\magnification 1100
\def\di{\displaystyle}
\def\d{\displaystyle}
\def\P{\hbox{\bf{P}}}
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\def\Omo{$\overline{\Omega}$}
\def\v{\par \noindent}
\def\R{I\!\!R}
\def\Z{Z\!\!\!\!Z}
\centerline{ \bf STRICT INEQUALITY FOR CRITICAL PERCOLATION VALUES}
\centerline{\bf IN FRUSTRATED RANDOM CLUSTER MODELS }
\vglue 0.2cm
\vglue 1.0cm
\parindent 10pt
\centerline{ Massimo Campanino\footnote{$^1$}
{Work supported by Italian G.~N.~A.~F.~A and EC grant
SC1-CT91-0695.}
\footnote{$^2$} {Investigation supported by University of Bologna.
Funds for selected
research topics.}}
\centerline{\it Dipartimento di Matematica,
Universit\`a degli Studi di Bologna,}
\centerline{\it piazza di Porta S.Donato 5, I-40126 Bologna, Italy}
\centerline{\it e-mail: campanin@dm.unibo.it}
\vskip 1.5cm
{\bf Abstract.}
We establish an inequality between the critical percolation value
of frustraded random cluster models and that of an unfrustrated model
where the free occupation probabilities of some bonds are strictly
decreased. Using the FK representation this gives an inequality
between the critical temperatures
of the corresponding spin models. In this way
we can prove a strict inequality
between critical temperatures for symmetry breaking of some frustrated
Ising models and the corresponding ferromagnetic models.
\vfill \eject
\vskip 2cm
\noindent
{\bf \S 1. Introduction, notation and statement of the results.} \hfill
\v
{ \it Notation.} \hfill \v
Let us introduce some notation. In this paper we will work on the lattice $Z^d$ with
$d \ge 2$. Elements of $\Z^d$ will be called {\it sites}.
On $\Z^d$ we shall use the distance $ d(x, y) = | x-y|= \sup _{i} |x_i -y_i| $.
Unordered pairs of nearest neighbour sites of $\Z^d$ will be called { \it bonds}.
Given a subset $\Lambda \subset \Z^d$ we denote by $\hat \Lambda$ the set of
bonds made up with sites in $\Lambda$. Given a set $R$ of bonds $R \subset \hat \Z^d$
we denote by $\tilde R$ the set of sites belonging to some bond $b \in R$.
We need also to consider ordered pairs of nearest neighbour sites of $\Z^d$: these
will be called {\it ordered bonds}. Given an ordered bond $b=(x, y)$ we denote by $-b$
the ordered bond $(y, x)$.
A sequence $p=(b_1, \ldots, b_k)$ of ordered bonds such that for $1 \le i \le k-1$ the second site
of $b_i$ is equal to the first site of $b_{i+1}$ is called a {\it path}; the first site of $b_1$
(respectively the second site of $b_k$) will be called the {\it initial site} (respectively the
{ \it end site}) of $p$.
A path $(b_1, \ldots, b_k)$ such that the second site of $e_k$ is equal
to the first site of $b_1$ is called a {\it loop}.
Given $\Lambda \subset \Z^d$ we say that a site $x$ is {\it in the $k$-interior} of $\Lambda$
where $k$ is a nonnegative integer if $d(x, \Z^d \backslash \Lambda) \ge k$. A bond, a path
or a loop are in the $k$-interior of $\Lambda$ if all their sites are.
Given a path $p=(b_1, \ldots, b_k)$,
we denote by $-p$ the path $(-b_k, \ldots, -b_1)$. Given two paths $p=(b_1, \ldots, b_k)$,
$r=(c_1, \ldots, c_{l})$ such that the second site of $e_k$ is equal to the first site of $c_1$,
we denote by $(e, r)$ the path $(b_1, \ldots, b_k, c_1, \ldots, c_{l})$. If $p$, $r$, $s$ are
paths, the symbol $(p, r, s)$ denotes the path $((p,r), s)$ provided the expression is defined,
i.~e. the paths have the required properties. Similarly for $(p_1, \ldots, p_k)$ if $p_1
\ldots, p_k$ are paths with the required properties.
