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\title{Phase Uniqueness and Correlation Length in Diluted-Field Ising Models}
\author{{L. R. G. Fontes \hspace{10 mm} E. J. Neves}\\
Instituto de Matem\' atica e Estat\'\i stica, USP}
\date{November 18, 1994}
\maketitle
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%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
The Diluted-Field Ising Model, a random nonnegative field
ferromagnetic model, is shown to have a unique Gibbs measure
with probability 1 when the field mean is positive.
Our methods involve comparisons with ordinary uniform field
Ising models. They yield as a corollary a way of obtaining
spontaneous magnetization through the application of a
vanishing random magnetic field. We also establish existence
and nonrandomness of the correlation length of this model,
defined as $\left(\lim_{n\to\infty}-(1/n)\log\ctin\right)^{-1}$,
where $\n$ is the site on the first coordinate axis at distance
$n$ from the origin and $\ctin$ is the origin to $\n$ two point
truncated correlation function,and derive an upper bound for it
in terms of the correlation length of an ordinary nonrandom
model with uniform field related to the field distribution of
the diluted model.
\end{abstract}
\ep
{\em Mathematics Subject Classification 1991:} 60K35, 82B44.
\noindent{\em Key words and phrases:} Phase uniqueness, exponential decay
of correlations, spontaneous magnetization, disordered systems,
diluted systems.
\ep
%%%%%%%%%%%%%%%%%% SECAO 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\setcounter{equation}{0}
\label{sec:int}
In this work we introduce what we call the Diluted-Field Ising Model
(DFIM) which is a quenched random field
ferromagnetic pair interaction Ising model in $\{-1,+1\}^{\zzd}$
with the
the i.i.d. fields having a distribution which is concentrated
on the nonnegative extended real numbers. That is,
$\h\equiv\{\hi,\, i\in\zd\}$, the fields of the model, form a
family of i.i.d. random variables with $\Pr(\hi\geq0)=1.$
We adopt this terminology in order to distinguish this model
from the usual Random Field Ising Model (RFIM), for which the
fields distribution support also negative values, usually
having mean zero. Another reason is that the randomness in the DFIM
is closer in nature to that in diluted-site and bond models
(although some features of the DFIM, e.g. absence of first order
phase transition --- see Theorem~\ref{teo:uni} below ---,
differ from those of these other diluted models), so much so that
techniques used in analyzing site and bond diluted models
work for the DFIM as well (\cite{kn:Fo}).
Important somewhat recent works in the RFIM are those of
Bricmont and Kupiainen (\cite{kn:BK}) and Aizenman and
Wehr (\cite{kn:AW}) who established respectively the existence
of first order phase transition in at least three dimensions
and its absence in dimension two.
Dreifus, Klein and Perez recently discussed the RFIM (\cite{kn:DKP})
in high field, no condition on the mean, and, improving on work by
Beretti (\cite{kn:B}), derived results similar
to ours for the DFIM (see Theorems 1 and 2 below).
Their argument needs the condition that the sites with field zero
do not percolate, which appears to be necessary in general. In the context
of the DFIM, although there has been some consideration that percolation might
also play a role, indeed it does not. The competing heuristics prevails
which says that the volume contribution of the diluted field, no matter
how small its distribution, as long as it has positive mean, dominates the
boundary effects.
Our approach is to put rigor in the (physical) intuition that
macroscopically the DFIM should behave as an ordinary Ising
model with uniform field related to the diluted field mean
(or, more precisely, distribution).
By means of correlation inequalities leading to convexity
properties of correlations as functions of $\h$ and the use
of Jensen's inequalities, we arrive at comparison inequalities
between the DFIM and ordinary uniform field Ising models from
which we get the results for the DFIM.
A corollary of our argument provides alternative 'random' ways to
spontaneously magnetize an Ising system below the critical
temperature other than the usual 'deterministic' way.
