non-negative continuous function
$f : \O_\L\, \to \, {\bf R}$ the following inequality holds:
$$\nu(f^2 \log(f))\;\leq\;c\nu ( \sum_{x\in
\L}(\partial_{\s_x}f(\s))^2)\;+\;\nu(f^2) \log((\nu (f^2))^{1\over
2}),$$
where
$$\partial_{\s_x}f(\s)\;=\;{f(i+1)\,-\,f(i)\over 2}, $$
if $\s_x\,=\,i$, with $N+1\,\equiv\,1$.\medskip
We define the {\it decimation transformation} of spacing $b$, $T_b$,
as follows. Let
$$\Z (b)\,\equiv \, \{y\in
\Z\,;\;y\,=\,bx\;\hbox{for some }x\,\in \, \Z\}.$$
Then
$$T_b\,\mu_\L^\t\;\equiv \;\mu_{\L,\ \Z (b)\cap\L}^\t,$$
i.e., $T_b\,\mu_\L^\t$ is the relativization to $\Z (b)\cap\L$ of
$\mu_\L^\t$. \medskip
Now we recall the Dobrushin - Shlosman finite size condition
for uniqueness. Given a metric $\varrho$
on the single spin space $ S$ let
$$\varrho_{\L_o} (\s ,\eta )\,=\,\sum_{x\in \L_o}\varrho (\s (x)
,\eta (x) ).$$
Then we say that condition $DSU_{\varrho} (\Lambda_o, \delta)$ is
satisfied if there exists a finite set $\Lambda_o
\subset \subset \Z$, and a number $\delta > 0$ such that
%to every $y \in \partial ^+_r \Lambda_o$ we can associate a number
to every $y \in \Lambda_o^c$ we can associate a number
$\alpha_y$ which satisfies the following two inequalities:
$ \forall \tau , \tau' \in \Omega ^c_{\Lambda _o}$ with
$\tau'_x = \tau_x \; \forall x \neq y$
$$ \hbox{KROV}_{\varrho_{\Lambda_o}}(\mu^\tau_{\Lambda _o},
\mu^{\tau'}_{\Lambda _o})\ \leq\ \alpha_y\varrho(\tau_y, \tau'_y),
\Eq(3.3)$$ \noindent
and
$$\sum_{y\in\partial^+_r\Lambda _o}\alpha_y \ \leq \
\delta\vert
\Lambda_o \vert. \Eq(3.4)$$
We simply say that $DSU (\Lambda _o, \delta)$ is
satisfied if \equ(3.3) and \equ(3.4) hold with $\varrho (i,j)$
%\bar \varrho (i,j)$ given by the Kronecker $ \d_{ij}$.
given by (1.8).
\bigskip
{\bf Theorem 3.1}\quad(Dobrushin - Shlosman [DS1])\par
{\it If $DSU_{\varrho} (\Lambda _o, \delta)$ is satisfied for some
$\varrho$, $\Lambda _0$ and $\delta < 1$, then there exist positive
constants $C$ and $\gamma$ such that condition
$WM_\varrho(\Lambda , C, \gamma)$
(defined by (1.12)) holds for every $\Lambda \subset \subset \Z$.
In particular, if $\varrho (i,j)$ is defined by (1.8), we obtain
$WM(\Lambda , C, \gamma)$ for all $\L \subset \subset \Z$.}
\bigskip
We are now in a position to state the main consequence of Theorem
1.1 :\bigskip
{\bf Theorem 3.2}\par
{\it In 2 dimensions the following are equivalent:\medskip
\item{(i)} There exist positive constants $C$ and $\gamma$ such that
$WM(\Lambda , C,\gamma)$ holds for every $\L\subset \subset \Zt$.
\item{(ii)} There exist positive constants $C$ and $\gamma$ such that
$SM(\Lambda_{L} , C,\gamma)$ holds for every square $\L_L$, $L\,>\,0$.
\item{(iii)} There exists a positive constant $m$ such that
$ECU(\G,m)$ holds, with $\G$ the family of all squares in $\Zt$.
\item{(iv)} There exists a positive constant $m$ such that
$gap(\L_L,\t )\,\geq \,m$ for every square $\L_L$, $L\,>\,0$.