Given a positive integer $L$ we want to give
a precise meaning to the concept of {\it subdivision} of $\Z^d$ or some subset of $\Z^d$
into cubes with side $L$. There is no problem in attributing sites to cubes.
For what concerns bonds that connect sites of different cubes we make the convention to
attribute them to the cube containing the site with larger coordinates.
We say that $V$ is a cube of bonds (with side $L$) if it is an element of a subdivision as
previously defined. As we will be actually interested in configurations of bonds,
we will say that a volume seen as a set of bonds
can be exactly subdivided into cubes with side $L$ when it is the disjoint union of cubes
of bonds with side $L$.
Given a set $R$ set of bonds the set of {\it bond configurations } in $R$ is the set
$S_{\d R} =\{ 0, 1 \} ^{ \d R }$.
If $e \in S_{\d R}$ we say that the bond $ b \in R$ is { \it empty}
(respectively { \it occupied}) in the configuration $e$ if $e_b=0$ (repectively $e_b=1$).
If $p$ is a path of bonds in $R$ we say that $p$ is {\it occupied} in the configuration
if $e_b=1$ for all bonds $b$ corresponding to ordered bonds of $p$.
Given a configuration $e \in S_{\d R}$ the set of sites
$ \tilde R$ can be partitioned into {\it clusters}, i.~e. connected
components, where two sites $x$ $y$ are said to be connected in $e$
if there is an occupied path
$p$ with $x$ and $y$ respectively initial and end sites of $p$.
For $e \in S_R$ $|e|$ denotes number of occupied bonds in $e$, $\#(s)$ number
of clusters in $e$. Given a finite set of bonds $R \subset \hat \Z ^d$ we say that $x$
is a boundary point of $R$ if there is a bond $b$ such that $x \in b$ and
$b \in \hat \Z ^d \backslash R$. Following [4] we define {\it boundary conditions}
on the set of boundary points of a finite set of bonds $R$. We define actually two kinds of
boundary conditions corresponding respectively to ``ferromagnetic'' and ``frustrated''
random cluster models that we are going to consider. A boundary condition on $R$ of the first
kind is simply a partition $\pi$ of the boundary sites of $R$.
A boundary condition on $R$ of the second kind is a pair $(\pi, \sigma)$ where $\pi$
is a partition of the set boundary sites of $R$ and $\sigma_{x, y}$ is an assignation of a
signature $+1$ or $-1$ to each unordered pair $\{ x, y \} $ of boundary points of $R$ belonging
to the same element of the partition $\pi$.
If $u$ is a boundary condition on $R$ of the first or the second kind, i.~e.
$u= \pi$ or $u=(\pi, \sigma)$ and $e \in S_R$
we denote by
$\#(e)_u$ the number of clusters in $e$ if declare connected two sites of the boundary
of $R$ that belong to the same element of the partition $\pi$.
Given a set of bonds $R \subset \hat \Z ^d $, an {\it interaction} $(J_b)_{ \d b \in R}$ is
a map from $R$ to $\R$. A loop $p=(b_1, \ldots, b_k)$ in $R \subset \hat \Z ^d $
is said to be frustrated with respect to the interaction $(J_b)_{ \d b \in R}$ if
$\prod _{i=1} ^{k} J_{\d b_i} < 0$. Given a boundary condition $u=(\pi, \sigma)$
of the second kind
a path $p=(b_1, \ldots, b_k)$,
with initial and end sites $x$ and $y$ boundary
sites of $R$, $x \neq y$, is said to be frustrated with respect to the interaction $J$ if
$x$ and $y$ belong to the same element of $\pi$ and
$\sigma_{x, y} \prod _{i=1} ^{k} J_{\d b_i} < 0 $.
