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\begin{document}
\title{ An Ordered Phase with Slow Decay of Correlations in
Continuum $1/r^2$ Ising Models}
\author{ Luiz R. G. Fontes \thanks{Supported by FAPESP,
proc. no. 87/2629-7.}\\
Courant Institute of Mathematical Sciences
\thanks{On leave from Instituto de Matem\' atica e Estat\'\i stica,
USP, Caixa Postal 20570,
S\~ ao~Paulo, SP, 01498, Brazil.\hfill\break
\indent{\it AMS 1980 subject classifications.} 60K35.\hfill\break
\indent{\it Keywords and phrases.} Ising model, intermediate phase.}\\
251 Mercer Street \\
New York, NY 10012 \\
internet: fontes@acf9.nyu.edu }
\maketitle
\begin{abstract}
For continuum $1/r^2$ Ising models, we prove that the critical
value of the long range coupling constant (inverse temperature),
above which an ordered phase occurs (for strong short range cutoff),
is exactly one. This leads to a proof of the existence of an ordered
phase with slow decay of correlations. Our arguments involve
comparisons between continuum and discrete Ising models, including
(quenched and annealed) site diluted models, which may be of
independent interest.
\end{abstract}
\vfill\eject
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\section{Introduction}
\setcounter{equation}{0}
\label{sec:introd}
The model to be discussed below is an Ising model in one dimension
with long range, translation invariant, ferromagnetic pair interaction.
However, unlike the usual case, its configurations are $\pm1$ valued
functions {\em on the real line} {\bf R} rather than on the discrete
one dimensional lattice {\bf Z}. Thus, it is a continuous time
stochastic process, to be more precisely defined in the next
section.
Such a process, which is called {\em continuum Ising model},
arises in the study of a quantum mechanical model of the motion
of a particle subjected to a field. The quantum mechanical energy
operator, known as spin-boson Hamiltonian for this model, can be
analysed by Feynman-Kac techniques, leading to the re-expression
of quantities related to the quantum model in terms of continuum
Ising quantities
(see \cite{kn:SD},
\cite{kn:L}, \cite{kn:S}).
Two parameters, $\a$ and $\e$, and a function, $W$, enter the model.
The parameter $\a$ is the long range coupling constant which can also
be interpreted as the inverse temperature;
$\e$ can be related to the inverse of the short
range coupling strength.
$W=W(r)$ is defined on $[0,\infty]$, nonnegative
and decays at infinity
as $1/r^2$.
In previous work (\cite{kn:SD}, \cite{kn:S}), the case of $1/r^2$
long range interactions (corresponding to the {\em ohmic} case
of the quantum model) has been studied with, among other results,
the following rigorous description of the phase diagram
(Theorem 2 in \cite{kn:S}).
For $\a\leq 1$ and any $\e > 0$, the model shows no
spontaneous magnetization,
whereas if $\a > 2$, then for small $\e$ (large short range coupling
force), there is spontaneous magnetization.
The strategy applied to get these results is to use the FK
representation of the Ising model, which in the continuum
case leads to a continuum bond percolation model, and then adapt
the results existing for the discrete FK model, obtained in
\cite{kn:AN}, \cite{kn:NS} and \cite{kn:ACN}.
Here, we establish the existence of long range order
(in the strong form known as {\em long long range order},
which implies spontaneous magnetization) for $\a > 1$, thus closing
a gap in the phase picture.
Indeed, what we do is establish a comparison between
the continuum model and the discrete $1/r^{2}$ Ising model
at inverse temperature ${\a}_{\e}$ and nearest neighbor
coupling $J_{\e}$, with ${\a}_{\e}$ close to $\a$ and $J_{\e}$
large when $\e$ is small. We then quote the results for
the discrete model obtained in \cite{kn:IN}.
As in the discrete case, long long range order for
$\a > 1$ leads to the existence of an intermediate
phase (at least for $1 < \a < 2$) with slow decay of correlations.
Here, we prove lower bounds for the
decay of the truncated two-point function in the
ordered phase. In the disordered phase,
lower and upper bounds for the
two-point function were obtained in~\cite{kn:S}.
Upper bounds in the ordered phase remain to be
obtained for the continuum model, unlike for
the discrete case, for which they were derived
in \cite{kn:IN}.
Our results are stated and proved in the next $5$ sections,
one for the description of the model and statement of results,
one for each of three steps of the
comparison with the discrete model and the last one for
the lower bounds on the truncated $2$-point function.
\vskip .25cm
{\bf Acknowledgements.}
This paper is part of the author's PhD research.
Thanks are due to
Charles M. Newman, the advisor, for his
insights and ideas during this and
previous work, and to Herbert Spohn,
for pointing out this problem to us.
\section{The Model}
\setcounter{equation}{0}
\label{sec:mod}
For $T$ positive, let ${\o}_{T}$ be the space of functions
$\s_t$ defined in the interval $\lb -T,T\rb$ and taking
values in $\{-1,+1\}$, which have only a finite number
of flips (and are right continuous, say). $\o_{T}$ is
the the set of configurations of the continuum system.
Let $P^f_{\e,T}$ be the measure on $\o_{T}$ such that the
flip points form a Poisson process with rate $\e > 0$
and $\s_{-T}$ equals $+1$ or $-1$ with equal probabilities
(free boundary conditions). $P^+_{\e,T}$ will denote the
measure on $\o_{T}$ such that the flips form a Poisson process
in $\lb -T,T\rb$ with rate $\e$, {\em conditioned to having only an
even number of
flips in} $\lb -T,T\rb$, and starting at $+1$ (ie, $\s_{-T} = +1$),
which corresponds to plus boundary conditions.
Now, let $W(t)$ be a nonnegative bounded
(piecewise) continuous function decaying like
$1/t^2$ at infinity, i.e. $t^2W(t)\ar 1$, as $t\ar\infty$.
It defines the continuum ferromagnetic
couplings and will be kept fixed throughout.