\item{(v)} There exists a positive constant $c_o$ such that
$\sup_{\t}c_s(\mu_{\L_L}^\t)\,\leq \,c_o$ for every square $\L_L$,
$L\,>\,0$.
\item{(vi)}
$DSU(\Lambda _o, \delta)$ is satisfied for some
$\Lambda _o \subset \subset \Zt$ and {\bf $\delta < 1$}.
\medskip
Moreover each one of the above conditions implies that the renormalized
measure $T_b\,\mu$ converges (exponentially fast in $b$) to a
product measure and that it is possible to carry out a convergent
cluster expansion for $\mu_\L^\t$.}\bigskip
{\bf Remarks.} For simplicity we did not state the above result
in its strongest version; namely in (i) we could have assumed
weak mixing only for all subsets of a large enough square or, even,
only for all rectangles with large enough shortest side inside a large
enough square. Similarly, in the remaining conditions, the collection
of all squares could have been replaced by the collection of all
subsets of $\Z$ multiple of a large enough square.
It is also worthwhile to stress the fact that from items (iii), (iv)
and (v) of the theorem, analogous statements follow for the system on
the infinite lattice $\Z$.
There is actually only one step in the proof of Theorem 3.2 where
we use the hypothesis that $d=2$: to assure that Theorem 1.1 can
be used and hence (i) implies (ii). In arbitrary dimension it is
still true (changing, of course $\Zt$ to $\Z$ and squares to cubes
in the statements) that (ii), (iii), (iv) and (v) above are equivalent,
that they imply (vi) and that (vi) implies (i).
\bigskip
{\bf Proof}\par
That (ii), and (iv) are equivalent was proved
in [MO1]. Moreover (iii) clearly implies (iv) and (ii) implies (v)
(see [MO2]) which, because of hypercontractivity (see e.g. [G] or
[SZ]), implies (iii). That (i) implies (ii) was proved in Theorem
1.1, while the Dobrushin-Shlosman uniqueness theorem above, Theorem 3.1,
tells us
that (vi) implies (i). Therefore we only have to show that (ii)
implies (vi). The simple argument that we present below to prove
this is borrowed from the proof of Theorem 3.3 in [DS1]; we reproduce
it here, for the reader's convenience, because that theorem was stated
in a weaker form in [DS1]. The argument is presented in arbitrary
dimension. Let $\L_o$
be a cube of side $L_o$ and, for $y\in \partial^+_r\Lambda _o$
and $A > 0$ set
$$B_A(y)\,\equiv\,\{x\in \L_o \, ;\,\norm{x-y}\,\geq \,
A\log(L_o)\}.$$
%where $A$ is a positive constant to be chosen later on and let
Suppose that $\tau'=_y \tau$. Because of
the validity of $SM(\L_o,C,\gamma )$ there
exists a joint representation $\nu$ of
$\mu^\tau_{\Lambda _o,B_A(y)}$ and $\mu^{\tau'}_{\Lambda _o,B_A(y)}$
such that
$$ \sum_{z\in B_A(y)}\nu ((\sigma,\eta)\ ; \ \s (z)\neq \h (z))\;
\leq\; L_o^d \ \nu ((\s, \h)\ ; \ \s \neq \h) \;
\leq\;L_o^d C\hbox{exp}(-\gamma A\log(L_o)).\Eq(3.4bis)$$
Let now $q(\s_{\L_o\setminus B_A(y)},\s_{\L_o\setminus B_A(y)}'\; \vert\;
\s_{B_A(y)},\s_{B_A(y)}')$ be any joint representation of
$\mu^{(\s\tau)_{B_A(y)}}_{\Lambda _o\setminus B_A(y)}$ and
$\mu^{(\s'\tau')_{B_A(y)}}_{\Lambda _o\setminus B_A(y)}$.