{\it The Ising spin model.}
Given an interaction $( J_b)_{\d b \in \hat \Z ^d}$, a finite volume $\Lambda \subset \Z^d$
and a configuration $\eta \in \{ -1, 1 \}^{ \d \Z^d}$ the finite volume Ising Gibbs
measure $\mu ^{(\Lambda)}$ in $\Lambda$ corresponding
to the interaction $(J_b)$ and boundary condition
$\eta$ is the probability measure on $\{ -1, 1 \}^{ \Lambda}$ given by
$$\mu ^{\Lambda}( {\bf s } ) = Z^{( \beta, \Lambda )-1 }_{\eta}
\exp \left(- \beta \left(\sum_{ \langle x, y\rangle } J_{x, y} s_x s_y+
\sum_{ \langle x, y \rangle } J_{x, y} s_x \eta_y \right) \right),\eqno(1.1) $$
where $Z^{(\Lambda)}_{ \eta}$ is the normalization constant, the first sum in the
exponential is over bonds in $ \hat \Lambda$,
the second sum is over bonds $ \{ i, j \}$ with $i \in \Lambda$
and $j \in \Z^d \backslash \Lambda$.
A {\it Gibbs state} for the interaction $( J_b)_{\d b \in \hat \Z ^d}$ at inverse
temperature $\beta$ is any weak limit
of finite volume Ising Gibbs measures for sequences of volumes $\Lambda_n$ and boundary
conditions $\eta^{(n)}$ with $\Lambda_n \uparrow \Z^d$. Alternatively Gibbs states can
be defined through DLR equations for conditional probabilities (see [8]).
{\it Random cluster models.}
Given a finite volume of bonds $V$, an interaction $( J_b)_{\d b \in V}$
with $J_b \ge 0$
and a boundary condition of the first kind
$u=\pi$, we
define the corresponding finite volume random cluster
measure as the probability measure on $S_{\d V}$ given by
$$ P_{u}^{( \beta, V)}( {\bf e} ) = Z^{( \beta, V)-1 }
\left( \prod_{e_b=1} p_b \right)
\left( \prod_{e_b=0} (1- p_b) \right) q^{ \# ({\bf e} )_u},\eqno(1.2)$$
where $q$ is a real number $q \ge 1$, $\chi_u$ is the indicator of the event that there
is no occupied frustrated loop and no occupied frustrated path, $p_b = 1 - \exp ( - 2 \beta J_b)$
and $Z^{( \beta , V)}$ is the normalization constant.
The superscript $( \beta , V)$ will be omitted when there is no danger of confusion (see [3]).
{\it Frustrated random cluster models.}
Given a finite volume of bonds $V$, an interaction $( J_b)_{\d b \in V}$
and a boundary condition of the second kind
$u=(\pi, \sigma)$, we
define the corresponding finite volume frustrated random cluster
measure as the probability measure on the
configurations in $\hat\Lambda$ given by
$$ P^{( \beta, V)}_{u}( {\bf e} ) = Z^{( \beta, \Lambda )-1 }
\left( \prod_{e_b=1} p_b \right)
\left( \prod_{e_b=0} (1- p_b) \right) q^{ \# ({\bf e} )_u} \chi_u(e)\eqno(1.3)$$
where $q$ is a real number $q \ge 1$, $\chi_u$ is the indicator of the event that there
is no occupied frustrated loop and no occupied frustrated path,
$p_b = 1 - \exp ( - 2 \beta | J_b|)$
and $Z^{( \beta ,V)}$ is the normalization constant (see [7]). As before the superscript $( \beta , V)$
will be omitted when there is no danger of confusion.
In the following the contant $q$ will be fixed once for all so that there will be well
definite finite volume (frustrated) random cluster volumes corresponding to
an interaction, a boundary condition and a value of $\beta$.
{\it Infinite volume (frustrated) random cluster measures} for an interaction $J_b$,
$b \in \hat \Z^d$, are weak limits of the corresponding finite volume measures.
As in the case of Gibbs measures they can be alternatively defined in terms of
DLR type equations for conditional probabilities.
Let $(J_b)$ be an interaction defined for $b \in \hat\Z^d$,
the set of bonds of $\Z^d$.
We say that a cube of bonds $C$ is
frustrated with respect to $(J_b)$ if there is
a loop $p=(b_1, \ldots, b_k)$ in the $3$- interior of $C$ which is frustrated
with respect to $(J_b)$.