The finite volume continuum Ising measures with free and plus
boundary conditions are defined as follows.
\beq
\label{eq:mod}
d\rst(\s) = \frac{1}{\zst} dP^{\ast}_{\e,T}(\s) \exp^{-\a H^\ast(\s)},
\eeq
where $ \ast = f$ or $+$, and
\beqn
\label{eq:ham}
H^f(\s) \= -\frac{1}{4} \intit W(|t-s|)\sit\sis dt ds\\
\label{eq:hamm}
H^+(\s) \= -\frac{1}{4} \intinf W(|t-s|)\sit\sis dt ds
\eeqn
are the Hamiltonians and $\zst$ is a normalizing constant.
(In~(\ref{eq:hamm}), $\s\equiv+1$ in {\bf R}$\backslash\tt$.)
We denote by $\rs$ the infinite volume limit ($T\ar\infty$) of
$\rst$ (which exists by standard arguments) and by
$\lab\cdot\rab^{\ast}$ or
$\lab\cdot\rab^{\ast}_{T}$ the expectations w.r.t. $\rs$ or $\rst$.
\begin{defin}
\label{defin:llro}
For given $\a$ and $\e$ {\em long long
range order} is said to occur if there are positive constants $\n$ and $\m$
such that for all $T>0$,
$$
\lab\s_0\s_t\rab^{f}_{T}\geq\n^{2}\,\,\,\mbox{\rm for all } |t|\leq\m T.
$$
\end{defin}
Let $M$ denote the spontaneous magnetization of \mp, i.e.
$$
M=\lab\s_0\rab^{+},
$$
and $G^T(t)$ its truncated two-point function, i.e.
$$
G^T(t)=\lab\s_{0}\sit\rab^{+} - M^{2}.
$$
We now state the main results.
\begin{theo}
\label{theo:llr}
If $\a>1$, then long long range order occurs
for $\e$ small enough.
\end{theo}
\begin{theo}
\label{theo:iph}
If for given $\a$ and $\e$
long long range order occurs, then,
for any $\d>0$,
there exists some $C>0$ so that
\beq
\label{eq:iph}
G^T(t)\geq C/|t|^{2\g},\,\,\,\mbox{\rm for all } t\geq 1,
\eeq
where
$\g=\min(1,\a-1+\d).$
\end{theo}
We prove Theorem~\ref{theo:llr} in three steps within the next three sections.
The proof of Theorem~\ref{theo:iph} is presented
in the last section.
\section{ Comparison to an Annealed Site-Diluted Model }
\setcounter{equation}{0}
\label{sec:st1}
We begin this section by representing the continuum Ising
measure~(\ref{eq:mod}) introduced in the last section as a weak limit
of discrete Ising measures.
(This is a well known result, the arguments for which we sketch
here for completeness.)
This representation is then used to
derive a comparison between the continuum model and a sort of annealed
site diluted Ising model.
Consider a discrete Ising model on the lattice
$\L\equiv\d {\bf Z}$, where {\bf Z} is the set of integers
and $\d$ is a positive number, with interactions $\j$ given by
\beqnn
J_{i,i+\d}\=\ha|\log\e\d|,\,\,i\in\L,\,\\
\j\=\a\d^2W(i-j),\,\,i,j\in\L,\,|i-j|>\d,
\eeqnn
and Hamiltonian
\beq
\label{eq:alt}
\hd(\s)=-\qt\sum_{i,j}\j\sii\sij.
\eeq
The finite volume (in $\tt$) Ising measure so defined is denoted by
$\ntd$ and its expectation by $\ctd$, $\ast=f, +$, where the
appropriate boundary conditions are used.
We make the discrete configurations into continuum ones by
setting $\sit=\sii$ for $t\in[i,i+\d),\,i\in\L.$
We can write $\hd$ as the sum $\hud+\htd$, where
\beqnn
\hud(\s)\=-\qt\sum_{|i-j|=\d}\j\sii\sij\\
\htd(\s)\=-\qt\sum_{|i-j|>\d}\j\sii\sij.
\eeqnn
Notice that $\htd(\s)\ar H^\ast(\s)$ as $\d\ar0$,
with $H^\ast$ the Hamiltonian of the continuum system given
by~(\ref{eq:ham}) and~(\ref{eq:hamm}). Also, the measure
$$\frac{e^{-\hud(\s)}\times\mbox{\rm counting measure}}
{\mbox{\rm normalization}}$$
is that of a Markov chain which, as $\d\ar0$, converges
weakly to the Poisson measure $P^\ast_{\e,T}$ entering
the continuum Ising measure~(\ref{eq:mod}).
It follows that $\ntd$ converges weakly to $\rst$, as $\d\ar0$.
In particular, $\csstd\ar\csst$ as $\d\ar0$.
>From now on, we write $H^f(\s)$ as $H(\s)$.
For K, N positive integers, let $L=T/K$ and
$\d=L/N$. Consider the discrete Ising model in $\L$
with interactions given by
$$\jo=\j,$$
for $|i-j|>\d$ and for $|i-j|=\d$, but
$i\not = kL$, and given by
$$J^\circ_{kL,kL+\d}=0$$for $ k\in\{-K,\ldots,K\}$.
Denote it by $\notd$ and its expectations by
$\cotd$. Since $\j\geq\jo,\,\forall \,i,j,$ we have by the GKS inequalities
(see~\cite{kn:G} and~\cite{kn:KS}) that
\beq
\label{eq:gks}
\cstd\geq\csod.
\eeq
Now, as $N\ar\infty$, the measures $\notd$ converge weakly to the measure
\beq
\label{eq:prean}
d\rot(\s)=\frac{1}{Z'}e^{-\a H(\s)}\prod_{k=-K}^K dP_k(\sik),
\eeq
where H is given by~(\ref{eq:ham}), $\sik$ is a continuum configuration
in the interval $I_k\equiv[kL,(k+1)L)$ and $P_k$ is the measure in the set
of those configurations such that the flips form a Poisson process
of rate $\e$ and the initial distribution assigns equal probabilities
to $\pm 1$. (By an abuse of notation, we will omit the subscript
from $P_k$ from now on.)