One of them goes as follows. As in the usual case, turn on an
external field $h$ but {\em only} for sites chosen at random
each with positive probability $p$. Do not turn any field on for
the sites not chosen. Then turn the external
field down slowly. Another way is assign to each site a uniform in [0,1]
random variable independent of the other sites. If for a given
site its uniform random variable is less than a positive number $p$,
then turn on an external field $h$ for this site. Do not turn any external
field on if the random variable is greater than $p$. Then make $p$ go to zero.
In both ways (and in any combination of them),
the spontaneous magnetization achieved is the same attained
by the usual way of turning on the external field $h$ for {\em all} sites
and then turning it off slowly.
Let $\mu$ denote the common field mean, i.e. $\mu=E(\hi).$
Of course then $\mu\geq0.$ Our main results are stated below.
The first one establishes insensitivity to boundary
conditions of the (infinite volume) magnetization at site $i$
(the {\em minus} and {\em plus} cases, denoted $\csm$ and
$\csp$ respectively, are well defined by ferromagnetism).
\bteo
\label{teo:uni}
If $\mu>0$, then $\csm=\csp$ with probability 1.
\eteo
\brm
By ferromagetism, the above result implies a.s. uniqueness of
the infinite volume Gibbs state (which of course nevertheless
depends on $\h$). By the same reason, there is also no extra
magnetization gained or lost by turning on an extra positive
or negative uniform
field and subsequently turning it off slowly.
\erm
The next result concerns the correlation length of the DFIM
and its relation to the fields distribution.
Let $\n=(n,0,\ldots,0)$ be the site on
the first coordinate axis of $\zd$ at distance $n$
from the origin.
We define the correlation length $\xi$ by
\beq
\label{eq:col}
\xi^{-1}=\lim_{n\ar\infty}-\on\log\ctin,
\eeq
when the limit exists (otherwise replace $\lim$ by $\liminf$).
\bteo
\label{teo:col}
If $E(-\log\ctio)<\infty$, then with probability 1
the limit in (\ref{eq:col})
exists, is finite and nonrandom. Moreover,
$$\xi\leq\xi(\d),$$
where $\xi(a),\,a\in\r,$ denotes the correlation length of the Ising
model with uniform field $a$ and $\d$ is a function of
the distribution of $\h$ which is positive whenever $\mu$ is positive.
\eteo
In the next section we present some basic auxiliary results
ingredient to the proofs of the above theorems which appear in
Section~\ref{sec:pro}. In the concluding Section~\ref{sec:ext}
we discuss briefly the case of field distributions
including negative values. The methods which were successful
in the diluted-field case
are seen to go only halfway in establishing uniqueness of the
Gibbs measure, stopping short of saying anything about the
correlation length. The extension of Theorem~\ref{teo:col}
to other diluted models is considered.
\ep
%%%%%%%%%%%%%%%%%%%% AGRADECIMENTOS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{\bf Acknowledgements.} This paper originated from discussions
with H. v. Dreifus and A. C. Paiva, whom we gladly thank. We benefitted
also from discussions with D. H. U. Marchetti and R. H. Schonmann.
This research is part of FAPESP "Projeto Tem\'atico" Grant number
90/3918-5. Partially supported by CNPq.
\ep
%%%%%%%%%%%%%%%%%%%%%%%% SECAO 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Correlation Inequalities and Convexity}
\setcounter{equation}{0}
\label{sec:cor}
Consider the (ferromagnetic) Ising model in $\o=\{-1,+1\}^{\zzd}$
with Hamiltonian $H$ given by
\beq
-H(\s)=\sum_{\ij}\sii\sij+\sum_i\hi\sii,
\eeq
where the first sum is taken, as notation indicates, over
nearest neighbor sites, each pair appearing only once.
The fields $\h=\{\hi,\,i\in\zd\}$ are in principle any
extended real numbers.
Let $\corr$ as usual denote the expectation with respect to the Ising
measure
\beq
\cof=\frac{1}{Z}\sum_\s f(\s)e^{-\beta H},
\eeq
where $\beta$ is a positive real parameter which is usually interpreted as
the inverse of the temperature, $Z$ is the usual normalization factor
\beq
Z= \sum_\s e^{-\beta H},
\eeq
and $f$ is a local function.
