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\title
Discontinuity of the Spin--Wave Stiffness in the two--dimensional XY Model
\endtitle
\bigskip
\bigskip
\leftheadtext { L. Chayes}
\rightheadtext\nofrills {SW Stiffness in the $2d$--$XY$--Model}
\author
\hbox{\hsize=5.4in
\vtop{\centerline{L. Chayes}
\centerline{{\it Department of Mathematics}}
\centerline{{\it University of California, Los Angeles}}}}
\endauthor
\address
L. Chayes
\hfill\newline
Department of Mathematics
\hfill\newline
University of California
\hfill\newline
Los Angeles, California 90095-1555
\endaddress
\email
lchayes\@math.ucla.edu
\endemail
\keywords
Kosterlitz--Thouless transition, Wolff representation
\endkeywords
\bigskip
\abstract
\baselineskip = 20pt
Using a graphical representation based on the Wolff algorithm,
the (classical)
$d$--dimensional $XY$ model and some related spin--systems are studied.
It is proved that in
$d=2$, the predicted discontinuity in the spin--wave stiffness indeed
occurs. Further, the critical properties of the spin--system are related to
percolation properties of the graphical representation. In particular, a
suitably defined notion of percolation in the graphical representation
is proved to be the necessary and sufficient condition for positivity
of the spontaneous magnetization.
\endabstract
\endtopmatter
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\subheading
{Introduction} Among the most noted early achievements of the
renormalization group was the analysis of the defect (vortex) unbinding
transition in two--dimensional systems with Abelian symmetries \cite {B},
\cite {KT}. The definitive (and experimentally accessible) prediction
of this analysis is the occurrence of discontinuities at the edge of the
low--temperature phase. Such a phenomenon is remarkable in and of the fact
that the transition itself, by any other criterion is continuous. In the
language of superfluid systems, the above mentioned discontinuity occurs
in the superfluid density; for spin--systems, it is the spin--wave
stiffness; sometimes known as the helicity modulus. This prediction has
been born out by
theoretical, numerical and experimental (and analog/experimental)
tests; cf. the review articles \cite{N}
and \cite {M} and references therein. In this note, a complete mathematical
proof for the (classical) $2d$--$XY$ model is provided.
The method of proof employs the graphical representation -- or cluster
representation -- due to Wolff \cite{W}. (More precisely, the graphical
representation that is implicit in the Wolff algorithm.) The importance of
understanding this representation was stressed in \cite{PS} and this
representation was exploited in
\cite{A} in the study of the ``vortex--free'' $XY$ model. In \cite{CM$_{\text{II}}$},
critical properties of the spin--system and the graphical representation
were shown to be related. Here some characterizations are presented: Up to
constant
factors constant factors the magnetization in the spin--system is equal to the
percolation density in the Wolff--representation and the susceptibility is
``equal'' to the average size of the connected clusters. Of more immediate
relevance is the fact that the spin--wave stiffness tested in finite volume
is directly related to crossing probabilities in the graphical
representation and in particular, a small stiffness implies and is implied
by a small crossing probability. If this probability is ``too small'' then,
using elementary rescaling ideas borrowed from rigorous percolation theory,
it tends to zero exponentially at larger scales (which furthermore implies
exponential decay of correlations). Thus, the stiffness is either uniformly
positive at all scales or it is zero. The existence of a low temperature
phase with power law decay of correlations (proved in \cite{FS}) thus implies
a discontinuity of the stiffness at a positive temperature.
A related class of problem -- in the sense that the RG equations turn out to
be nearly identical -- are the one dimensional long--range discrete models,
e\.g\. $1/r^2$ Ising model. In this context, the magnetization at the critical
point plays the role of the Spin--Wave stiffness and it was predicted in
\cite{T}
to be discontinuous at
$T_c$ (the Thouless effect). This was rigorously established in \cite{ACCN}
by vaguely similar methods: graphical representations and ``real space
renormalization group'' inequalities. However, in the rigorous as well as
in the renormalization group arenas the deeper relationship
between these two problems is still unclear.
The remainder of this paper is organized along the following lines: Below, the
definition of the spin--wave stiffness used in this note is provided. In
the next
section, the Wulff representation is developed. Here, the key relationship
between the
spin--wave stiffness and appropriate crossing probabilities is derived.
This will be
followed by the section in which the main result -- the discontinuity of
the spin--wave
stiffness in $d=2$ -- is established. In the final section, some auxiliary
results will
be stated (but not proved) and in the appendix, complete proofs of these
results and
various properties of the Wolff representation will be provided.
\subheading {Spin--Wave Stiffness}
The spin--wave stiffness is the appropriate notion of a leading
correction to the bulk free energy when the surface tension is zero.
It may be defined as follows: Consider a regular finite volume $d$--
dimensional shape $V$ with two (separated) boundary components.
Let $V_L$ denote the lattice approximation to this shape at scale $L$
i.e. the intersection of $\Bbb Z^d$ with the image of V that has been
uniformly scaled by a factor of $L$. The general strategy is to
consider the difference in free energies of the system with uniform
boundary conditions and twisted boundary conditions on $V_L$. For
typical ferromagnetic spin--systems, ``uniform'' means that all the
boundary spin are aligned and ``twisted'' means that the two
boundary components are individually aligned but are anti--parallel.
For the purposes of this note, the above is sufficient.
In more generality, one may consider cylindrical or
even toroidal geometries which, in other contexts, are
arguably a better choice. C\.f\. the discussion in \cite{FJB}.
Modulo constants, for $L\gg1$, the log of the ratio of the
twisted and uniform partition functions serves to ``define'' the
spin--wave stiffness $K$. Let us proceed more cautiously and define
this
ratio as $e^{-\beta K_L(V,\beta)g(V)L^{d-2}}$ with $\beta$ the
inverse
temperature and $g(V)$ a geometric constant (which is essentially the
capacitance) to
be described below. A spin--wave stiffness may be defined via the limiting
behavior of
$K_L(V,\beta)$; since there is no general proof that the limit exists,
let alone is independent of $V$, the matter will be left as it stands.
Suffice it to say that if for any $V$ of a roughly annular shape,
$K_L(V,\beta)$ tends to zero then all possible $K_L$'s tend to zero (And
similarly, in $d >2$, if any $K_L(V,\beta)L^{d- 2}\to 0$, then they all
do.)
Let us tend to the constant $g(V)$. The models under consideration will
have spins with bounded values in $\Bbb R^2$; let us assume that the bound
is one. Furthermore (and here rather vaguely) let us assume that if the
Hamiltonian is expressed in ``deviation'' variables, the leading
non--constant term is quadratic with coefficient 1/2.
Let
$\phi_V$ be the solution to Laplaces' equation with boundary values
$\pm 1$ on the two components. Then
$$
g = \int_V|\nabla \phi_V|^2d^dx.
\tag 1
$$
With this definition, it is an elementary exercise to show, for the
standard
$XY$ model on $\Bbb Z^d$ (e.g. as defined in Equation (3.a) with unit
couplings between neighboring sites) that
$$
\lim_{L\to\infty} \lim_{\beta \to \infty} K_L(V,\beta) = 1.
\tag 2
$$
In this paper, all that is needed is the simplest of annular shapes:
Consider, in $d = 2$, the square of size 3,
$S_{(3)} = \{x_1,x_2\mid -\frac 32 \leq x_1 \leq + \frac 32, -\frac 32
\leq x_2 \leq +
\frac 32 \}$ and $S_{(1)}$ defined accordingly. The shape of interest is
$A\equiv S_{(3)}\setminus S_{(1)}$. In $d > 2$ the corresponding
generalization is used: a hypercube of side 3 with the central hypercube of
side 1
removed.