Denote expectations w.r.t.~(\ref{eq:prean}) by $\cl$ (the dependence on T
is omitted). From~(\ref{eq:gks}) we conclude that
\beq
\label{eq:coan}
\csft\geq\cls.
\eeq
(Concerning the above discussion, see also~\cite{kn:SD}.)
Now, let $N_i$ denote the number of jumps of $\s_i$ in the interval
$I_i$. We can write the measure $P$ as
\beqnn
dP(\s_i)&=&dP(\s_i|N_i=0)\,P(N_i=0) + dP(\s_i|N_i>0)\,P(N_i>0)\\
\= (1-\e')dP_1(\s_i)+\e'd\bp(\s_i),
\eeqnn
where
\beqnn
dP_1(\s_i) \= dP(\s_i|N_i=0) = \frac{1}{2}(\d_{-1}(\s_i)
+ \d_1(\s_i)),\\ d\bp(\s_i) \= dP(\s_i|N_i>0),
\eeqnn
$\e'=1-e^{-\e L}$ is approximately $\e L$ as
$\e\da0$ and
$\d_u(\cdot)$ is the Dirac delta at the constant function $u$.
We rewrite $P$ further as
$$
dP(\s_i) = (1-2\e')dP_1(\s_i) + 2\e' dP_0(\s_i),
$$
where $P_0 = \frac{1}{2}(P_1 + \bp)$, and $\e$ is so small that
$2\e'\leq1$.
Now,
we
can write the correlations
\beqnn
\lab\s_A\rab '&=&\frac{1}{Z'}\int\s_A e^{-\a H(\s)}\prod_i dP(\s_i)\\
&=&\frac{1}{Z'}\int\s_A e^{-\a H(\s)}\prod_i
((1-\r)dP_1(\sii) + \r dP_0(\sii)),
\eeqnn
where $A$ is any set of (distinct) points $\{t_1,\ldots,t_k\}$
in $\lb -T,T\rb$ and $\sia = \prod_{i=1}^k\s_{t_i}$. We have
rewritten $2\e'$ as $\r$.
Expanding the product, we see that the integral can be viewed as
an expectation with respect to a family of i.i.d. random
variables $\l = (\l_i)$, where $\l_i$ has a Bernoulli distribution
with parameter $1-\r$, as follows.
\beq
\label{eq:anneal}
\lab\s_A\rab ' = \frac{1}{Z'}E_\l(\int\s_A e^{-\a H(\s)}\prod_i
dP_{\l_i}(\s_i))
\eeq
This model is a (sort of) annealed site diluted Ising model.
In this case, dilution applies to configurations in
an interval and means that there can be flips in the configuration
in that interval.
We have thus shown that the correlations for \mf are bigger than
the corresponding ones of an annealed model.
\section{ Comparison to a Quenched Site-Diluted Model }
\setcounter{equation}{0}
\label{sec:st2}
In this section we derive a comparison between the annealed
site diluted model of the last section and a regular quenched one.
>From now on, we restrict attention to sets $A = \{t_1,\ldots ,t_n\}$
such that there are no two $t_i$'s in the same $I_j$.
We write the expectation in (\ref{eq:anneal}) as
$$
E_\l(\int\sia e^{-\a H(\s)}\prod_i dP_{\li}(\sii)) =
E_\l(\frac{\int\sia\ex\prodi \pi}{\zl(H)} \zl(H)),
$$
where $\zl(H) = \int\ex\prodi \pi$. We denote the quotient inside the
expectation sign above by $\csi$.
\blem
\label{lem:pos}
$$\int\sia\ex\prodi\pi \geq 0.$$
\elem
\bprop
\label{prop:inc}
$\zl(H)$ is increasing in $\l$ (w.r.t. the usual partial ordering).
\eprop
\bprop
\label{prop:comp}
$$\csi \geq \csib,$$
where
\beq
\bar{H} (\s) = \sum_{i,j}\li\lj\intij W(|t-s|)\sit\sis dt ds.
\eeq
\eprop
{\bf Proof of Lemma \ref{lem:pos}:} Expanding the exponential, we
obtain
$$
\int\sia\ex\prodi\pi = \sum_n C_n\int\sia H^n(\s)\prodi\pi,
$$
where $C_n$ are positive numbers.
Expanding the $n$-th power, the r.h.s. can be expressed as
$$
\sum_n C_n\int\sia\intt\cdots\intt
\prod_{j=1}^n (W(|\tj - \ssj |)
\sitj\sisj d\tj d\ssj)
\prodi\pi.
$$
Moving the integral w.r.t. $\s$ inside, we get
\beq
\label{eq:rhs}
\sum_n C_n \mbox{$\int\int\cdots\int\int$}\prod_j
(W(|\tj -\ssj |)d\tj d\ssj)
(\int\sia\prod_j\sitj\sisj
\prodi\pi).
\eeq
The expectation $\int\sia\prod_j\sitj\sisj\prodi\pi$ factors into
$\prodi\int\siai\pi$, where $A_i$ is a set of points in the interval
$I_i$, for all $i$.
Now, if $|A_i|$ is odd ($|\cdot|$ denotes the cardinality),
then $\int\siai\pi = 0$, by the symmetry $\sii\ar -\sii$ of
$P_0$ and $P_1$. If $|A_i|$ is even and $\li = 1$, then
$\int\siai\pi = 1$. If $\li = 0$, we have
\beqnn
\int\siai\po \= \ha\int\siai\pu +\ha\int\siai d\bp\\
\= \ha(1 + \int\siai d\bp).
\eeqnn
Now, $\int\siai d\bp\geq -1$. Therefore, $\int\siai\po\geq 0$.
We conclude that $\int\siai\pi\geq0$, for all $i$. Thus, we
see that (\ref{eq:rhs}) is nonnegative and the lemma is
proved.