The quantities to be studied here include the magnetization
at site $i$, $\csi$, the truncated two point function at sites
$i$ and $j$,
$$\ctij\equiv\csij-\csi\csj$$ and the truncated three point function
at sites $i$, $j$ and $k$,
$$\ctijk\equiv\csijk-\csij\csk-\csik\csj-\csjk\csi+2\csi\csj\csk.$$
We keep the volume and boundary dependence of $\corr$ implicit for
awhile. The volume is in principle finite, but the thermodynamic
limit will be taken eventually.
The following {\em GHS} type inequalities will be
useful for us. Their proof can be found in~\cite{kn:Sy}.
Let us consider the duplicated Ising model
($\s,\t$) in $\o^2$ with two independent copies of the model
considered initially. Define the transformed variables
$$t_i=\inv(\sii+\t_i)\quad q_i=\inv(\sii-\t_i),\quad i\in\zd.$$
For a subset $A$ of $\zd$ (with multiplicity of elements allowed),
let $t_A$ denote the product
$\prod_{i\in A}t_i$ and similarly for $q$.
\bprop
\label{prop:ghs}
For any $\h$,
\beq
\label{eq:ghs1}
\lab q_A\rab\geq0.
\eeq
For any $\h$ such that $h_l\geq0$, for all $l$,
\beqn
\label{eq:ghs2}
\lab q_At_B\rab\geq0,\\
\label{eq:ghs3}
\lab q_At_B\rab-\lab q_A\rab\lab t_B\rab\leq0,\\
\label{eq:ghs4}
\lab q_Aq_B\rab-\lab q_A\rab\lab q_B\rab\geq0.
\eeqn
\eprop
We have the following consequent results, respectively
of (\ref{eq:ghs1}) (making $A=\{i,j\}$)
and (\ref{eq:ghs3}) (making $A=\{i,j\}$ and $B=\{k\}$).
\bcor
\label{cor:ghs13}
For any $\h$, $i$ and $j$,
\beq
\label{eq:ghs5}
\ctij\geq0.
\eeq
For any $\h$ such that $h_l\geq0$, for all $l$, $i$, $j$ and $k$,
\beq
\label{eq:ghs6}
\ctijk\leq0.
\eeq
\ecor
The following is a corollary to (\ref{eq:ghs2}) and (\ref{eq:ghs4}).
\bcor
\label{cor:ghs24}
For any $\h$ such that $h_l\geq0$, for all $l$, $i$, $j$ and $k$,
\beq
\label{eq:ghs7}
\ctij\geq\ha\ctik\ctjk.
\eeq
\ecor
\noindent{\bf Proof:}
>From $(q_k+t_k)^2=2$, we have
\beqnn
2\lab q_iq_j\rab&=&\lab q_i(q_k+t_k)^2q_j\rab=\lab q_iq_k^2q_j\rab+
2\lab q_iq_jq_kt_k\rab+\lab q_iq_jt_k^2\rab\\
&\geq&\lab q_iq_k^2q_j\rab
\geq\lab q_iq_k\rab\lab q_jq_k\rab,
\eeqnn
where the two inequalities are due respectively to (\ref{eq:ghs2})
and (\ref{eq:ghs4}).
The proof is concluded by the observation that
$\ctij=\lab q_iq_j\rab$ for all pair of sites $i$ and $j$. $\bo$
\ep
Consider now $\csh$ and $\cth$ as functions of $\h$.
The above inequalities imply the following
properties of these functions, which are well known and
will be of later use.
\bprop \mbox{ }
\label{prop:prop}
$\csh$ is (coordinatewise) nondecreasing in $\{\h:\hi\in\r,\,
\mbox{\rm for all $i$}\}$.
$\cth$ is (coordinatewise) nonincreasing in $\{\h:\hi\geq0,\,
\mbox{\rm for all $i$}\}$.