Then, as one can easily verify, the measure
$$\tilde \nu (\s ,\s ')\;=\;q(\s_{\L_o\setminus B_A(y)}
,\s_{\L_o\setminus B_A(y)}'\; \vert\;
\s_{B_A(y)},\s_{B_A(y)}')\nu(\s_{B_A(y)},\s_{B_A(y)}')\Eq(3.4tris)$$
is a joint representation of $\mu^\tau_{\Lambda _o}$ and
$\mu^{\tau'}_{\Lambda _o}$ and its relativization
to the set $B_A(y) \times B_A(y)$ coincides with the measure
$\nu$. Thus, because of the definition of
$\hbox{KROV}_{\varrho_{\L_o}}$ and using \equ(3.4bis) and \equ(3.4tris),
we have, when $\varrho$ is given by (1.8),
$$\hbox{KROV}_{\varrho_{\Lambda_o}}(\mu^\tau_{\Lambda _o},
\mu^{\tau'}_{\Lambda _o})\leq\ \ L_o^d C\hbox{exp}(-\gamma
A\log(L_o))\;+\;3^d (A\log(L_o))^d.\Eq(3.5)$$
Clearly, if we take $A\,=\,d/\gamma$ and if $L_o$ is taken large
enough, the r.h.s. of \equ(3.5) is bounded above by
$(3^d + 1) (A\log(L_o))^d\;\equiv\; \a_y$. Thus, in this case,
$$\sum_{y\in\partial^+_r\Lambda _o}\alpha_y\ \ \leq
(3^d +1) (A\log(L_o))^d\vert \partial^+_r\Lambda _o\vert\;<<\;\vert
\L_o\vert, \Eq(3.6)$$
if $L_o$ is large enough. This finishes the proof that (ii) implies (vi).
\par\bigskip
%The fact that each one of the conditions (i)$\dots$ (v) implies that
The fact that condition (ii) implies that
the renormalized measure $T_b\,\mu$ converges (exponentially fast in
$b$) to a product measure and that it is possible to carry out a
convergent cluster expansion for $\mu_\L^\t$
follows directly from the methods developed in [O],
[OP], [MO1] ( this was already noticed in [MO3] in the particular case of
3D Ising model but the argument is valid for the
class of interactions considered in this paper).\par
In fact in [O], [OP] a polymer expansion was performed via a block decimation
procedure on a sufficiently large scale. The fact that the convergence of the
corresponding cluster expansion immediately follows from
the validity of $SM(\L,C,\g)$ on a suitable cube $\L$ was shown in
Appendix 2 of [MO1].\eop \par
\bigskip
From this point on, we restrict ourselves to the
case $S\,=\,\{-1,+1\}$ and
consider the (formal) Ising-type Hamiltonian :
$$H(\s )\;=\;-{1\over 2}\sum_{x\neq y} J(x-y)\s_x\s_y\;-\,
%{1\over 2}\sum_{x}h\s_x, \Eq(3.6')$$
\sum_{x}h\s_x, \Eq(3.6')$$
%where $J(\vert x-y\vert)\,\geq \,0$ and
%$J(\vert x-y\vert)\,=\,0$ if $\vert x-y\vert \;>\;r$.
where $J(\cdot)$ satisfies the following conditions for arbitrary
$z \in \Z$: $J(-z) = J(z)$, $J(z) \geq 0$
and in case $||z|| > r$, then $J(z)=0$.
More formally, the corresponding potential
$U \,=\, \{U_{X},\ X \subset \subset \Z\}$
%is given by $ U_{X}(\s) = -(1/2) h\s_x $ for
is given by $ U_{X}(\s) = - h\s_x $ for
$X=\{x\}$, $\, U_{X}(\s) =
%-(1/2) J(|x-y|)\s_x\s_y $ for $X=\{x, y\}$,$\,\hbox{and}
-J(x-y)\s_x\s_y $ for $X=\{x, y\}$, and $U_{X} = 0$ otherwise.
For arbitrary $\L \subset \subset \Z$ we will use the notation
$\mu^+_{\L}$ ({\it resp.} $\mu^-_{\L}$) to denote the Gibbs measure
in $\L$ with b.c. identically $+1$ ({\it resp.} $-1$).
By $\beta_c$ we will denote the critical inverse temperature,
i.e., the supremum of the
values of the inverse temperature $\beta$ (in the definition (1.4))
for which there is a unique Gibbs measure on the infinite lattice $\Z$,
when $h=0$. The following theorem shows that the weak mixing condition
is verified in most of the phase space (the $(\beta,h)$ plane)
of these models. As a consequence,
in $d=2$ the equivalent statements (i) -- (vi) in Theorem 3.2 all
hold for these same values of $\beta$ and $h$.