{\bf Proof of Proposition~\ref{prop:inc}:} Do the same steps
as in the last proof and notice that
$$
\int\siai\po\leq\int\siai\pu.
$$
Indeed, both integrals are zero if $|A_i|$ is odd and,
if $|A_i|$ is even, the r.h.s. is $1$, which is the
most the l.h.s. can be. This and positivity proves
the proposition.
{\bf Proof of Proposition~\ref{prop:comp}:} We will use
the following terminology. Let $\L = \{ i:\li = 1\}$ and
$\Lc = \{i:\li = 0\}$. We call an interval $I_i$ either an $1$-interval
or a $0$-interval depending on whether $i\in\L$ or $i\in\Lc$.
Also, let $\j$ denote the integral $\intij W(|t-s|)\, dt ds$.
First, notice that if any of the elements of $A$, say $t_{j_o}$,
belongs to a $0$-interval, say $I_{i_0}$, then $\csib =0$.
This is because under $\bar{H}$, all the $0$-intervals get
disconnected from the rest of the system, so that $\csib$ factors into
terms, one of which is $\int\s_{t_{j_0}}dP_0(\s_{i_0})$ (it
is here that the restriction on A made at the beginning of the section
enters). This integral vanishes (by symmetry), making the
product vanish.
Thus, we need only consider $A$'s all of whose elements belong
to $1$-intervals.
We change notation here and write $S_t$ instead of $\sit$ for
$t$'s belonging to $1$-intervals. Further, since $S_t$ is
constant in each $1$-interval $I_i$, we write $\si$ instead.
Notice that $\csb$ is the correlation of a discrete Ising
model (in {\bf Z}$\cap\lb -K,K\rb$) with interactions $\jb\equiv\li\lj\j$.
Now, the proof:
\beqnn
\zl(H)\,\cs \= \int\sa\, e^{-\a(H_1(S)+H_2(\s)+H_{12}(S,\s))}
\prod_{i\in\L}dP_1(\si)\prod_{i\in\Lc}dP_0(\sii)\\
\= \int\sa\, e^{-\a H_{12}(S,\s)} dQ(S) d\tilde{Q}(\s),
\eeqnn
where
\beqnn
H_1(S) \= -\qt\sum_{i,j\in\L}\j\si\sj (=-\qt\sum_{i,j}\li\lj\j\si\sj),\\
H_2(\s) \= -\qt\sum_{i,j\in\Lc}\intij W(|t-s|)\sit\sis dt ds,\\
H_{12}(S,\s) \= -\qt\sum_{i\in\L}\si\psii (\s),\,\,\mbox{\rm with}\\
\psii (\s) \= \sum_{j\in\Lc}\intij W(|t-s|) \sit dt ds,\\
dQ(S) \= e^{-\a H_1(S)}\prod_{i\in\l}dP_1(\si),\\
d\tq (\s) \= e^{-\a H_2(\s)}\prod_{i\in\Lc} dP_0(\sii).
\eeqnn
Notice that $Q$ is the unnormalized Ising measure with interactions $\jb$.
We expand the exponential.
\beqnn
\zl(H)\,\csb\=\int\sa\exp(\frac{\a}{4}\sum_{i\in\L}\si\psii(\s))dQ(S)d\tq(\s)\\
\= \sum_n C_n \int\sa(\sum_{i\in\L}\si\psii (\s))^n dQ(S)d\tq(\s)\\
\= \sum_n C_n\sum_{\me\in\gn} \cm \int\sa\prodi\si^{\mi} dQ(S)
\int\prodi\psii^{\mi} (\s) d\tq (\s),
\eeqnn
where $\gn = \{(m)\equiv (m_1,\ldots ,m_n)|m_i\geq0,
\sum_i m_i = n\}$ and $C_{\me}$ are positive numbers.
Now, by GKS inequality, the first integral is bigger
than
$$
\frac{1}{Z''}\int\sa\, dQ(S)\int\prodi\si^{\mi} dQ(S),
$$
where $Z'' = \int dQ(S)$ is a normalizing factor.
Also, the second integral is positive (this is proved like
Lemma~\ref{lem:pos} by expanding $\prodi\psii^{\mi} (\s)$
as well as the exponential in $\tq$ and checking that
everything is positive, which is done as before).
Notice now that $\frac{1}{Z''}\int\sa dQ(S) = \csb$
and that this quantity does not depend on $\me$ or $n$.
We thus have
\beqnn
\zl(H)\,\cs &\geq& \csb\sum_n C_n\sum_{\gn} \cm\int\prodi(\si\psii (\s))^{\mi}
dQ(S) d\tq(\s)\\
\= \csb\int\ex\prodi\pi.
\eeqnn
And Proposition~\ref{prop:comp} follows by the fact that the last integral is $\zl(H)$.
Proposition~\ref{prop:comp} and Lemma~\ref{lem:pos} imply the inequality
\beq
\label{eq:alqu}
\el (\csi\zl(H))\geq\el (\csib\zl(H)).
\eeq
We need one last result in this section.
\bprop
\label{prop:cinc}
$\csib$ is increasing in $\l$.
\eprop
{\bf Proof:} Since $\csib$ is always nonnegative and vanishes whenever
there is an element of $A$ in a $0$-interval, we need only check the
proposition for $A$'s without elements in $0$-intervals. But this
case follows by monotonicity in the couplings for correlations of the
ordinary discrete Ising ferromagnet.
Now, Propositions~\ref{prop:inc} and~\ref{prop:cinc} imply, via
Harris-FKG inequality (see~\cite{kn:H}),
the following inequality for~(\ref{eq:anneal}):
\beqnn
\frac{1}{Z'}\el (\csi\zl(H)) &\geq&\frac{1}{Z'}\el (\csib\zl(H))\\
&\stackrel{\mbox{\rm H-FKG}}{\geq}&\frac{1}{Z'}\el (\csib)
\el (\zl(H))\\
\= \el(\csib),
\eeqnn
since $\el (\zl(H))$ is by definition $Z'$.