$\csh$ is (coordinatewise) concave in $\{\h:\hi\geq0,\,
\mbox{\rm for all $i$}\}$.
\eprop
\noindent{\bf Proof:}
$\ds\csh=\beta\ctik\geq0,$ by (\ref{eq:ghs5}), proves the first assertion.
$\ds\cth=\beta\ctijk\leq0,$ by (\ref{eq:ghs6}), proves the second one.
$\dd\csh=\beta^2\ctikk\leq0,$ by (\ref{eq:ghs6}), proves the third one. $\bo$
\ep
Next we derive convexity properties of $\csh$ and $\cth$ when $\h$ is
restricted to a hyperretangle.
For $a$ and $b$ two real numbers such that $a**\d$, which is proved as (\ref{eq:abo}) using the
monotonicity of $\ctij(\h)$ (Proposition~\ref{prop:prop})
as an extra ingredient. Just take the limits $n\uparrow\infty$,
$\D\uparrow\infty$, $\G\nearrow\zd$ and $\L\nearrow\zd$
(in this order) together with the Dominated Convergence Theorem and
FKG on the \lhs and the absence of first order phase transition
in the uniform positive field regime on \rhs to get the result.
$\bo$
\ep
\brm
As anticipated in the introduction,
it follows from the above proof that one can spontaneously magnetize
a ferromagnetic Ising system below its critical temperature
by first selecting randomly with arbitrary uniform positive probability
the sites to be submitted to a constant field, the non-selected
sites receiving no direct field, and then turning the field off slowly.
This is because the expected magnetization of the system is bounded
below by that of an ordinary system with positive uniform field
at the same temperature.
Another way, by the same reason,
is to assign to each site a standard uniform random variable
independent of those of the other sites and turn on an external field
$h$ for this site if its random variable is less than a positive number
$p$, leaving it with no direct field if the random variable is bigger
than $p$. Then make $p$ go to zero.
\erm
\noindent{\bf Proof of Theorem~\ref{teo:col}:}
Let $\lij$ denote $-\log\ha\ctij$. By Corollary~\ref{cor:ghs24}
the following holds
\beq
\label{eq:sub}
\lmpn\leq\lm+\lmn.
\eeq
This together with the ergodicity of $\h$ and the finite expectation
hypothesis allows the application of Kingman's Subadditive Ergodic
Theorem so that we have
\beqn
\xi^{-1}\=\lim_{n\to\infty}\on\ln=\lim_{n\to\infty}\on E(\ln)\\
\label{eq:jcol}
\=\lim_{n\to\infty}\on E(-\log\ctin).
\eeqn
To prove the last assertion of the theorem (with a somewhat optimal $\d$)
we apply Propositions~\ref{prop:prop}~and~\ref{prop:conv}
and Jensen to get
\beq
E(-\log\ctin(\h))\geq E(-\log\ctin(\hm)\geq-\log\ctin(\dmb),
\eeq
for all $m$ and $n$, with $\hm$ as defined prior to~(\ref{eq:dlc}) and
$\dmb=\eta_{0m}(E[\hi^{(m)}]).$
We can take then $\d=\sup_m\dmb,$ so that
\beqnn
\xi^{-1}\geq\lim_{n\to\infty}-\on\log\ctin(\d)=\xi^{-1}(\d).
\eeqnn
(Notice that the above limit exists by subadditivity
\footnote{We note that the limit
$\lim_{n\to\infty}-\on\log E(\ctin))$
also exists by subadditivity, which is a consequence
of Proposition~\ref{prop:prop}, Corollary~\ref{cor:ghs24}
and the Harris-FKG inequality.}.)
$\bo$
\ep
\brm
A special case where the condition
$E(-\log\ctio)<\infty$
is not met is when $\Pr(\hi=\infty)>0$, for then $\ctio=0$
with positive probability. It is clear that in this situation
one can find \wp a (random) subsequence $(n_k)$ along which
$\ctink=0$, so that \wp
$$\limsup_{n\to\infty}-\on\log\ctin=\infty.$$
\erm
%%%%%%%%%%%%%%%%%%%%%% SECAO 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extensions and Open Questions}
\setcounter{equation}{0}
\label{sec:ext}
We use this section to briefly discuss some
extensions and limitations of the previous arguments
and methods to other related models, basically the
RFIM with field distribution including negative values
and diluted-bond ferromagnetic models.