By ``standard Ising model'' below we refer, as usual, to the particular
case in which $J(z)=1$ iff $z$ is one of the $2d$ nearest neighbors
(in the usual sense, i.e., w.r.t. the Euclidean norm) of
the origin in $\Z$, and $J(z)=0$ otherwise.
\bigskip
{\bf Theorem 3.3} {\it
For every Hamiltonian of the form \equ(3.6'), in arbitrary dimension:
\item{a)}
The following two statements
are equivalent:
a.i) There exist positive constants $C$ and $\gamma$ such that the
weak mixing condition $WM(\L,C,\gamma)$ holds for every
$\L \subset \subset \Z$.
a.ii) There exist positive constants $C$ and $\gamma$ such that for
every $L$,
$$ | \mu^+_{\L_L}(\sigma_0) - \mu^-_{\L_L}(\sigma_0) |
\leq C e^{-\gamma L}. $$
\item{b)}
For all $\beta \,<\,\beta_c$ and all $h$ there exists positive constants
$C$ and $\gamma$ such that $WM(\L ,C,\gamma)$ holds for every
$\L \subset \subset \Z$.
\item{c)}
For the standard Ising model there exists a finite $\beta_o$
such that for all $\beta > \beta_o$ and all $h \neq 0$
there exists $C$ and $\gamma$ such that $WM(\L ,C,\gamma)
$ holds for every $\L \subset \subset \Z$.}
\bigskip {\bf Proof}\par
%\item{a)}
a)
That (a.ii) follows from (a.i) is a tautology, so we turn to the
proof of the converse statement.
Let $\L$ be given and let $ \tau , \tau' \in \Omega_{\Lambda^c}$
be two boundary conditions.
%such that for some $y \in \partial ^+_r \Lambda _o$
%$$\tau'_x = \tau_x \quad \forall \;x \neq y,
%\quad\tau'_y\, =\,+1,\; \; \tau_y\, =\,-1.$$
By the F.K.G.-Holley inequalities, (see [H] or Chapter 4 of [L]),
one can construct a probability space and a family, $\{\s^\tau \ ;
\ \tau \in \Omega_{\L^c} \}$, of $\Omega_\L$-valued
random variables on this probability space so that
i) For each $\tau$, $\s^\tau$ has law $\mu^\tau_\L$.
ii) If $\tau \leq \tau'$, then $\s^\tau \leq \s^{\tau'}$ (both
inequalities in the coordinatewise sense).
\noindent Using the relation between the total variation distance and the
Kantorovich - Rubinstein - Ornstein - Vasserstein distance
(with $\varrho$ given by (1.8)), one concludes that
%(see e.g. Appendix 1 of [MO1]),
for any $\Delta \subset \Lambda $,
$$\eqalign{Var(\mu^\tau_{\Lambda , \Delta}\ , \
\mu^{\tau'}_{\Lambda , \Delta})\, & \leq
\, P(\s_\Delta^\tau \neq \s_\Delta^{\tau'}) \, \leq
\, P(\s_\Delta^- \neq \s_\Delta^+) \cr
& \leq
\,\sum_{x\in\Delta} P(\s^-_x \neq \s^+_x) \, =
\,\sum_{x\in\Delta}|\,\mu_\L^{+}(\s_x)\,-\,\mu_\L^-
(\s_x)\,|.} \Eq(3.7)$$
Let now, for each $x\in \D$, $Q_x$ be the largest cube centered
at $x$ and entirely contained in $\L$. Then, again by F.K.G.-Holley,
the r.h.s. of \equ(3.7) can be bounded from above by:
$$ \sum_{x\in\Delta}|\,\mu_{Q_x}^+(\s_x)\,-\,\mu_{Q_x}^-(\s_x)\,|
\leq C \sum_{x \in \Delta} e^{-\gamma \hbox{dist}(x, \L^c)}.
\Eq(3.8)$$
$WM(\L,C,\gamma)$ follows from \equ(3.7) and \equ(3.8).
%\par \item{b)}
b)It follows from part (i) of Theorem 2 in a paper by Higuchi,
[Hi], that for all $\beta \,<\,\beta_c$ and all $h$
%there exists positive $C$ and $\gamma$ such that
(a.ii) above is satisfied. Therefore (b) follows from (a).