\v
{ \bf Definition.}
Let $J_b$ be an interaction on $\hat \Z ^d$. The
{ \it critical percolation value} for $J_b$
is defined to be the supremum of those values
$\beta$ such that for any infinite (frustrated) random
cluster measure the probability of
the existence of an infinite cluster is $0$.
\v
{ \it Statement of the results.} \hfill
In [6] Newman proved a weak inequality between the critical point
of a frustrated
random cluster model with interaction $(J_b)$ and that of a random
cluster model
with interaction $(|J_b|)$ (see also [7]). Using the FK representation
with $q=2$ this translates into a weak inequality for critical temperatures of
Ising models with the same inter\-
actions. In situations where one has a ``uniformly distributed frustration''
one would
expect a strict inequality. Here we prove a result of this kind. We can deal
the case
when the frustration is uniformly distributed such as a periodic interaction
with
a frustrated loop. In this case we can get a strict inequality for critical
percolation
points for random cluster models. By using FK representation
in the case $q=2$
this translates into a strict inequality for critical temperatures
for spin flip symmetry
breaking of Ising models.
We have the following results:
\v
{ \bf Theorem 1.1}
Let $(J_b)$ be an interaction on $ \hat \Z ^d$ such that
$ 0 < K_1 \le |J_b| \le K_2 < \infty $ $\forall b \in \hat \Z^d$,
for some constants $K_1$, $K_2$.
There is a constant
$\delta > 0$ such that the critical percolation
value $\beta_c$ for $(J_b)$ is larger than or equal to the critical
percolation value
for the interaction $(I_b)$ where $I_b= |J_b| - \delta$ if $b$ belongs to
the interior of a frustrated cube $C \in G$ and $I_b = |J_b|$ otherwise.
\vskip 0.5 cm
By using the FK representation for $q=2$ (see [4]), Theorem 1.1 gives the following
result for
spin models.
\v
{ \bf Theorem 1.2}
Let $(J_b)$ be an interaction on $ \hat \Z ^d$ such that
$ 0 < K_1 \le |J_b| \le K_2 < \infty $ $\forall b \in \hat \Z^d$,
for some constants $K_1$, $K_2$. There exists a constant $ \delta > 0$
such that for $\beta < \beta_c$ all Gibbs states correspoding to
$(J_b)$ are invariant
by spin flip, where $ \beta_c $ is the critical value
for the interaction $I_b$
$$I_b = \cases{ |J_b| - \delta & if $b$ belongs to the interior of a
frustrated cube $C \in G$ \cr
|J_b| & otherwise. \cr } $$
Theorem 1.2 can be applied to give a strict inequality between the critical
temperature
for spin flip symmetry breaking of some frustrated models
and that of the corresponding
ferromagnetic models. The following theorem follows immediately
from combined application of
theorem 1.2 and the results of [1].
\v
{ \bf Theorem 1.3}
Let $(J_b)$ be an interaction on $ \hat \Z ^d$ such that
$ 0 < K_1 \le |J_b| \le K_2 < \infty $ $\forall b \in \hat \Z^d$,
for some constants $K_1$, $K_2$. Assume that for some $L$ there
is a subdivision of $\hat \Z^d$ into frustrated cubes with side $L$
(this is in particular the case of a periodic interaction with
some frustrated loop).
Then
\v \item{i)}
the critical
percolation value for the random cluster model corresponding
to the interaction $(J_b)$
is strictly larger than that for the interaction $(|J_b| )$;
\item{ii)}
the critical temperature for symmetry breaking of the Ising model
with interaction $(J_b)$
is strictly larger than that for the Ising model corresponding
to the interaction $(|J_b|)$.
\smallskip
\v
{ \bf Remark 2.}
Recently G. R. Grimmett has proved strict inequalities for random cluster
models that allow to extend the results of Theorem 1.3 to frustrated
models with random interactions ([4]).
\v
\vskip 0.5 cm
\v
{\bf \S 2. Proofs of the results.}
\vskip 0.2cm
The proof of Theorem 1.1 is based on the following Lemma 2.1.
\v
Let us start with some definitions.
Let $V$ be a cube of bonds. We say that $E$, $E \subset S_{\d V}$,
is a connection event
of $V$ if it refers to connections of boundary sites of $V$.