We have thus obtained the following comparison.
\beq
\label{eq:quen}
\lab\sia\rab '\geq\el (\csib )
\eeq
The latter expression is the correlation of a quenched site diluted
Ising model.
{\bf Remark:} For the case of regular (discrete) annealed
site diluted models (i.e. those where the spins at the sites
are independently randomly diluted), a similar inequality
follows by the same arguments.
\section{ Comparison to a Standard Discrete Ising Model }
\setcounter{equation}{0}
\label{sec:up}
In this section, a general inequality relating quenched
(site) diluted Ising model and an ordinary undiluted
Ising model is derived.
We then apply it to the quenched model at the end of last
section to complete the comparison of the continuum
to the discrete models and prove Theorem~\ref{theo:llr}.
For an ordinary (discrete) Ising model in
a finite volume $\L$ with ferromagnetic
pair interactions $\j$
and Hamiltonian
$$H(\s)=-\qt\sum_{i,j}\j\sii\sij,$$
let $\lab\cdot\rab_{\a,J}$ denote
the expectation w.r.t. the Ising measure (with free or $+$ b.c.)
at inverse temperature $\a$.
Let $(\l_i)_{i\in\L}$ be a sequence of independent
nonnegative random variables which are less than
one and have means $(\csii)$. Let $\mjb$ denote the (random)
expectation w.r.t. the Ising
measure with inverse temperature $\a$ and interactions
$\bar{J}_{i,j} = \li\lj\j$. We define the quenched site diluted Ising measure
with expectation $\mq$
to be the mean of the random measure w.r.t. the distribution of
the $\l$'s, i.e.
$$
\mq \equiv\el (\mjb).
$$
We prove the following result.
\bprop
\label{prop:quench}
Let $A$ be a set of points in $\L$. Then
\beq
\label{eq:quench}
\csiq\geq\csitj,
\eeq
where $\jt = \eti(\csii)\etj(\csj)\j$, and $\eti$ is an increasing
continuous function in $[0,1]$ which is $0$ in $0$ and $1$ in $1$,
for each $i\in\L$. ($\etj$ does not depend on the distribution of the
$\li$'s, it does depend on $\a$ and the $\j$'s, especifically on
$\frac{\a}{2}\sum_j\j$ --- see~(\ref{eq:eta}).)
\eprop
{\bf Proof:}
For $i\in\o$, let $\eti (x)$ be a nonnegative function
in $[0,1]$ such that $\eti(x)\leq x$. (The specific form
of $\eti$ will be chosen below.)
By monotonicity of Ising correlations in the couplings,
we have
\beq
\label{eq:mon}
\csijb\geq\csijst,
\eeq
where $\jst = \eti (\li)\etj (\lj)\j$.
Let $N=|\o|$ and write $\csijst$ as
$$
\csij (\eta_1 (\l_1),\ldots,\eta_N (\l_N)).
$$
Now suppose that
\beq
\label{eq:convex}
\p\csij (\g_1,\ldots,\g_{i-1},\eti (x),\g_{i+1},\ldots,\g_N)\geq0
\eeq
for all $0\leq\g_i\leq 1$, $i\in\L$ and all $x$ in $(0,1)$.
Then we get~(\ref{eq:quench}) by
successive applying Jensen's inequality to $\csiq$.
By differentiating $\csij$ as above, we obtain the following expression.
\beqnn
\{\eti''-2(\eti')^2\lab\aq\sum_{j\in\L}\g_j\j\sii\sij\rab\}\cdot
[\lab\sia(\aq\sumjo)\rab - \lab\sia\rab\lab\aq\sumjo\rab]\\
+(\eti')^2[\lab\sia(\aq\sumjo)^2\rab -\lab\sia\rab\lab(\aq\sumjo )^2\rab],
\eeqnn
where primes mean differentiation w.r.t. $x$. We have ommited the argument of
$\eti$ and the subscripts of the Ising expectation signs.
The expressions in square brackets are nonnegative, by GKS inequality,
so that we only need the expression in braces to be positive. We use
the boundedness of the $\s$'s and $\g$'s to bound it below by
\beq
\label{eq:ode}
\eti''-\G_i(\eti')^2,
\eeq
where $\G_i = \frac{\a}{2}\sum_{j\in\L}\j$.
Setting (\ref{eq:ode}) to zero and
solving the differential equation with
boundary conditions $0$ in $0$ and $1$ in $1$, we obtain
\beq
\label{eq:eta}
\eti = \zeta(\G_i,x)\equiv\frac{1}{\G_i}\log(1-(1-e^{-\G_i})x)^{-1},
\eeq
which satisfies all the conditions above.
{\bf Remarks:}
\ben
\item The above proposition holds for both free and $+$ b.c..
\item A similar result is valid for a quenched
{\em bond} model with a similar proof.
\item These results can be used to derive lower bounds
for the critical temperature of diluted models
in cases more general than, for example, those
studied in \cite{kn:B}.
For the cases studied in this reference, our
bounds are weaker.
\een
We are ready now for the
{\bf Proof of Theorem~\ref{theo:llr}:}
The results of this and previous sections give us the following
comparison between the correlations of the continuum model (in $[-KL,KL]$,
configurations denoted by the letter $\s$)
and those of the discrete one (in $\{-K,\ldots,K\}$,
configurations denoted by $S$).
\beq
\label{eq:discont}
\lab\sia\rab^f\geq\lab S_{A^{\ast}}\rab^{f}_{\tJ},
\eeq
where $A$ is a set of points in $[-KL,KL]$ such that no two are in the
same interval $I_i$ $(=[iL,(i+1)L])$, $i\in \{-K,\dots,K\}$, and
$A^{\ast}$ is the set of integers $i\in \{-K,\dots,K\}$ such that there is
a point of $A$ in $I_i$. $\jt = (\zeta(\G,1-\r))^2\j$,
with $\j=\intij W(|t-s|)\,dtds$ and $\G=\at\sum_j\j$ (which does not
depend on $i$, by translation invariance, and is finite, due to
the decay of $W$ --- notice that by applying the proposition above
directly, we obtain~(\ref{eq:discont}) but with the finite sum for
$\G$; we can then replace it by the infinite sum due to the monotonicity of
both $\zeta$ and the Ising correlations).