For the RFIM with both positive and negative fields,
we do not have monotonicity nor convexity of the truncated
two point function (at least not in the form of
Propositions~\ref{prop:prop}~and~\ref{prop:conv}),
but we still have them for the magnetization.
This is sufficient for the following result.
Let $\csx$ denote the limit
\beq
\label{eq:ll}
\ll\csi(\hl,\xlc),
\eeq
where $x$ is any extended real number, when the limit exists
\bprop
\label{prop:rfi}
Suppose the distribution of $\hi$ is supported in a finite interval
$[a,b]$. Then the limit in (\ref{eq:ll}) exists for $x\leq\d=\eta_{ab}(\mu)$
and $x\geq\D=\zeta_{ab}(\mu)$ and
\beqn
\csm\=\csx,\,\mbox{ for $x\leq\d,$}\\
\csp\=\csx,\,\mbox{ for $x\geq\D.$}
\eeqn
\eprop
\noindent{\bf Proof:}
Use FKG and (\ref{eq:mct}), (\ref{eq:j1}), (\ref{eq:j2}) as
in~(\ref{eq:dlc}), which are all valid for any finite $a$ and $b$ $\bo$
\ep
Together with the FKG ordering, this proposition
implies that the Gibbs measure with minus boundary
conditions equals all with {\em $x$ boundary conditions}
for $x\leq\d$ and the {\em plus} Gibbs measure equals all with
{\em $x$ boundary conditions} for $x\geq\D$.
In the case of Theorem~\ref{teo:uni}, we proceeded to
prove that $\csd=\csde$ by using the convexity
properties of the truncated two point function,
which do not hold if $a<0****0$ and
the {\em minus} Gibbs measure equals one with positive field ($\d$)
boundary condition. This could be an evidence that $\csd=\csde$
and so that $\csm=\csp$, yielding uniqueness. But this seems
to be a problem not amenable to our methods.
\ep
Another issue raised by our methods is that the functions $\eta$
(and $\zeta$) depend on $\beta$ in a way that $\eta\downarrow0$
as $\beta\uparrow\infty$, making the positive field of the uniform field
model dominated by the DFIM not uniform with temperature, which
probably should not be the case.
\ep
We turn now briefly to the issue of the existence and nonrandomness
of the correlation length for {\em diluted-bond Ising models}, which are
quenched random Ising models with
\beq
-H(\s)=\sum_{\ij}\jij\sii\sij,
\eeq
where $\{\jij,\,\ij\}$ is a family of i.i.d. non-negative random variables.
We note that Theorem~\ref{teo:col} holds in this situation,
as well as its proof. It is worth mentioning that the more
common dilution variables (the $\jij$'s), for which $\Pr(\jij=0)$
is positive, do not satisfy the finite expectation hypothesis.
Another limitation is we do not have a convexity property of
$\ctij$ as a function of $\{\jij\}$ from which to get bounds
for $\xi$.
%%%%%%%%%%%%%%%%%%%%%% REFERENCIAS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{xxxxxx 89}
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Ising Ferromagnets,} J. Stat. Phys. {\bf 15}, 327-341 (1976)
\bibitem{kn:Fo} Fontes L.R.G., {\em An Ordered Phase with Slow Decay
of Correlations in $1/r^2$ Continuum Ising Models,}
Ann. Probab. {\bf 21}, 1394-1412 (1993)
\bibitem{kn:BK} Bricmont J., Kupiainen A., {\em Phase Transition in the
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\noindent Instituto de Matem\'atica e Estat\'\i stica --- %
Universidade de S\~ao Paulo \hfill\break
Cx.\ Postal 20570 --- 01452-990 S\~ao Paulo SP --- Brasil \hfill\break
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