Formally let $E_{x, y}$ be the
event that $x$ and $y$ are connected; $E$ is a connection event
of $V$ if it belongs to
the algebra generated by the events $E_{x, y}$ as $x$ and $y$
vary on the set of boundary
points of $V$.
Let $V \subset \hat \Z ^d$. An event $E$, $E \subset S_{\d V}$,
is said to be
{ \it positive} (resp. { \it negative })
if its indicator $\chi_{\d E}$ is a nondecreasing (resp. nonincreasing)
function on $S_{\d V}$.
Given $E \subset S_{\d V}$ and $b \in V$ we denote by $\delta_b E$
the event that
$e$ is pivotal for $E$, i.~e. the set
${ \{ e \in S_{\d V} | \chi_{\d E} (T_b e) \neq \chi_{\d E} ( e) \} },$
where $T_b e$ is the configuration with $(T_b e)_{b'}= e_{b'}$ for
$ b' \neq b$
and $(T_b e)_{b} \neq e_{b}$. We decompose $\delta_b E$ as
${\delta_b E = \delta^{(i)}_b E \cup \delta^{(e)}_b E}$ with
${ \delta^{(i)}_b E = \delta _b E \cap E}$ and
${ \delta^{(e)}_b E = \delta _b E \cap E^{c}}$ .
\v
{\bf Lemma 2.1.}
Let $V$ be a frustrated cube of bonds for the interaction $(J_b), b \in V$
with $J_b \neq 0 $ $\forall b \in V$.
Let $(\pi, \sigma)$ be a boundary condition for $V$ and let
$\P_{(\pi, \sigma)}$ ( $\P^{(F)}_{(\pi)}$ ) denote the frustrated random
cluster model
(respectively the random cluster model (here capital $F$ stays
for ``ferromagnetic''))
in $V$ interaction $(J_b), b \in V$
(resp. $(|J_b|), b \in V$ ), boundary condition
$(\pi, \sigma)$ (resp. $(\pi) $.
Then for any nontrivial positive connection event $E$ of $V$
we have $$\P_{(\pi, \sigma)} (E) <
\P_{(\pi)}^{(F)} (E)\eqno(2.1)$$.
\v
\noindent
\vskip 0.2 cm
\v
{\bf Proof of Lemma 2.1.}
Let $p=(b_1, \ldots, b_k)$ be a frustrated loop in the $3$-interior of $V$.
Let $U$ be the event that there
is no frustrated loop or path constituted of occupied bonds in $V$ with
respect to the boundary conditions $(\pi, \sigma)$ ; $U$ is clearly a negative event.
Let now $E$ be any nontrivial positive connection event
of $V$. Since $E$ is nontrivial event
there must be a bond $b \in V$
of $V$ such that
$\P_{(\pi)}^{(F)} ( \delta_b E) > 0$ and hence also
$\P_{(\pi)}^{(F)} ( \delta^{(i)}_b E ) > 0$.
Moreover since $E$ is a connection event it is easy to see that
$b$ can be taken in $\hat \Lambda$ where $\Lambda$is the cube of sites
corresponding to $V$. Let $b$ be chosen with
these properties.
We want to show that there exists a frustrated loop containing $b$.
If $b$ is one of the bonds of $p$ we have done.
Assume that this is not the case. It is easy to check that we
can build in the complementary of the support of $p$ two disjoint paths $q_1$ $q_2$
joining the extremes of $b =(y_1, y_2)$ with two
different sites $x_1$ and $x_2$ of $p$ respectively. $y_1$ and $y_2$ split $p$ into two paths
$p_1$ and $p_2$ from $ y_1$ to $y_2$ and from $y_2$ to $y_1$ respectively.
We can build two loops containing $b$ : $ r_1= (b, q_2, p_2, - q_1)$ and
$r_2= (e , q_2, - p_1, - q_1)$. It is easy to check that either $r_1$ or $r_2$
is a frustrated loop since the products of the $J$'s along $p_1$ and $p_2$
have opposite signs. We call $r$ this frustrated loop.