Notice that the model in the r.h.s.
of~(\ref{eq:discont}) is a (one dimensional)
$1/r^2$ Ising model at inverse temperature $\a\zeta^2(\G,1-\r)$.
Notice also that $\j =\j(L)$ is such that,
denoting $$\int_{i}^{i+1}\int_{j}^{j+1}\frac{1}{|t-s|^2}\,dtds$$
by $\jp$ for $|i-j|>1$, we have that, as $L\ar\infty$,
$\j/\jp$ converges to $1$
uniformly in $i,j$ such that $|i-j|\geq 2$, and $J_{i,i+1}$
(which does not depend on $i$) goes to $\infty$.
We want to use the result of \cite{kn:IN} (Theorem 3.4)
stating that,$\,{\rm as}\,J'_{i,i+1}\!\ar~\infty,$
$$
\lab S_0 S_x\rab^f_{J'}\ar 1
$$
uniformly in the volume and in $x$ inside the volume,
provided $\a>1$, to prove
the following corresponding continuum result. As $\e\ar 0$,
$$
\lab\s_0\s_t\rab^f_T\ar 1
$$
uniformly in the volume and in $t$, provided
$\a>1$.
We proceed as follows.
Given $\a>1$ and $\d>0$, let $\bar{\a}$ be such that $1<\bar{\a}<\a$.
By the discrete result just quoted, there exists $J$ such that for
the model with nearest neighbor interactions bigger than $J$,
long range interactions given by $\jp$ and inverse temperature
$\bar{\a}$, we have
$$
\lab S_0 S_x\rab^f_{J'}\geq1-\d
$$
uniformly in the volume and in $x$.
Now, let $L$ be so big that $J_1\equiv J_{i,i+1}>J$
and also $\a\j>\ba\jp$
for $i,j$
with $|i-j|\geq2$.
Next, make $\e$ so small that $\r$ is so small that
$\zeta$ is so close to $1$ that $\zeta^2\a J_1>\ba J$ and
$\zeta^2\a\j>\ba\jp$.
By applying the comparison~(\ref{eq:discont}), we get
\beq
\label{eq:lll}
\tptc_T\geq1-\d
\eeq
uniformly in the volume and for $|t|>L$. If necessary
we can take $\e$ smaller so that~(\ref{eq:lll}) holds
uniformly in $t$. The theorem is now proven.
\section{ Slow Decay of Correlations }
\setcounter{equation}{0}
\label{sec:decor}
In this last section, we use the FK representation of the continuum
Ising model to derive lower bounds for the truncated two point
function, proving Theorem~\ref{theo:iph}. It is done almost
exactly in the same way as has been done in \cite{kn:IN}
for discrete FK models, with a few modifications (to account
for the extra randomness of the continuum case). For this
reason we will be a bit sketchy, referring the reader to
the discrete results for missing details (also to
\cite{kn:S} for definitions and properties of continuum FK measures).
We start by defining the FK measures. Let $\tei,\, i=1,
2,\ldots$ be the points of a Poisson process of rate $\e$
on the real line and $\oj = (\ssj,\tj),\,j=1,2,\ldots$ the points of a
Poisson process in $\ro = \{(s,t):s\leq t\}$ with density
$\de=\a W(t-s)$. Denote these (random) sets of points by
$\te$ and $\om$, respectively, and call them configurations.
(We will alternatively use the terminology $\te$-points for $\te$.)
The $\oj$'s will be given the meaning of occupied bonds
linking $\ssj$ and $\tj$.
Consider the partition of $R$ (resp., of the interval $[-T,T]$,
for $T>0$) into intervals, produced by the points $\tei$ in
$[-T,T]$. Call those $\te$-intervals. Say that two disjoint
intervals I and J are linked (denoted $I\frown J$) if there is
an occupied bond linking two points, one in each interval.
(If there are none, we denote $I\not\!\fr J$. Notice that the
two infinite intervals of the partition of $R$ are linked
with probability $1$.)
Two $\te$-intervals I and J are connected if there is a sequence
of $\te$-intervals $I_0,\ldots,I_n$ with $I_0=I$ and
$I_n=J$, so that $I_i\fr I_{i-1},\,i=1,\ldots,n$.
Two points $s$ and $t$ are connected (denoted $s\leftrightarrow t$)
if either they belong to the same
$\te$-interval or belong to distinct connected $\te$-intervals.
A {\em cluster} is a maximal union of connected $\te$-intervals.
Let $\cw$ (resp., $\cf$) be the number of distinct connected clusters
obtained with the $\te$-intervals of the partition of $R$ (resp., of
$[-T,T]$). We define the finite volume,continuum FK measures
with parameter $q$ as follows
\beq
\label{eq:FK}
d\pqwts = \frac{1}{N}dP(\te)dP'(\om) q^{\cas},
\eeq
where $\ast = w $ or $f$ for the {\em wired} and {\em free}
cases (see \cite{kn:S}),
$P$ and $P'$ are the Poisson processes mentioned and
$N$ is the normalizing factor. (We will drop some subscripts
sometimes).
The infinite volume measure exists (by standard arguments) and
is denoted $\pqws$.
Notice that for $q=1$, $P_{1,\D}^w=P_{1,\D}^f$ is
an independent (continuum) percolation model.
We list the properties of the FK measures
we will need.