\subheading
{The Representation: Notation and Definitions}
Although the primarily concern is with the behavior of uniform systems on
regular $d$--dimensional lattices, the cluster representation is just as easily
formulated on an arbitrary (finite) graph. Indeed, there is a need for these
sorts of generalities in order to formulate the representation of these
systems in the presence of boundary conditions. Thus, let
$\Cal G$ denote finite graph with sites $\Bbb S_{\Cal G}$ and bonds $\Bbb
B_{\Cal
G}$. For each $i\in\Bbb S_{\Cal G}$, let $\vec s_i$ denote a $2d$ spin of
length one
and for each $\langle i,j \rangle\in \Bbb B_{\Cal G}$, let $J_{i,j} > 0$
denote the
couplings. The $XY$--Hamiltonian is given by
$$
H^{XY}_{\Cal G} = -\sum_{\langle i,j \rangle}J_{i,j}\vec s_i\cdot\vec s_j.
\tag 3.a
$$
Writing $a_i$ and $b_i$ for the magnitude of the $Y$ and $X$ components
respectively, (here $0 \leq a_i, b_i \leq 1$) and allowing $\tau_i = \pm 1$ and
$\sigma_i = \pm 1$, $H^{XY}_{\Cal G}$ may be read
$$
H^{XY}_{\Cal G} = -\sum_{\langle i,j \rangle}J_{i,j}
[a_ia_j\tau_i\tau_j + b_ib_j\sigma_i\sigma_j].
\tag 3.b
$$
For most of what remains, we will have little use for the specifics of the
$XY$--model itself. Indeed, we might just as well allow the right hand side of
Equation (3.b) to define the model along with some constraint on the $(a_i,b_i)$
that makes one a decreasing function of the other and an {\it a priori}
distribution,
$f_i$, for the $b_i$ (which need not be continuous). For the purposes of
brevity we
will, however assume complete symmetry between the $a$'s and the $b$'s and
that these
objects are bounded.
The idea behind the Wolff representation is to develop one (or both) of the
Ising
systems in an FK \cite{FK} random cluster representation.
\footnote {In typical simulations one
does this for only one of the Ising variables -- as will most often be the
case here
-- but picking a direction at random. However, as argued in
\cite{CM$_{\text{II}}$}, it may be advantageous to use the full expansion
in conjunction
with the {\it Invaded Cluster} algorithm.} The partition is given by the usual
$$
Z(\Cal G, \underline J,\beta) =
\sum_{\underline\sigma,\ \underline\tau}\int\prod_i df_i(b_i)
e^{\beta\sum_{\langle i,j \rangle}J_{i,j}[a_ia_j\tau_i\tau_j +
b_ib_j\sigma_i\sigma_j]}.
\tag 4
$$
In the above, $\underline\sigma$ and $\underline \tau$ are notation for the
Ising configurations on $\Cal G$ while $\underline J$ denotes the
collection of couplings. And similarly,
$\underline a$ and
$\underline b$ will be notation for configurations of the magnitude of the spin
components with the $a_i$ understood to be a function of the $b_i$.
Let us start by writing the Ising portion of the Hamiltonian in Potts form:
$\sigma_i\sigma_j = 2\delta_{\sigma_i\sigma_j} - 1$, etc. For fixed
$\underline b$, let us trace over the
$\underline
\tau$ variables and then trade the $\underline \sigma$ degrees of freedom
for those
of an FK expansion. Thus let $Z^I_{\underline a}(\beta)$ denote the Ising
partition
function according to an Ising Hamiltonian written in Potts form:
$$
H^I_{\underline a} = -\sum_{\langle i,j \rangle}
J_{i,j}\ a_ia_j(\delta_{\tau_i\tau_j} - 1)
\tag 5.a
$$
$$
Z^I_{\underline a}(\beta) = \sum_{\underline \tau}e^{-\beta H^I_{\underline a}}.
\tag 5.b
$$
Here, the dependence of these quantities on $\Cal G$,
and the $(\underline J)$ has been temporarily suppressed. Unfortunately, the
relevant $\beta$ is twice what appears in
Equation (5.b) so to avoid confusion, this parameter
will stay with us. Performing the afore mentioned trace and expansion, we
arrive
at the weights (or density function) of a joint distribution for the
$\underline b$ and bond configurations $\omega\subset \Bbb B_{\Cal G}$:
$$
V^W_\beta(\underline b, \omega) = Z^I_{\underline a}(2\beta)
\prod_{\langle i,j \rangle}e^{\beta J_{i,j}(a_ia_j + b_ib_j)}\
W_{\underline b;2\beta}(\omega)
\tag 6
$$
where $W_{\underline b;2\beta}(\omega)$ are the usual ($q = 2$) FK weights with
couplings $J_{i,j}b_ib_j$ and inverse temperature $2\beta$:
$$
W_{\underline b;2\beta}(\omega) =
q^{C(\omega)}\prod_{\langle i,j \rangle \in\omega}p_{i,j}
\prod_{\langle i,j \rangle \notin\omega}(1 - p_{i,j}),
\tag 7
$$
$p_{i,j} = 1 - e^{2\beta J_{i,j}b_ib_j}$ and $C(\omega)$ the number of
connected components of $\omega$. The measures defined by the weights in
Equation
(6) will be denoted by $\nu^W_\beta(-)$
Let us consider the two marginal distributions: (i) Integrate out the
$\underline
b$ degrees of freedom to obtain a measure on the bond configurations $\omega$.
These will be denoted by $\Bbb M_\beta(-)$ -- or $\Bbb M^*_{\beta,\Cal G \dots\
}(-)$, with $*$ signifying possible boundary conditions to be discussed
later. (ii)
Integrate out the $\omega$ degrees of freedom (i.e. skip the FK step and
trace the
$\underline \sigma$ variables). The associated density will be denoted by
$\rho_\beta(-)$ -- or $\rho^*_{\beta,\Cal G \dots\ }(-)$ when the need arises.
Finally, let us consider the conditional FK measures, $\mu^{FK}_{\underline
b}(-)$
determined by the weights in Equation (7). These distributions allow for a
convenient
decomposition of $\Bbb M_\beta(-)$
$$
\Bbb M_\beta(-) = \int_{\underline b}
d\rho_\beta(\underline b)\mu^{FK}_{\underline b}(-).
\tag 8
$$
Some immediate applications of these measures have been discussed in
\cite{A} and
\cite{CM$_{\text{II}}$}. For example, in the usual isotropic XY case, if
$T_{i,j}$ is the
(bond) event that $i$ is connected to $j$ then, e.g. in free boundary
conditions,
$$
2\Bbb M_{\beta, \Cal G}(T_{i,j}) \geq
\langle \vec s_i\cdot\vec s_j\rangle_{\beta, \Cal G}
\tag 9
$$
with $\langle - \rangle_{\beta, \Cal G}$ denoting expectation with respect
to the
canonical distribution. This has been supplemented by a
lower bound proportional to a power of $\Bbb M_{\beta, \Cal G}(T_{i,j})$. Here
we will obtain a lower bound of a constant times $\Bbb M_{\beta, \Cal
G}(T_{i,j})$. Of direct relevance to the present work is the following:
Let $K_L(A,\beta)$ denote the spin wave stiffness as discussed in the
introduction.
Explicitly, let $Z^{\imath^+o^+}(A_L, \beta)$ denote the partition function on
the annulus $A_L$ with boundary conditions obtained by setting all boundary
spins
on the inner boundary ($\imath$) and the outer boundary ($o$) to the
$X$--direction.