Now let us consider a configuration $c$ of bonds in $ V$ with all bonds
of $r$ occupied and all other bonds vacant. We have $\P_{(\pi)}^{(F)} ( c) > 0$.
But $c$ has been built so that $c \subset \delta^{(e)}_e U$. Therefore
$\P_{(\pi)}^{(F)} ( \delta^{(e)}_e U ) > 0$.
The probability measure $\P_{(\pi)}^{(F)}$ satisfies the conditions of FKG inequality.
By applying to the events $E$ and $F$ Theorem 1.1 of [2] we get
$$ \eqalign{
&-\left( \P _{(\pi)}^{(F)}(E \cap U) - \P _{(\pi)}^{(F)} (E) \P _{(\pi)}^{(F)}(U) \right) =
\P _{(\pi)}^{(F)}(E \cap U ^c) - \P _{(\pi)}^{(F)} (E) \P _{(\pi)}^{(F)}( \tilde U^c) \geq \cr
& { \d \P _{(\pi)}^{(F)}( e_b = 0) \over \d \P _{(\pi)}^{(F)}( e_b = 1)}
\P_{(\pi)}^{(F)}( \delta^{(i)}_b E) \P_{(\pi)}^{(F)}( \delta^{(i)}_b U^c)= \cr
& { \d \P _{(\pi)}^{(F)}( \omega_e = 0) \over \d \P _{(\pi)}^{(F)}( \omega_e = 1)}
\P_{(\pi)}^{(F)}( \delta^{(i)}_b E) \P_{(\pi)}^{(F)}( \delta^{(e)}_b U) >0. \cr } \eqno(2.2)$$
We get therefore from (2.2 )
$$\P_{(\pi, \sigma)} (E) =
\P_{(\pi)}^{(F)} (E | U)= { \d \P_{(\pi)}^{(F)} (E \cap U) \over \P_{(\pi)}^{(F)} ( U) }
< \P_{(\pi)}^{(F)} (E)\eqno(2.3)$$
Q.~E.~D.
\vskip 0.5 cm
\v
{\bf Proof of Theorem 1.1.}
Let $V$ be a frustrated cube of bonds with side $L$ with boundary conditions $ (\pi, \sigma) $.
The probabilities of events with support in $V$ are continuous functions of the interactions
and of $\beta$ . Therefore
there exists $ \delta > 0$ such that for every nontrivial positive connection event $E$
$$ \P_{(\pi, \sigma)} (E) \le \P_{(\pi)}^{'(F)} (E),\eqno(2.4) $$
where $ \P_{(\pi)}^{'(F)} $ is the random cluster Gibbs measure with interaction
$I_b = |J_b| - \delta $ for each bond $ b \in V $. We have assumed the interaction
$(J_b)$ to be bounded. The constant $\delta$ can therefore be chosen uniformy
for $\beta$ in a bounded interval $I \subset [0, \infty) $. The inequality holds
immediately also for trivial positive events. We can then apply a theorem by
Strassen ( [9], see also e.~g. [5] Theorem 2.4 p. 72) and obtain that there is a coupling between
the measures $\P_{(\pi, \sigma)}$ and $\P_{(\pi)}^{'(F)}$ restricted on
connection events supported on pairs of connection realizations $(c_1, c_2)$ with
$c_1 \le c_2$ in the natural ordering.
A similar coupling also exists
between
the measures $\P_{(\pi, \sigma)}$ and $\P_{(\pi')}^{'(F)}$ if $\pi$ is finer than
$\pi'$ by FKG ordering of the measures $\P_{(\pi, \sigma)}$ with respect to
boundary conditions (see [7]). In the case that $V$ is nonfrustrated we have
a coupling between
the measures $\P_{(\pi, \sigma)}$ and $\P_{(\pi')}^{(F)}$, i.~e. with the corresponding
model with interactions $ |J_b|$, if $\pi$ is finer than $\pi'$ (see [7]).
Let now $V$ be a volume of bonds decomposable into cubes $C_1, \ldots, C_k$ with side $L$.