1) $\pqws$ is a strong FKG measure, i.e., for any
region A in $R\times\ro$ and $f, g$
increasing functions in the configurations (w.r.t. the
partial order $(\te,\om)\prec(\te',\om')$ whenever
$\te\supset\te'$ and $\om\subset\om'$), we have
$$
\pqws(fg|{\cal A})\geq\pqws(f|{\cal A})\pqws(g|{\cal A}),
$$
where ${\cal A}$ is the $\s$-algebra generated by the
configurations in A.
For the properties below, we use the notation $P(\cdot)$ for
the expectation w.r.t. the measure P.
2) $\pqws(f)\geq P^{\ast}_{q',\de'}(f)$, for $q'\geq q, \de\geq \de'$
and $f$ increasing.
3) $P^w_{q,\de}(f)\geq P^f_{q,\de}(f)$, for $f$ increasing.
4) We have the following representation of continuum
Ising correlations (where, the notation $0\rl\infty$
means that the cluster of the origin is infinite):
\beqnn
M\=\lab\s_0\rab^{+}=\ptw(0\rl\infty),\\
\lab\sis\sit\rab^{\ast}\=\pts(s\rl t).
\eeqnn
We conclude
\beqnn
\tau(t)\=\lab\s_0\s_t\rab^{+}-M^2=\ptf(0\rl t)-(\ptw(0\rl\infty))^2\\
\ge\ptw(0\rl t,0\nrl\infty,t\nrl\infty)\equiv\tau'(t).
\eeqnn
So, all we need to prove Theorem~\ref{theo:iph} is to derive the
same bounds for $\tau'$. We do that in the propositions below.
As in \cite{kn:IN}, we begin with an estimate for the {\em self
similar} percolation case, i.e., the $q=1$ case with
\beq
\label{eq:sim}
W(t)=\tw(t)\equiv\frac{1}{t^2}1_{\{t>1\}}.
\eeq
For $\xi$ a real number, let
$$ T_{\xi}=\inf\{\tei:\tei\geq\xi\},\,\,S_{\xi}=\sup\{\tei:\tei\leq\xi\}.$$
Define
\beqnn
\m^\xi_1\=T_\xi\\
\m^\xi_{n+1}\=T_{\mxn},\,n\geq1\\
\n^\xi_1\=S_\xi\\
\n^\xi_{n+1}\=S_{\nxn},\,n\geq1,\,\mbox{\rm i.e. }
\eeqnn
$\mxn$ is the $n$-th $\te$-point after $\xi$ and $\nxn$ is the
$n$-th $\te$-point before $\xi$.
Notice that $\mxn=\xi+Y_n$ and $\nxn=\xi-W_n$, where $Y_n$ and
$W_n$ are random variables each having a Gamma distribution
with parameters $n$ and $\e$. In particular,
$E\mxn=\xi+n/\e$ and $E\nxn=\xi-n/\e$.
We say that an interval $[\xi',\xi]$ is {\em dissociated} if there is
no occupied bond from $[S_{\xi'},T_{\xi}]$ to its complement. Below, we
use the notation $\{I\nrl\infty\}$ for an interval I none of whose points
are in an infinite cluster.
\bprop
\label{prop:sim}
For L a positive integer, let
$
F_L=\{\exi$ an integer $k\in [1,L)|[0,\mkl)\not\!\fr(\mkl,\infty)\}.$
If $\a>1$, then there exists constants $C$ and $C'$ so that
in the self similar case~(\ref{eq:sim})
\beqn
\label{eq:lem}
P_1(F_L)\ge C/L^{\a-1}\,\,\,\mbox{ for all L, }\\
\label{eq:res}
P_1([0,L]\not\!\rl\infty) \ge C'/L^{2(\a-1)}\,\,\,\mbox{ for all L. }
\eeqn
\eprop
{\bf Proof:} Define
\beqnn
\fls\=\{\exi\,\,\mbox{\rm integer } k'\in [1,L)|
[\nok,L]\not\!\fr(-\infty,\nok)\},\\
\hl\=\{(\nol,0)\not\!\fr(L,\infty)\},\\
\hls\=\{(L,\mll)\not\fr(-\infty,0)\}.
\eeqnn
Then
\beqnn
P_1([0,L]\nrl\infty) \ge P_1(\fl\cap\fls\cap\hl\cap\hls)\\
\ge P_1^2(\fl)\,P_1^2(\hl),
\eeqnn
with the second inequality due to the FKG property (all of
the events are decreasing).
Now,
\beqnn
P_1(\hl)\=E\{\exp(-\a\int_{\nol}^0\int_L^\infty(t-s)^{-2}\,dtds)\}\\
\=E\{\exp(-\a\int_{-Y}^0\int_1^\infty(t-s)^{-2}\,dtds)\}\\
\ge\exp(-\a\int_{-EY}^0\int_1^\infty(t-s)^{-2}dtds),
\eeqnn
where the last inequality is Jensen's inequality and the expectation E
is w.r.t. a Gamma random variable Y.
The last expression is positive and does not depend on L.
It follows that~(\ref{eq:lem}) implies~(\ref{eq:res}).
To derive~(\ref{eq:lem}), let
$${\cal N}=\#\{k\in[1,L)\cap {\bf Z}:[0,\mkl]\nfr(\mkl,\infty)\}.$$
$\fl$ is the event that ${\cal N}>0$. We compute
the expected value of ${\cal N}$.
\beqnn
E_1({\cal N})\=\sum_{k=1}^{L-1}P_1([0,\mkl]\not\!\fr(\mkl,\infty))\\
\=\sum_1^{L-1}E\{\exp[-\a(\int_0^{\mkl}\int_{\mkl+1}^\infty(t-s)^{-2}\,
dtds\\&&+\int_{\mkl}^{\mkl+1}\int_0^{t-1}(t-s)^{-2}\,dsdt)]\}\\
\ge const.\sum_1^{L-1}E(\frac{1}{(\mkl)^\a})\geq const.\sum_1^{L-1}
\frac{1}{(E\mkl)^\a}\\
\=const.\sum_1^{L-1}\frac{1}{(L+k/\e)^\a}\geq const L^{1-\a},
\eeqnn
where the second inequality follows by Jensen's inequality
and E is expectation w.r.t. $\mkl$.