(Or, in the language of Equation (3.b), all the $b_i$'s are set to their
maximum values and $\sigma_i \equiv 1$ on the boundary.) Similarly let
$Z^{\imath^-o^+}(A_L,
\beta)$ be the
partition function for the setup in $A_L$ where the spins on the outer
boundary are pointing in the positive $X$--direction and the spins on the inner
boundary pointing in the negative $X$--direction. Thus
$$
e^{-\beta g(A)K_L(A,\beta)L^{d-2}}
\equiv Z^{\imath^-o^+}(A_L, \beta)/Z^{\imath^+o^+}(A_L, \beta).
$$
Concerning the ``$\imath^+o^+$'' system, it is clear that we can treat this
setup
along the lines already described: the boundary spins act as a single spin
albeit
with a concentrated distribution. Let us denote by
$\Bbb M^{\bold 1^{++}}_{\beta, A_L}(-)$ the bond measure associated with these
boundary conditions and let $T_{\imath,o}$ denote the event of a connection
between the inner and outer boundaries of $A_L$. The first claim is
\proclaim{Proposition 1}
$$
1 - Z^{\imath^-o^+}(A_L, \beta)/Z^{\imath^+o^+}(A_L, \beta) =
\Bbb M^{\bold 1^{++}}_{\beta, A_L}(T_{\imath,o}).
$$
In particular, the spin--wave stiffness is related in a simple way to the
probability of a connection between the boundary components of $A_L$.
\endproclaim
\demo{Proof}
As is well known, in random cluster measures corresponding to Potts systems with
spins on the boundary set to some fixed value, the weights for the graphical
configurations are given by the standard one with the interpretation that
$C(\omega)$ counts only the components that are disconnected from the boundary.
(Equivalently, up to an irrelevant constant, one counts all the sites that are
attached to the boundary as part of the {\it same} component.) Thus if we write
$$
Z^{\imath^+o^+}(A_L, \beta) =
\sum_{\omega}\int_{\underline b} dV^{W,\bold 1^{++}}_\beta(\underline b,
\omega),
\tag 10
$$
the sum contains terms both with and without connections between the boundary.
On the other hand, in an situation where two separate boundary components in the
Potts system are set to different values, the rule for counting clusters is the
same but now bond configurations containing connections between these
components are assigned zero weight. Thus for fixed $\underline b$, the
formula for
the Wolff weights $V^{W,\bold 1 ^{+-}}_\beta(\underline b, \omega)$
corresponding to
the twisted boundary condition is seen to be identical except for the proviso
that $\omega$ does not connect $\imath$ with $o$ -- and here these
configurations
are discounted. The desired result is established.
\qed
\enddemo
It is plausible that these measures enjoy various monotonicity properties
but in any case, this will not be easy to prove. In particular it
turns out that the joint measure is not strong FKG. What can be proved
is that for a certain class of boundary conditions -- that are called the
$\odot$--boundary conditions -- the $\rho$--measures {\it do} have the FKG
property.
The precise definition of a $\odot$--boundary condition is somewhat
intricate but this class
includes every boundary condition of physical interest where one could
expect the
FKG property to hold e\.g\. free, periodic and setting all the boundary
spins to the positive $X$--direction. Furthermore, among all boundary
specifications in the $\odot$--class, this latter mentioned is {\it maximal}
in the sense of FKG. The same dominance therefore holds over the
$\overline \odot$--class of specifications which is defined as superpositions of
specifications from the $\odot$--class. This larger class has the property that
its restrictions to smaller sets are also in the $\overline \odot$--class
relative
to the ``larger'' boundary component. The relevant consequences of the above is
summarized in the form of a Lemma:
\proclaim{Lemma 2} Let $\Cal G$ denote a graph. Then for every $\Bbb
L\subset \Bbb
S_{\Cal G}$, there is a class of specifications on $\Bbb L$ called the
$\overline \odot$--class such that: (1) If $\Bbb K \supset \Bbb L$ and $*$ is a
$\overline \odot$--specification on $\Bbb L$ then the restriction of the
various
measures, $\nu^{W,*}_\beta(-)$, $\Bbb M^*_{\beta,\Cal G}(-)$, etc. to the
complement
of $\Bbb K$ is itself a $\overline \odot$--class specification on $\Bbb K$.
(2)
Setting all spins of $\Bbb L$ to the $X$--direction constitutes a
$\overline \odot$--class specification on $\Bbb L$; this is denoted by the
$\bold 1^+$ boundary conditions on $\Bbb L$. If $*$ is any other $\overline
\odot$--specification on $\Bbb L$ then
$$
\Bbb M^{\bold 1^+}_{\beta, \Cal G}(-)
\underset\text{FKG} \to\geq
\Bbb M^{*}_{\beta, \Cal G}(-).
$$
\endproclaim
A proof (including relevant definitions) will be supplied in the appendix. The
important point is that among all possible relevant boundary conditions, on
$A_L$,
the one that maximizes the probability of $T_{\imath,o}$ is precisely
$\Bbb M^{\bold 1 ^{++}}_{\beta, A_L}(-)$.
\subheading{Main Results}
With the identity of Proposition 1 and the inequalities of Lemma
2, the main
argument reduces to a standard routine in percolation theory:
\proclaim{Theorem 3}
There is an $\epsilon_0 = \epsilon_0 (d)$ such that if for any
$L_0$,
$\Bbb M^{\bold 1 ^{++}}_{\beta, A_{L_0}}(T_{\imath,o}(L_0)) < \epsilon_0$ then
$$
\lim_{L\to\infty}\Bbb M^{\bold 1 ^{++}}_{\beta,
A_L}(T_{\imath,o}(L)) = 0.
$$
In particular, under these conditions, $\Bbb M^{\bold 1 ^{++}}_{\beta,
A_L}(T_{\imath,o})$ tends to zero exponentially fast in $L$.
\endproclaim
\demo{Proof}
Suppose that
$\Bbb M^{\bold 1 ^{++}}_{\beta, A_{L_0}}(T_{\imath,o}(L_0)) \leq
\epsilon < \epsilon_0$ with
$\epsilon_0$ to be specified below. Let $N \gg 1$ and consider
the event
$T_{\imath,o}(NL_0)$ for the annulus $A_{NL_0}$. Divide
$A_{NL_0}$ into a grid of scale $L_0$ so as to have the appearance of
an
$A_N$ on the large scale lattice. If
$\Cal P: \imath \to o$ is a path in $A_{NL_0}$, each ``site'' on the
large scale
lattice that is visited by $\Cal P$ has achieved an event like
$T_{\imath,o}(L_0)$
-- with the possible exception of the sites next to the boundary. Let us
denote a ``site''
of $A_N$ to be ``occupied'' if the analog of the $T_{\imath,o}(L_0)$
occurs and vacant otherwise. For the sake of being definitive, let us deem
all sites
neighboring the boundary of $A_N$ to be occupied. It is clear that
$\Bbb M^{\bold 1 ^{++}}_{\beta, A_{NL}}(T_{\imath,o}(NL))$ does not
exceed the
probability of a connection between the $\imath$ and the $o$ of
$A_N$ in the
large--scale problem.
Now of course, these site variables are not independent.
However let us regard a
sublattice consisting of a fraction -- $1/3^d$ -- of these sites as
sitting in the center of a translate of
$A_{L_0}$ with these translates of $A_{L_0}$ situated in
such a way that they tile the lattice. With the maximizing boundary
conditions on these
translates of
$A_{L_0}$, the sublattice of site occupation variables {\it are}
independent and
their probability is bounded above by $\epsilon$. There are $3^d$
possible ways to
design such sublattices (depending on which sites are chosen as the
centers) such
that each site of $A_N$ is a central site on one of these $3^d$
sublattices. Thus an
``occupied cluster'' of consisting of $K$ interior sites of $A_N$ must have
at least
$1/3^d$ of these sites on (at least) one of the sublattices. Therefore, the
probability of a given occupied cluster with $K$ interior sites is less than
$(\epsilon)^{K/3^d}$. The minimum sized cluster that permits the
possibility of an
actual path is essentially $N$ and
there are only of
the order of $N^{d-1}$ starting points on the inner boundary. Hence
$$
\Bbb M^{\bold 1 ^{++}}_{\beta, A_NL}(T_{\imath,o}) \leq
C_2N^{d-1}\sum_{K > N - C_1}[\lambda(d)\epsilon^{1/3^d}]^{K}
\tag 11
$$
with $C_1$ and $C_2$ constants of the order of unity and
$\lambda(d) < (d-1)$ the
connectivity constant. It is evident that if $\epsilon < \epsilon_0 =
1/\lambda^{3^d}$,
the stated result follows.