It follows easily from Lemma 2.1 arguments that for any
boundary condition $(\pi, \sigma)$ of $V$ and for $\pi$ finer than $\pi'$
we have that if $E$ is an event that can be expressed in terms of connections
between sites on the boundaries of cubes $C_i$
$$\P_{(\pi, \sigma)}(E) \le \P^{'(F)}_{(\pi')}(E), \eqno(2.5) $$
where $\P^{'(F)}_{(\pi')}$
is the random cluster measure with interaction given by
$$I_b = \cases{ |J_b| - \delta & if $b$ belongs to some frustrated cube $C_i$ \cr
|J_b| & otherwise. \cr } \eqno(2.6) $$
An event $E$ of this kind is the event that the boundary of a cube $C_i$ is connected
to the boundary of $V$. It contains the event that a site in $C_i$ is connected to
the boundary of $V$.
We remark that on random cluster models the boundary conditions on a volume
of bonds $C \subset V$ are determined just by the connections of boundary
sites of $C$ in $V \backslash C$.
Inequality (2.5) can be established by constructing a coupling between connection
realizations on the sites of the boundaries of the cubes $C_i$.
One way to check it is to use the couplings between conditional
probabilities to couple two "Glauber type" dynamics on connection realizations
on the boundary sites that have as invariant measures
$\P_{(\pi, \sigma)}$ and $\P^{'(F)}_{(\pi')}$ respectively and such that in the dynamics
the first component is always less than or equal to the second. This can be achieved
by choosing at random one of the cubes $C_i$ and ajourning the two connection
configurations on the boundary of $C_i$
while leaving invariant the connection configurations on the boundaries
of other cubes. The transition probabilities are just the coupling between conditional
probabilities that have been shown to exist if the partition $\pi$ corresponding
to the first component is finer than the partition $\pi'$ corresponding to the
second. Last condition is always verified since it is implied by the ordering
between the two components.
Let now $V$ be cube centered at the origin and let $E$ be the event that the
boundary of the cube $C_i$ that contains the origin is connected to the
boundary of $V$. By taking the volume $V$ tending to $\hat \Z^d$ we get that
if the r.~h.~s. of (2.5) tends to $0$ so does the l.~h.~s.. This can be easily extended
to any sequence of volumes (not necessarily decomposable into cubes with
side $L$) tending to $\hat \Z^d$. This implies the inequality between critical
percolation values.
Q.~E.~D.
\vfill \eject
{\bf References:}
\smallskip
\item{[1]}
C.~E.~~Bezuidenhout, G.~R.~Grimmett and H.~Kesten ``Strict inequality
for critical
values of Potts models and random-cluster processes'' Commun.~Math.~Phys.
{\bf 158}1-16 (1993).
\smallskip
\item{[2]} M. Campanino ``A lower bound for the covariance of positive events''
Preprint of the Department of Mathematics of the University of Bologna
n.15 (1997)
mparc 97-529 to appear in Forum Mathematicum.
\smallskip
\item{[3]} G. R. Grimmett ``The stochastic random-cluster process and the uniqueness
of random-cluster measures'' Annals of Probability {\bf 23} 1461-1510 (1997).
\smallskip
\item{[4]} G.~R.~Grimmett ``Strict inequalities for quenched random-cluster
models'' preliminary draft, private communication (1997).
\smallskip
\item{[5]} T. M. Liggett. Interacting Particle Systems. Springer-Verlag, Berlin (1985).
\smallskip
\item{[6]} C.~M. Newman ``Disordered Ising systems and random cluster
representations''
in { \it Probability and Phase Transitions}, Geoffrey Grimmett ed.,
NATO ASI Series C: Mathematical and
Physical Sciences - Vol. {\bf 420} 247-260 (1994), Kluwer Academic Publishers,
Dordrecht.
\smallskip
\item{[7]} C.~M. Newman. ``Topics in Disordered Systems'' Birkh\"auser.
Basel (1997).
\smallskip
\item{[8]}
D. Ruelle. Thermodynamic formalism. Addison-Wesley, Reading Mass (1978).
\smallskip
\item{[9]}
V. Strassen ``The existence of probability measures with given marginals''
Ann. Math. Statist., {\bf 36} 423-439 (1965).
\bye
\end