Now,
$$P_1(\N>0)=\frac{E_1(\N)}{E_1(\N|\N>0)}.$$
Let $X=\inf\{k'\in[1,L)\cap {\bf Z}:[0,\mkll]\not\!\fr(\mkll,\infty)\}.$
Then,
\beqnn
E_1(\N|\N>0)\=\sum_{k'=1}^{L-1}\int_L^\infty P_1(X=k',\mkll\in dt|\N>0)
E_1(\N|X=k',\mkll=t)\\
\=\sum_{k'=1}^{L-1}\int_L^\infty P_1(X=k',\mkll\in dt|\N>0)
\\&&\hspace{.5in}\cdot(1+\sum_{k=1}^{L-k'}
P_1((t,\mkt]\not\fr(\mkt,\infty))\\
\le\sum_{k'=1}^{L-1}\int_L^\infty P_1(X=k',\mkll\in dt|\N>0)\\&&\hspace{.5in}
\cdot(1+\sum_{k=1}^\infty E\{\exp(-\a\int_t^{\mkt}\int_{\mkt+1}^\infty
(r-s)^{-2}drds)\})\\
\le\sum_{k'=1}^{L-1}\int _L^\infty P_1(X=k',\mkll\in dt|\N>0)
(1+\sum_{k=1}^\infty E(\frac{1}{(\mkt-t+1)^\a}))\\
\=\sum_{k'=1}^{L-1}\int _L^\infty P_1(X=k',\mkll\in dt|\N>0)
(1+\sum_{k=1}^\infty E(\frac{1}{(Y_k+1)^\a}))\\
\le 1+const\sum_1^\infty\frac{1}{k^\a}=const.
\eeqnn
Combining all these inequalities, we get~(\ref{eq:lem}).
In the next proposition, we omit the subscript $\de$ in $P^\ast$.
\bprop
\label{prop:wifree}
For $|t|\leq L$,
$$
\tau'(t)\geq P^f_{2,L}(0\rl t)\,\ptw([-L,L]\not\rl\infty).
$$
\eprop
{\bf Proof:} Let $\ll=$ bonds $s,t$ with $|s|,|t|\leq L$. Then,
$$
\tau'(t)\geq\ptw(0\rl t\,\, \mbox{\rm by bonds in }\ll|\linf)
\ptw(\linf).$$
The first probability on the r.h.s. can be estimated by first
noticing that the conditioning event only depend on bonds
in $\ll^c$ and points of $\te$ outside $[-L,L]$. Proceed
now exactly as in the proof of Proposition 2.1 in \cite{kn:IN},
by conditioning further
on such configurations, expressing the infinite volume measure
as the proper limit of the finite volume ones, and then
using the strong FKG property to conclude that
\beqnn
&\ptw(0\rl t\,\, \mbox{\rm by bonds in }\ll|\linf)\\
\geq&\lim_{L'\ar\infty}
P^w_{2,L'}(0\rl t\,\,\mbox{\rm by bonds in }\ll|\not\!\exi\,\,
\mbox{\rm occupied bonds in }
\ll^c)\\ =& \ptf(0\rl t).
\eeqnn
\bprop
\label{prop:resu}
If $\a>1$, then for any $\d>0$, there exists some $C'>0$ so that
$$P^w_{2,\de}(\zinf)\geq C'/L^{2(\a-1)+\d},\,\,\,\mbox{\rm for all }L\geq 1.$$
\eprop
{\bf Proof:} Let $\hat{W} = W\chi(|t|R)$, for $\tw$ given
by (\ref{eq:sim}), where $\chi$ is the indicator function of a set.
Given $\d$, choose $\hata>\a$ so that
$2(\hata-1)=2(\a-1)+\d$, and R so that $\hata W>\a\tw$, for $t>R$.
Let $\hata\hat{W}=\hd$.
We then have
$$
\ptww(\zinf)\geq P_{1,\hat{\de}}(\zinf),$$
by the monotonicity properties of the FK measures.
Exactly as in Proposition~\ref{prop:sim}
$$\puhw(\zinf)\geq const\puhw^2(\fl).$$
Define $\flh =\{\exi\,\, \xi\in[L,2L)\cap {\bf Z}|$ there is no
occupied
bond longer than R linking $[0,\txi]$ to $(\txi,\infty)\}.$
Since $\flh$ does not involve the short bonds distinguishing between
$\puhw$ and $\putw$, we have
$$
\puhw(\flh)=P_{1,\hata\tw}(\flh)\geq P_{1,\hata\tw}(\fl)
\geq C/L^{\hata-1},$$
by Proposition \ref{prop:sim}.
Now, conditioning
\beqnn
\puhw(\fl)\=\puhw(\flh)\puhw(\fl|\flh)\\
\ge\puhw(\flh) E\{e^{-\hata\int_{\txi-R}^{\txi}\int_{\txi}^{\txi+R}
W(t-s)\,dtds}\},\\
\= const\puhw(\flh).
\eeqnn
To complete the proof of Theorem~\ref{theo:iph} we need the
following result.
\bprop
\label{prop:dsq}
There is a constant $C>0$ such that
$$\tau'(t)\geq C/t^2,\,\,\,\mbox{\rm for $t$ large.}$$
\eprop
{\bf Proof:} We consider the event $\lt$ that the $\te$-intervals
$\io$, $\ite$ containing the origin and the point $t$ respectively
are connected to each other but to no other $\te$-interval.
Condition on $\te$ and observe that the resulting measure is a
discrete FK measure. Follow the steps of \cite{kn:IN} to find
\beqnn
\ptw(\lt|\te)\ge (1-e^{-\a\intot})\\
&&\cdot e^{-\a\intoo}e^{-\a\intet}\\
&\equiv&f(\io,\ite).
\eeqnn
Now, there exist constants $0