\qed
\enddemo
\proclaim{Corollary}
For the 2d models, the spin--wave stiffness does not go
continuously to zero at any temperature.
In any dimension, if the conditions of Theorem 3 hold for some finite
$L_0$, there
is exponential decay of correlations in any limiting $\overline \odot$--state.
\endproclaim
\demo{Proof}
According to Lemma 2, the $\overline \odot$--state that maximizes
the probability
of $T_{i,j}$ is always the $\bold 1^+$--state. Under the conditions
stated in
Theorem 3, it is clear that the probability of $T_{i,j}$ tends to zero
exponentially in any limiting $\overline \odot$--state. (Later we
will show that
under these conditions there is in fact a unique limiting $\odot$--state.)
Using a bound along the lines of
Equation (9), exponential decay for the 2--point function is readily
established:
The factor of 2 in this inequality is for the $X$ and
$Y$--component pieces of $\vec s_i \cdot \vec s_j$. Indeed, in {\it
any} boundary
condition $*$,
$$
\langle s_i^{[X]} s_j^{[X]} \rangle_{\beta, \Cal G}^*
\equiv \langle b_i\sigma_i b_j\sigma_j \rangle_{\beta, \Cal G}^*
\leq \Bbb M^*_{\beta, \Cal G}(T_{i,j})
\tag 12
$$
with connections through the boundary included in the definition of
$T_{i,j}$.
Since, among limiting $\overline \odot$--states this is maximized in
the
limiting $\bold 1^+$--state, the correlation among the $X$--
components goes to
zero. The correlations between the $Y$ components (in $\overline
\odot$--states)
would be maximized in the analog of the $\bold 1^+$--state and
hence, by the
symmetry between $X$ and $Y$ components, is also (in any
$\overline
\odot$--state) always bounded by the probability of $T_{i,j}$ in the
$\bold 1^+$--state. Thus we actually recover Equation (9) for the
$\bold
1^+$--states and the conclusion about exponential decay is
immediate.
The statement concerning the spin wave stiffness is a tautology,
however c.f.
Remark 2 below.
\qed
\enddemo
\remark{Remark 1} If $\beta_{c}$ is {\it defined} by the infimum
over temperatures at which $K_{\infty}(\beta)$ is zero, then, by an
obvious continuity argument, $K_{\infty}(\beta_c) > 0$ in $d = 2$.
For the $XY$--model, the results of \cite {FS} (concerning the existence of a
region of power law decay of correlations) rather easily imply that such a
discontinuity occurs at a finite $\beta$.
\endremark
\remark{Remark 2}
Starting with \cite {NK}, detailed renormalization group studies of this
``class''
of problems predicts a {\it universal} value of
$\beta_c K_{\infty}(\beta_c)$. Although the present derivation is far cry
from a
proof of any such statement, it is worth observing that the same set of
results proved
in Theorem 3
hold for a variety of models with ``O(2)'' characteristics -- e.g. the
$\Bbb Z_{4n}$--clock models -- using the {\it same} value of
$\epsilon_{0}$.
Thus we have a universal lower bound on $\beta_c K_{\infty}(\beta_c)$. This is
analogous to (and borrowed from) the current situation in percolation
theory: various
crossing probabilities -- even the one used here -- which at the critical point
are believed to converge to universal values at large
length scale, can at least be shown to satisfy uniform bounds with
universal constants.
\endremark
\subheading {Additional Results}
Some further results will be stated below but all the remaining proofs have been
relegated to the appendix.
The usual definition of {\it percolation} in correlated models
starts, in finite volume, with the probability of a connection to the
boundary in
the boundary conditions that optimize this probability. (C.f.
\cite{CM$_{\text{I}}$}, definition following Equation (II.11).) Here,
let us define:
\definition{Definition}
Let $\Lambda\subset\Bbb Z^d$ be a finite connected set that
contains the origin
and let $T_{0,\partial \Lambda}$ denote the event that the origin is
connected to the
boundary. Let
$$
\Pi_{\Lambda}(\beta) =
\Bbb M^{\bold 1 ^{+}}_{\beta, \Lambda}(T_{0,\partial \Lambda})
\equiv
\max_{{*\in\odot}}
\Bbb M^{*}_{\beta, \Lambda}(T_{0,\partial \Lambda})
\tag 13.a
$$
and
$$
\Pi_{\infty}(\beta) =
\lim_{\Lambda\nearrow \Bbb B^d}\Pi_{\Lambda}(\beta).
\tag 13.b
$$
(In light of Lemma 2, the existence of this limit is not hard to
establish.)
The actual {\it percolation} probabilities, denoted by $P$'s instead of
$\Pi$'s
is defined as in Equations (13) but with the maximum taken over
all boundary
conditions.
\enddefinition
\proclaim{Theorem 4}[A]
Let $m(\beta)$ denote the spontaneous magnetization. Then there
are finite
non--zero constants, $c_1$ and $c_2$ (that depend only on minor the details
of the model) such that
$$
c_2 \Pi_{\infty}(\beta) \leq m(\beta) \leq c_1 \Pi_{\infty}(\beta).
$$
[B]\ If $m(\beta) = 0$, there is a unique limiting $\odot$--state.
\endproclaim
\demo{Proof} A proof will be provided in the appendix.
\enddemo
\remark{Remark} The results concerning uniqueness are hardly an
improvement over the existing results which apply to most of these models
considered
here -- uniqueness among translation invariant
states when the magnetization vanishes \cite{MMPf}.
Of greater concern (to the author) is the connection between
phase transitions in the spin--systems and
percolation in the corresponding graphical representation. This is
further underscored by the final result:
\proclaim{Theorem 5}
Let $*$ denote any finite volume $\overline \odot$--measure or
infinite volume limit
thereof and let $\langle s_i^{[X]}s_j^{[X]} \rangle^{*}_{\beta} \equiv
\langle b_i\sigma_ib_j\sigma_j \rangle^{*}_{\beta}$ denote the
(untruncated)
correlation function for the $X$--components. Then,
$$
c_{1}^{2}\Bbb M_{\beta}^*(T_{i,j}) \geq
\langle s_i^{[X]}s_j^{[X]} \rangle^{*}_{\beta}
\geq c_{2}^{2}\Bbb M_{\beta}^*(T_{i,j})
$$
with $c_1$ and $c_2$ as in Theorem 4. In particular, if $m(\beta) =
0$ and
$\Cal X$ is defined by
$$
\Cal X(\beta) = \sum_j \langle s_0^{[X]}s_j^{[X]} \rangle^{}_{\beta}
$$
evaluated in the unique limiting $\odot$--state then
$$
c_{1}^{2}\Bbb E_\beta(|C_0|) \geq \Cal X(\beta)
\geq c_{2}^{2}\Bbb E_\beta(|C_0|)
$$
where $\Bbb E_\beta(|C_0|)$ is the expected size of the connected
cluster of the
origin in the graphical representation.
\endproclaim
\demo {Proof} The upper bound for the correlation function was
derived in
\cite{A}, the rest will be proved in the appendix.
\enddemo
Theorems 4 and 5 provide complete justification for the use of
``percolation''
as the critical criterion in the Wolff algorithm \cite{W} or the
Invaded Cluster
version of this algorithm \cite{CM$_{\text{II}}$}.
\subheading
{Appendix: Monotonicity Properties of the Wolff Measures} For reasons
that are primarily of a technical nature, this appendix will be
concerned with generalizations of the types of models already
discussed (even though such generalizations are ``unphysical'' from
the perspective of systems with $O(2)$ symmetry). Thus consider a
graph $\Cal G$ and let $H_{\Cal G}$ denote the Hamiltonian
$$
H_{\Cal G} =
-\sum_{\langle i,j \rangle}
(K_{i,j}a_ia_j\tau_i\tau_j + J_{i,j}b_ib_j\sigma_i\sigma_j)
\tag A.1
$$
with $K_{i,j}, J_{i,j} \geq 0$. As discussed previously, the single site {\it a
priori} measures and the range of the $a_i$ and $b_i$ as well as the constraint
between them may be regarded as fairly arbitrary: It is enough to assume
that they
are non--negative, uniformly bounded and that $a_i$ goes down when $b_i$
goes up.