(Actually Higuchi, in [Hi], discusses only the
standard Ising model. Nevertheless in his
proof of part (i) of his Theorem 2
the only ingredients used are the GHS and
Lebowitz inequalities,
and the results by Aizenman, Barski and Fernandez [ABF] on the absence of
intermediate phase.
All these results hold in our more general situation. For the
particular case $h=0$,
one can also find an argument which shows that (a.ii) holds for all
$\beta < \beta_c$ in [CCS], in the derivation of equation (2.7) in that
paper.)
%\item{c)}
c)Again, it is sufficient to check that the condition in (a.ii) above
holds.
This was proved in Corollary 5.1 in [MO1] using
a result by Martirosyan, [M], on the low temperature
$d$-dimensional standard Ising model under arbitrary non-null
external field. A stronger version of Martirosyan's result appears
as Theorem 3 in [S], where the reader can find a self-contained proof
in arbitrary dimension as well as a greatly simplified proof in case
$d=2$. \eop
\par \bigskip
We conclude now with a discussion of some of the nice consequences of the
results presented in this paper.
From Theorems 3.2 and 3.3, and the perturbative theory
developed in [O], [OP], one
can immediately deduce, for 2D Ising models, in the one-phase
region, analyticity
properties of thermodynamic functions and correlation functions in terms of
perturbation parameters, for a wide class
of possible complex perturbations.\par
For simplicity only the results concerning the free energy, in the case of
translation invariant complex perturbation of the Hamiltonian, are precisely
stated in the following Theorem 3.4. Similar statements hold also for the
expectations of local observables even in the
case of non-translationally invariant
complex perturbations when some spatial uniformity condition is satisfied.
The following theorem is a corollary to our results above and Theorem 1.1
in [OP]. \par
\bigskip
{\bf Theorem 3.4}\par
{\it Consider a ferromagnetic Ising model in 2D, described by the
Hamiltonian given in \equ (3.6').
%The corresponding potential
%$U \,=\, \{U_{X},\ X \subset \subset \Z\}$
%is given by $ U_{X}= -1/2 h\s_x $ for
%$X=\{x\}$, $\, U_{X}= -1/2 J(|x-y|)\s_x\s_y $ for $X=\{x, y\}$,$\,U_{X} = 0$
%otherwise.
%\par
Let $ U_1,\dots,U_l$ be $l$ finite range, translationally invariant real
potentials (i.e. $U_j $ satisfy H1 and H2 of Section 1 for $j=1,\dots,l$).
Consider the complex partition function
$ Z(\L,\b,\tilde U)$ defined as in \equ
(1.5) with $\tilde U = U + \sum_{j=1}^l
\l_j U_j$ in place of $U$ and $\l_j \in
{\bf C} ,j=1,\dots,l$. Let $f_{\L}^{\b} : {\bf C ^l \to \bf C}$ be given by :
$$
f_{\L}^{\b} = {1\over |\L|} \log Z(\L,\b,\tilde U).
$$
Then, for $\b$ and $h$ as in parts (b) or (c) of Theorem 3.3 above,
there exists a neighborhood $V$
of the origin in ${\bf C ^l}$ such that the limit
$$
\lim_{\L \to {\bf Z^2}} f_{\L}^{\b}
$$
exists and is a holomorphic function of $\l_j ,j=1,\dots,l$ in $V$ }.\par\bigskip
%{\bf Proof}\par
%\bigskip
%It is a corollary of Theorem 1.1 in [OP].\par \bigskip
We notice that, in particular, for the standard 2D Ising model we
get, from Theorem 3.4, that the free energy is a real analytic function of $h$,
at $h=0$ and {\it every} inverse temperature $\beta < \beta_c$.
To the best of our knowledge, this result, certainly expected on
physical grounds, had been proven before
{\it only} for $ \beta << \beta_c $.\par
\vskip 1cm
{\bf Acknowledgments}\bigskip\noindent
We would like to thank the Dipartimento di Matematica
dell'Universita' di ``Tor Vergata'' and the Mathematics Department of
the University of California Los Angeles for their hospitality
during the early and final stage of this work. \bigskip
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\end
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