Finally, it will be assumed that if $b_i$ achieves its maximum value then the
corresponding $a_i$ is zero. Most of these assumptions can be removed but
with an
unreasonable cost of labor and space. To
avoid spurious notational provisos, let us assume that the single site
measures are
discrete. (Indeed, since we will always start in finite volume, the
``general'' case
can be recovered from the discrete by a limiting procedure.) Thus we let
$\rho^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$ denote the
measure on configurations $\underline b = (b_i\mid i\in\Bbb S_{\Cal G})$
defined by
the weights
$$
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(\underline b)
= Z^I_{\underline a,\underline K}(2\beta)
Z^I_{\underline b,\underline J}(2\beta)
\prod_{\langle i,j \rangle \in \Bbb B_{\Cal G}}
e^{\beta[K_{i,j}a_ia_j + J_{i,j}b_ib_j]}
\prod_{i \in \Bbb S_{\Cal G}}f_i(b_i)
\tag A.2
$$
where $f_i(b_i)$ is the {\it a priori} probability of $b_i$, $\underline f
\equiv
(f_i\mid i\in\Bbb S_{\Cal G})$,
$\underline K\equiv (K_{i,j}\mid \langle i,j \rangle \in \Bbb B_{\Cal G})$
and all
other notation has been defined elsewhere.
\proclaim{Proposition A.1} The measures
$\rho^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$
are (strong) FKG.
\endproclaim
\demo{Proof}
Let $\underline b$ denote a fixed configuration and let $u$ and $v$ denote any
distinct pair of sites in $\Cal G$. Let $\Delta_u > 0$ and $\delta_u$
denote the
configuration that is zero except at the site $u$ where it is equal to
$b_u + \Delta_u$. Similarly for $\delta_v$ with some $\Delta_v > 0$. It
may as well
be assumed that $f_u(b_u + \Delta_u)$ and $f_u(b_v + \Delta_v)$ are positive.
Thus, the configuration $\underline b\lor \delta_u\lor \delta_v$ has been
``raised''
at the sites $u$ and $v$ while $\underline b\lor \delta_u$ has been raised
only at
$u$, etc. Similarly, if $\Gamma_u$ is the corresponding amount that $a_u$
has to
be lowered (determined by the constraint at $u$, the value of $b_u$ and
$\Delta_u$)
then let $\underline a\land \gamma_u$ denote the configuration of
$\underline a$'s
that has been lowered at $u$ etc. (Formally, $\gamma_u$ is $a_u -
\Gamma_u$ at the
site $u$ and infinite elsewhere.) To prove the desired claim, it is
sufficient (and
necessary) to show
$$
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}
(\underline b\lor \delta_u\lor \delta_v)
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(\underline b)
\geq
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}
(\underline b\lor \delta_u)
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}
(\underline b\lor \delta_v)
\tag A.3
$$
After cancellation of all manifestly equal terms (assumed non-zero) the
purported
inequality boils down to
$$
\align
[e^{\beta\Delta_u\Delta_v}
Z^I_{\underline b\lor \delta_u\lor \delta_v,\underline J}(2\beta)Z^I_{\underline
b,\underline J}(2\beta)] (e^{\beta\Gamma_u\Gamma_v}
Z^I_{\underline a\land \gamma_u\land\gamma_v,\underline K}(2\beta)
Z^I_{\underline a,\underline K}(2\beta)) \geq \\
\geq
[Z^I_{\underline b\lor \delta_u,\underline J}(2\beta)
Z^I_{\underline b\lor \delta_v,\underline J}(2\beta)]
(Z^I_{\underline a\land \gamma_u,\underline K}(2\beta)
Z^I_{\underline a\land\gamma_v,\underline K}(2\beta)).
\tag A.4
\endalign
$$
It is claimed that the term in the square bracket on the rhs does not exceed the
corresponding term on the left and similarly for the terms in the round
bracket.
Indeed, a moments reflection will show that these two inequalities are of an
identical form. Let us therefore focus on proving
$$
[e^{\beta\Delta_u\Delta_v}
Z^I_{\underline b\lor \delta_u\lor \delta_v,\underline J}(2\beta)Z^I_{\underline
b,\underline J}(2\beta)] \geq
[Z^I_{\underline b\lor \delta_u,\underline J}(2\beta)
Z^I_{\underline b\lor \delta_v,\underline J}(2\beta)].
\tag A.5
$$
and the same derivation will hold for the $\underline a$--pairs.
It turns out that the derivation is far easier without the annoyance of the
$\Delta_u\Delta_v$ cross terms. Let us thus define
$$
H^{(0)} = - \sum_{\langle i,j \rangle}J_{i,j}
(\delta_{\sigma_i,\sigma_j} -1)b_ib_j,
\tag A.6a
$$
$$
H^{(U)} = - \sum_{\langle i,u \rangle}J_{i,j}
(\delta_{\sigma_i,\sigma_u} -1)\Delta_ub_i,
\tag A.6b
$$
and similarly for $H^{(V)}$. In these terms
$Z^I_{\underline b\lor \delta_u\lor \delta_v,\underline J}(2\beta)$
is given by
$$
Z^I_{\underline b\lor \delta_u\lor \delta_v,\underline J}(2\beta)
= Tr[e^{-2\beta H^{(0)}}e^{-2\beta H^{(U)}}e^{-2\beta H^{(V)}}
e^{2\beta J_{u,v}\Delta_u\Delta_v(\delta_{\sigma_u,\sigma_v} -1)}].
\tag A.7
$$
To get rid of the cross terms, it will be shown that
$$
\align
e^{\beta\Delta_u\Delta_vJ_{u,v}}
&Tr[e^{-2\beta H^{(0)}}e^{-2\beta H^{(U)}}e^{-2\beta H^{(V)}}
e^{2\beta J_{u,v}\Delta_u\Delta_v(\delta_{\sigma_u,\sigma_v} -1)}] \geq\\
\geq
&Tr[e^{-2\beta H^{(0)}}e^{-2\beta H^{(U)}}e^{-2\beta H^{(V)}}].
\tag A.8a
\endalign
$$
Indeed, dividing both sides of the purported inequality (A.8a) by the right hand
side and denoting by $\Bbb E^I_{H,\beta}(-)$ the expectation with respect to the
Ising Hamiltonian $H$ at inverse temperature $\beta$, the desired (8.Aa) reads
$$
e^{\beta\Delta_u\Delta_vJ_{u,v}}
\Bbb E^I_{H^{(0)} + H^{(U)} + H^{(V)},2\beta}
(e^{2\beta J_{u,v}\Delta_u\Delta_v(\delta_{\sigma_u,\sigma_v} -1)})
\geq 1.
\tag A.8b
$$
Expanding the integrand in the usual FK fashion, this reduces to showing that
$$
e^{-\beta\Delta_u\Delta_vJ_{u,v}} +
2\text{sh}(\beta\Delta_u\Delta_vJ_{u,v})
\Bbb E^I_{H^{(0)} + H^{(U)} + H^{(V)},2\beta}(\delta_{\sigma_u,\sigma_v}) \geq 1
\tag A.8c
$$
Here is one of the few places where the fact that the underlying model has an
Ising structure is used: $\Bbb E^I_{H,\beta}(\delta_{\sigma_i,\sigma_u})
\geq 1/2$ so the
left hand side of (A.8) is at least as big as ch$\beta\Delta_u\Delta_vJ_{u,v}$.
For the remainder of the proof, it might just as well be assumed that the
underlying
model is the
$q$--state Potts model.
The remainder of this proof reduces to showing
$$
\Bbb E^I_{H^{(0)} + H^{(U)},2\beta}(e^{-2\beta H^{(V)}}) \geq
\Bbb E^I_{H^{(0)},2\beta}(e^{-2\beta H^{(V)}}).
\tag A.9
$$
This is, very similar to the sorts of inequalities that were established in
\cite{C} so here the derivation will be succinct. Let
$\epsilon_{i,v} = 1 - e^{2\beta J_{i,v}b_i\Delta_v}$ and let $\Cal N_v$
denote the
collection of sets in $\Bbb S_{\Cal G}$ each of which contains $v$ and some
subset
of the sites in $\Cal G$ that are connected to $v$. Expanding
$e^{-2\beta H^{(V)}}$ in the usual FK fashion, it is seen that
$$
e^{-2\beta H^{(V)}} = \sum_{\Cal F \in \Cal N_v}r_{\Cal
F}\delta_{\sigma_{\Cal F}}
\tag A.10
$$
with $r_{\Cal F} = \prod_{i\in \Cal F}\epsilon_{i,v}\prod_{j\notin \Cal F}
(1 - \epsilon_{j,v})$ and where $\delta_{\sigma_{\Cal F}}$
is one if all the spins in $\Cal F$ agree and zero otherwise. However,
using an FK
expansion of the $q$--state Potts system with Hamiltonian $H$, it is not hard to
show
$$
\Bbb E^I_{H,\beta}(\delta_{\sigma_{\Cal F}}) =
\Bbb E^{FK\ (q = 2)}_{H,\beta}((\frac 1q)^{C_{\Cal F} -1})
\tag A.10
$$
where $C_{\Cal F}$ is the number of connected components of the set $\Cal F$.
This is the expectation of an FKG increasing function and thus the desired
inequality follows -- term by term -- from the fact that random cluster
model that
comes from the ``bigger'' Hamiltonian (i.e. $H^{(0)} + H^{(U)}$) is FKG
dominant.
\qed
\enddemo
\proclaim{Corollary I} Consider two systems on the same graph $\Cal G$
with parameters $\underline J$, $\underline J'$ and single site measures
determined
by the collections $\underline f$ and $\underline f'$ respectively.
Suppose that
$\underline J \succ \underline J'$ meaning that for each
$\langle i,j \rangle \in \Bbb B_{\Cal G}$, $J_{i,j} \geq J'_{i,j}$ and further
suppose that $\underline f \succ \underline f'$ in the sense that for each $i$,
$f_i(b_i)/f'_i(b_i)$ is an increasing function of $b_i$. Then
$$
\rho^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)
\underset\text{FKG} \to\geq
\rho^{\underline J',\underline K,\underline f'}_{\beta,\Cal G}(-)
$$
\endproclaim
\demo{Proof}
This is an immediate consequence of the FKG properties of these measures and the
previous derivation. First, if $f' \prec f$, then
$$
\prod_{i\in \Bbb S_{\Cal G}}f_{i}(b_i) =
[\frac {\prod_{i\in \Bbb S_{\Cal G}}f_{i}(b_i)}{\prod_{i\in \Bbb S_{\Cal
G}}f'_{i}(b_i)}]
\prod_{i\in \Bbb S_{\Cal G}}f'_{i}(b_i)
\tag A.11
$$
so the $\underline f$--weights are of the form [increasing function]$\times$
$\underline f'$--weights. To establish the desired result for
$\underline J \succ \underline J'$ it is sufficient to consider one bond at
a time.
Thus let $\langle u,v \rangle \in \Bbb B_{\Cal G}$ and suppose that
$J_{u,v} = J_{u,v}' + L_{u,v}$ (with $L_{u,v} > 0$) and all other $J$'s equal.
Then
$$
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(\underline b)/
R^{\underline J',\underline K,\underline f}_{\beta,\Cal G}(\underline b)
= e^{\beta L_{u,v}b_ub_v}
\Bbb E^I_{H^I_{\underline b},2\beta}
[e^{2\beta L_{u,v}b_ub_v(\delta_{\sigma_u,\sigma_v} -1)}]
\tag A.12a
$$
where the Ising Hamiltonian $H^I_{\underline b}$ was defined in Equation (5.a)
-- and the $\underline J$ dependence has been suppressed.
After a few manipulations along the lines of those in the previous proposition,
Equation (A.12a) reduces to
$$
R^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(\underline b)/
R^{\underline J',\underline K,\underline f}_{\beta,\Cal G}(\underline b)
= \text{ch}(\beta L_{u,v}b_ub_v) + \text{sh}(\beta L_{u,v}b_ub_v)
\Bbb E^{FK\ (q = 2)}_{H^I_{\underline b},2\beta}(\Cal X_{T_{u,v}})
\tag A.12b
$$
where $\Cal X_{T_{u,v}}$ is indicator of the event that $u$ is connected to
$v$. The
sines and cosines are manifestly (non--negative) increasing functions of
$\underline
b$, while the random cluster term is the expectation of a {\it positive}
event and is
therefore an increasing function of all couplings in the Hamiltonian --
including
the
$\underline b$'s
\qed
\enddemo
Let us now turn to a discussion of boundary conditions. Let $\Cal G$
denote a graph and let $\Bbb L \subset \Bbb S_{\Cal G}$. The starting
point will be
the consideration of conditional measures for
$\nu^{W \underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$ -- the
measures
corresponding to the weights in Equation (6) cast in the more general
framework --
subject to specifications on $\Bbb L$ and the consequence of these
specifications
on the $\underline b$ marginals. A specification $*$ will be called a $\tilde
\odot$--specification if (i) the values
$(b_i\mid i\in \Bbb L)$ are specified: $b_i = b^*_i\ ; i\in \Bbb L$ and
(ii) $\Bbb
L$ is divided into disjoint components $\ell^*_1, \ell^*_2, \dots \ell^*_k$ such
that the counting rule in the FK expansion deems all the sites in
and connected to each $\ell^*_n$ to be part of the same cluster.
\remark{Remark}
Back in the spin--system, one interpretation of a
$\tilde\odot$--specification is
obvious: having determined the $b_i$ on $\Bbb L$, the signs of the
$X$--components
of the spins -- the $\sigma_i$'s -- are locked together within each
component and
they take on both values with equal probability. On the other hand, the same
graphical weights emerge if one (and only one) of the components is deemed to
represent spins pointing in the positive
$X$--direction. The reader is cautioned that at this stage, the
signs of the $Y$ components of the boundary spins still have all their
{\it a priori} degrees of freedom.
There is a natural partial order on
the set of all possible $\tilde \odot $--specifications:
$* \succ *'$ if (1) $\Bbb L \supset \Bbb L'$ and each $b_i$ on
$\Bbb L \setminus \Bbb L'$ is set to the maximum value, (2) each $b_i^* \geq
b_i^{*'}$, $i\in\Bbb L \cap \Bbb L'$ and (3), the components of $*$,
$\ell^*_1, \ell^*_2, \dots \ell^*_k$ ``contain'' the $*'$--components
$\ell^{*'}_1, \ell^{*'}_2, \dots \ell^{*'}_{k'}$ in the sense that if
$\ell^{*'}_{j'}\cap\ell^{*}_{j}\neq \emptyset$ then
$\ell^{*'}_{j'}\subset\ell^{*}_{j}$. The following is easily seen:
\proclaim{Corollary II} If $*$ is a $g$--specification and
$\rho^{* \underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$
is the associated measure on the remaining $\underline b$'s then
$\rho^{* \underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$
is (strong) FKG. Furthermore if $*\succ *'$ in the sense described above,
$\underline J \succ \underline J'$ and $\underline f \succ \underline f'$ then
$$
\rho^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)
\underset\text{FKG} \to\geq
\rho^{*'\underline J',\underline K,\underline f'}_{\beta,\Cal G}(-).
$$
\endproclaim
\demo{Proof}
The above is clear given the following mechanism to create a
$\tilde \odot$--specification:
to fix the values of $b_i$ on $\Bbb L$, concentrate the {\it a priori} measures.
To lock the components, introduce artificial $J$--type couplings between
all pairs
of sites in a given component and send these couplings to infinity; the desired
measure is recovered in the limit. If
$*\succ *'$ this procedure involves higher $J$'s and higher $b$'s.
\qed
\enddemo
\proclaim{Proposition A.2}
Let $\nu^{W\ *\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$ and
$\nu^{W\ *'\underline J',\underline K,\underline f'}_{\beta,\Cal G}(-)$
denote two Wolff measures with all primed quantities below unprimed
quantities in
the sense described. Let $\Bbb M^{*\underline J,\underline K,\underline
f}_{\beta,\Cal G}(-)$ and $\Bbb M^{*'\underline J',\underline K,\underline
f'}_{\beta,\Cal G}(-)$ denote the corresponding bond measures. Then
$$
\Bbb M^{*\underline J,\underline K,\underline
f}_{\beta,\Cal G}(-)
\underset\text{FKG} \to\geq
\Bbb M^{*'\underline J',\underline K,\underline
f'}_{\beta,\Cal G}(-)
$$
\endproclaim
\demo{Proof} Let $\Cal A$ denote an increasing bond event. Let us write as in
Equation (8)
$$
\Bbb M^{*\underline J,\underline K,\underline
f}_{\beta,\Cal G}(\Cal A) =
\sum_{\underline b}\rho^{*\underline J,\underline K,\underline
f}_{\beta,\Cal G}(\underline b)
\mu^{FK*}_{\underline J,\underline b}(\Cal A)
\tag A.13
$$
and similarly for
$\Bbb M^{*'\underline J',\underline K,\underline f'}_{\beta,\Cal G}(\Cal A)$.
The desired result follows immediately from the FKG properties of the usual
random
cluster measures: both
$\mu^{FK*}_{\underline J,\underline b}(\Cal A)$ and
$\mu^{FK*'}_{\underline J',\underline b}(\Cal A)$ are increasing functions of
$\underline b$ and furthermore, if $* \succ *'$ and
$\underline J \succ \underline J'$ then
$\mu^{FK*}_{\underline J,\underline b}(\Cal A) \geq
\mu^{FK*'}_{\underline J',\underline b}(\Cal A)$.
\qed
\enddemo
Thus far, the $Y$ degrees of freedom have been
left completely unspecified. Now the same sorts of specifications will be
considered
for these objects and this defines a $\odot$--specification: In addition
to a $\tilde
\odot$ specification, $\Bbb L$ is divided into disjoint components
$\jmath_1, \dots \jmath_m$ on which the $\tau$--variables act in unison. A
recapitulation of the previous arguments yields:
\proclaim{Proposition A.2} Let $*$ denote a $\odot$ specification and let
$\rho^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$ denote the
corresponding measure. Then
$\rho^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$
if FKG. Further, if $* \succ *'$ meaning the same as above regarding the
$\underline J$'s, the $\underline f$'s and the $\ell$--components while
$\underline K' \succ \underline K$ and the $\jmath'_1, \dots, \jmath'_m$
contain the
$\jmath_1, \dots, \jmath_m$ then
$$
\rho^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)
\underset\text{FKG} \to\geq
\rho^{*'\underline J',\underline K',\underline f'}_{\beta,\Cal G}(-)
$$
and accordingly
$$
\Bbb M^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)
\underset\text{FKG} \to\geq
\Bbb M^{*'\underline J',\underline K',\underline f'}_{\beta,\Cal G}(-).
$$
In particular, the FKG maximizing boundary condition (on $\Bbb L$) in the
$\odot$--class is the $b_i$ set to the maximum value, $\sigma_i \equiv 1$
and the
$\jmath_1, \dots, \jmath_m$ being the individual sites of $\Bbb L$. The latter
is, of course automatic if $b_i$ maximized $\Rightarrow $ $a_i = 0$
\endproclaim
\demo{Proof}
Follows the lines of the previous arguments along with the observation that any
increasing function of $\underline a$ is a decreasing function of
$\underline b$.
\qed
\enddemo
Superpositions of $\odot$--specifications do not constitute a
$\odot$--class boundary condition nor, in general, are they FKG measures.
This is
the usual situation in ferromagnetic systems and is of no serious
consequence since
we have knowledge of the maximizing measure in the $\odot$--class. In any
case, let us
define the $\overline \odot$--class as that which consists of
superpositions from
the $\odot$--class. The following is pivotal:
\proclaim{Lemma A.3}
Let $\Bbb L \subset \Bbb S_G$ and let $*$ denote a $\overline
\odot$--specification
on $\Bbb L$. Let $\Bbb K \supset \Bbb L$ and consider
$\rho^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}
(-)_{||_{\Bbb S_\Cal G\setminus \Bbb K}}$, the restriction of
$\rho^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$
to the remaining sites.
Then this restricted measure is of the $\overline \odot$--class.
\endproclaim
\demo{Proof}
It is sufficient to discuss the case where $*$ is itself a pure
$\odot$--specification. Consider the full Wolff measures
$w^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$
on configurations $(\omega,\eta,\underline b)$
where $\omega$ and $\underline b$ are as have been described and
$\eta$ denotes configurations of FK bonds in the random cluster
expansion of the $\tau$--system. Thus, e.g. the
$\nu^{\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$ measures
are obtained
by integrating out
the $\eta$--bonds. Now, to study the restricted measure,
let us may condition on an $(\omega,\eta,\underline b)$
configuration on $\Bbb K$ and sum over all $\eta$--configurations
(and, if desired, $\omega$--configurations) pertaining to the bonds of
$\Bbb S_\Cal G\setminus \Bbb K$. Having done so, a sum must be performed
over all the
external configurations with the appropriate weights assigned by
$w^{*\underline J,\underline K,\underline f}_{\beta,\Cal G}(-)$.
But, since $*$ is a $\odot$--specification, it is clear that each
$(\underline \omega,\underline \eta,\underline b)$
configuration on $\Bbb K$ provides a $\odot$--specification on
$\Bbb S_\Cal G\setminus \Bbb K$: Indeed, the $b$--values are fixed, the
components $\ell_1, \dots \ell_k$ are just the $\omega$--components while
the $\eta$--components constitute the $\jmath_i, \dots, \jmath_m$.
\qed
\enddemo
It is now straightforward to establish the various results claimed in
Theorems 4 and 5.
Indeed everything except the statements concerning uniqueness follow
immediately from
the existing machinery. Here, to simplify matters notationally, let us
again assume
that $\beta$, $\underline J$, and $\underline K$ and the graph $\Cal G$ are
fixed and
omit any further explicit reference. All of Theorem 5 amounts to the stated
bound of the correlation function in terms of the connectivity function.
Recalling
that in a $\odot$--state, the event $T_{i,j}$ includes connections via the
boundary
component, these bounds are easily proved:
\demo{Proof of Theorem 5}
If $*$ denotes a $\odot$ state, it is claimed that
$$
\langle s^{[X]}_{i}s^{[X]}_{j} \rangle =
\Bbb E^{*}_{\rho}[b_ib_j\mu^*_{\underline b}(T_{i,j})]
\tag A.14
$$
where $\Bbb E^{*}_{\rho}[-]$ denotes expectation with respect to the
$\rho^*(-)$ measure on the $\underline b$--configurations.
Indeed, fixing $\underline b$ and $\omega$, the Ising spins
are equal if $i$ is connected to $j$ -- either directly or via one of the
boundary
components -- and are uncorrelated with at least one of them having equal
probability of $\pm 1$ otherwise. Summing over all $\omega$ with $\underline b$
fixed and then summing over $\underline b$ yields the identity displayed in
Equation (A.14). But obviously, $b_i$ and $b_j$ cannot exceed their maximum
values and this provides the upper bound with $c_1$ equal any uniform bound
on these
values. On the other hand, $\mu^*_{\underline b}(T_{i,j})$, $b_i$ and
$b_j$ are all
increasing functions of $\underline b$ and hence, the FKG inequality,
provides the bound
$$
\langle s^{[X]}_{i}s^{[X]}_{j} \rangle \geq
\Bbb E^{*}_{\rho}[\mu^*_{\underline b}(T_{i,j})]
\Bbb E^{*}_{\rho}[b_i]
\Bbb E^{*}_{\rho}[b_j].
\tag A.15
$$
The quantities $\Bbb E^{*}_{\rho}[b_i]$ and $\Bbb E^{*}_{\rho}[b_j]$
may be estimated by considering the worst case $\odot$--boundary conditions
on the
neighborhoods of $i$ and $j$ which yields the uniformly positive constant
$c_2$. For the $d$--dimensional $XY$--model, we have $c_1 = 1$ and
$c_2 = (2/\pi)( e^{-2d\beta})$
\enddemo
\demo{Proof of Theorem 4\ [A]}
First observe that the lower bound follows because the magnetization can be
estimated from below by the average of the $s^{[X]}$'s in any state, and by
using the
$\bold 1^+$--state, this is obtained. In fact, for the $XY$ model, and
several other of the models under consideration, both of these bounds
follows because
it can be proved, via correlation inequalities, that the $\bold 1^+$ state is
exactly the state that produces the magnetization. For the general case,
consider the addition of the usual magnetic term:
$$
\sum_{i}hs^{[X]}_i \equiv
\sum_{i}2hb_i(\delta_{\sigma_i,+} -1) + hb_i
\tag A.16
$$
to the Hamiltonian. The effect of this additional term may be incorporated into
the present analysis by the addition of a single ``ghost'' spin connected to all other
spins with coupling $h$. (Here the ghost spin plays more the r\^ole of a
boundary
site than a full blown $XY$--degree of freedom.)
Now for a\.e\. $h$, the (thermodynamic) magnetization can be defined by
evaluating
the actual magnetization (the average of the $s^{[X]}$'s) in any convenient
state.
Thus, using the limiting state constructed from $\bold 1^+$ boundary
conditions, it
is clear that for a\.e\. positive $h$, the magnetization is bounded above
by the
(limiting) average fraction of sites
connected to the ghost site or the boundary. Let $\Lambda_L$ denote the box of
scale $L$ and define
$$
\pi_{L}(h,\beta) = \frac{1}{\Lambda_L}
\sum_{i\in\Lambda}\Bbb M^{\bold
1^+}_{\beta,h,\Lambda_L}(T_{i,\bold B})
\tag A.17
$$
where $T_{i,\bold B}$ is the event that the site $i$ is connected to the
boundary
or the ghost site and the sum includes the contribution from the boundary sites
themselves. The desired result follows from two elementary facts: First, by
continuity in finite volume,
$$
\lim_{h \to 0}\pi_{L}(h,\beta) = \pi_{L}(0,\beta).
\tag A.18
$$
Second, by a sequence of fairly standard manipulations,
$$
\Pi_{\infty}(\beta) \equiv \lim_{L\to\infty}\Pi_{\Lambda_L}(\beta)
= \lim_{L\to\infty}\pi_{L}(0,\beta).
\tag A.19
$$
Now, for $h > 0$ suppose we were to evaluate $m(h,\beta)$ starting on
$\Lambda_{NL}$
using $\bold 1^+$ boundary conditions and letting $N\to\infty$. Since, for
finite
$N$, this is a certified finite volume $\odot$--state, we increase the value by
conditioning on the event that the grid that divides $\Lambda_{NL}$ into small
copies of $\Lambda_L$ is fully occupied. Thus, at each stage it is learned that
$$
m_{\Lambda_{NL}}(h,\beta) \equiv
\frac{1}{|\Lambda_{NL}|}\sum_{i\in\Lambda_{NL}}
\langle s^{[X]}_i \rangle^{\bold 1^+}_{\beta,h,\Lambda_L}
\leq \pi_{L}(h,\beta).
\tag A.20
$$
Taking $h\downarrow 0$ (along a sequence of points of continuity) the desired
result follows from Equations (A.18) and (A.19).
\enddemo
\demo{Proof of Theorem 4\ [B]}
Let $\Cal G$ denote a graph, $\Bbb I\subset \Bbb S_{\cal G}$ and
$\Bbb K = \Bbb S_{\cal G} \setminus \Bbb I$. Let
$\gamma = \{\langle i,k \rangle \in \Bbb B_{\Cal G}\mid i \in \Bbb I,
k\in\Bbb K\}$
denote the connecting bonds and let $\Gamma(\gamma)$ denote the contour
event that
every $\omega$--bond in $\gamma$ is vacant. In what follows, it is assumed
that is there is any specification on $\Cal G$, it is of the $\odot$--type and
involves only the sites of $\Bbb K$.
It is claimed that if $\Gamma(\gamma)$ occurs then the measure on the
$(b_i \mid i \in \Bbb I)$ lies below, in the sense of FKG, the ``free measure''
on $\Bbb I$ that would be obtained if all the $J_{i,k}$ on $\gamma$ were zero.
Indeed, for any fixed $\underline b$ on $\Bbb K$ and $\eta$--configuration the
weights for the configurations $(b_i \mid i \in \Bbb I)$ are given by
$Z^{I, \eta}_{\underline a}(2\beta)
\prod_{\langle i,k \rangle \in \gamma}e^{\beta J_{i,k}(a_ia_k - b_ib_k)}
Z^{I,f}_{\underline b}(2\beta)$ where $Z^{I,f}$ denotes the free boundary
partition
function and $Z^{I, \eta}$ denotes the partition function with ($\odot$--type)
boundary specification provided by $\eta$. On the other hand, the free
weights are
given simply by $Z^{I,f}_{\underline a}(2\beta)Z^{I,f}_{\underline b}(2\beta)$.
Thus it is clear that irrespective of the information on the outside, the
conditional weights are a decreasing function times the free weights.
Now, supposing that $\Pi_{\infty}(\beta) = 0$, it is easy to establish
uniqueness of
the limiting $\rho$--measures among $\odot$--states:
Let $\Lambda\subset\Bbb Z^d$ be a finite connected set. Let $\Xi\supset\Lambda$
with $\Xi$ so large that the probability of an $\omega$--connection
between $\Lambda$ and $\partial \Xi$ in the $\bold 1^{+}$ state on $\Xi$ is
negligible. Under these circumstances, there are contours separating
$\Lambda$ from $\partial \Xi$; let $\gamma$ denote such a contour and
let $\tilde \Gamma(\gamma)$ the event that
$\gamma$ is the {\it outermost} such separating contour. These contour events form
a disjoint partition so, up to the negligible probability of a connection
between $\Lambda$ and $\partial \Xi$, the restriction of the maximal measure in
$\Xi$ to $\Lambda$ is below a superposition of free measures on various
separating
contours.
Now consider the lowest boundary condition on $\Xi$: setting all the boundary
$a_i$ to one and locking {\it their} spin directions. By $a\leftrightarrow b$
symmetry, the same outermost contours (in the $\eta$ expansion) appear with the
same probabilities and we find -- again up to negligible terms -- that this
worst
measure in $\Xi$ restricted to $\Lambda$ lies above the previously discussed
superposition. Evidently the two restricted measures coincide in the
$\Xi\nearrow\Bbb
Z^d$ limit and hence all the limiting $\odot$--measures coincide at least
as far as the
distributions of $\underline b$'s are concerned. However, the same argument
implies uniqueness for the various other Wolff--measures in the $\odot$--class
and, given the fact that all bond clusters are finite, uniqueness among all
Gibbs
measures of in the $\odot$--class follows easily.
\qed
\enddemo
\bigskip
I would like to thank Jon Machta who
provided help and encouragement in all stages of this
work and without whom this project would not have existed.
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