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\magnification=1200
\TagsOnRight
\define\sx{S_{_X}}
\define\sy{S_{_Y}}
\define\sz{S_{_Z}}
\redefine\sp{S^+}
\define\sm{S^-}
\define\Tr{\text{Tr}}
\define\cond{\,\bigg|\!\bigg|\,}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\topmatter
\title
Improved Lower Bound on the Thermodynamic Pressure\\
of the Spin 1/2 Heisenberg Ferromagnet
\endtitle
\author
B\'alint T\'oth
\footnote""{Work supported by the Hungarian National Foundation
for Scientific Research, grant No. 1902}
\footnote""{Submitted to {\sl Letters in Mathematical Physics},
9 March 1993}
\endauthor
\affil
Mathematical Institute of the\\
Hungarian Academy of Sciences
\endaffil
\address
{B. T\'oth\newline
Mathematical Institute of the\newline
Hungarian Academy of Sciences\newline
POB 127\newline
H-1362 Budapest, Hungary\newline
e-mail: h1222tot\@huella.bitnet}
\endaddress
\abstract
{
We introduce a new stochastic representation of the
partition function of the spin 1/2 Heisenberg ferromagnet. We
express some of the relevant thermodynamic quantities
in terms of expectations of functionals of so called
random stirrings on $\Bbb Z^d$. By use of this
representation we improve the lower bound on the
pressure given by Conlon and Solovej in [CS2].
\hbox{}
\noindent
{\bf AMS subject classification (1991):} 82D40, 82B20
}
\endabstract
\endtopmatter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\document
\bigskip
\noindent
{\bf 1. Introduction and Result}
\medskip
We consider the $\frac12$-spin isotropic quantum Heisenberg
ferromagnet (QHF) on the $d$-di\-men\-sio\-nal hypercubic
lattice. The Hamiltonian of the model is
$$
H_\Lambda=\frac12\sum_{|x-y|=1}\left[
{\left(\bold{S}(x)-\bold{S}(y)\right)}^2-1\right]
\tag 1.1
$$
where
$\bold{S}(x)=\left(\sx(x),\sy(x),\sz(x)\right),\,\,x\in\Lambda$,
are the local spin operators and the summation runs over nearest
neighbour pairs of lattice sites in the rectangular box
$\Lambda$, with periodic boundary conditions. The canonical
commutation relations satisfied by the spin operators are:
$$
\big[S_\alpha(x),S_\beta(y)\big]=i \delta_{\!x\!,y}
\epsilon_{\!\alpha\!,\beta\!,\gamma}
S_\gamma(x),\quad \sx^2(x)+\sy^2(x)+\sz^2(x)=\frac34.
\tag 1.2
$$
The grand partition function and the thermodynamic pressure
are defined in the usual way:
$$
\Xi_\Lambda(\beta,h)=
\Tr\bigg[\exp-\beta\left(H_\Lambda-h\sum_{x\in\Lambda}
\sz(x)\right)\bigg]
\tag 1.3
$$
and
$$
\align
\beta p_\Lambda(\beta,h)
&={|\Lambda|}^{-1}\log\Xi_\Lambda(\beta,h)
\tag 1.4
\\
p(\beta,h)
&=\lim_{\Lambda\nearrow\Bbb Z^d} p_\Lambda(\beta,h).
\tag 1.5
\endalign
$$
It is generally conjectured that in the thermodynamic limit, at
zero external field ($h=0$)
and very low temperatures the elementary excitations of the
QHF (i.e. the so called magnons) behave like noninteracting
bosons on the lattice $\Bbb Z^d$, (see e.g. [CS2], for
historical origins of the magnon approximation see [B] and [D]).
In particular it is expected that
$$
\lim_{\beta\to\infty}\beta^{\frac{d+2}{2}}p(\beta,0)=
C_d\overset\text{def}\to{=}
\frac{-1}{{(2\pi)}^d}\int_{\Bbb R^d}
\log\left(1-\text{e}^{-k^2}\right)\text{d}k.
\tag 1.6
$$
Conlon and Solovej considered this asymptotics in [CS2] and
proved the following bound:
$$
\liminf_{\beta\to\infty}\beta^{\frac{d+2}{2}}p(\beta,0)\ge
\frac12c_d\overset
\text{def}\to{=}\frac12\frac{1}{{(2\pi)}^d}\int_{\Bbb R^d}
\text{e}^{-k^2}\text{d}k.
\tag 1.7
$$
Using a new stochastic representation of the partition function,
which is of course closely related to that of [CS1] and [T2],
we improve this bound in the present Letter by proving the
following
\proclaim{Theorem 1}
In three and more dimensions
$$
\liminf_{\beta\to\infty}\beta^{\frac{d+2}{2}}p(\beta,0)\ge
\log2\cdot C_d.
\tag 1.8
$$
\endproclaim
\noindent
Remark: in one and two dimensions our proof yields the constant
$\log2\cdot c_d$ on the right hand side of (1.8) , which is
still better than that of [CS2].
In three dimensions the constants appearing in (1.6), (1.7) and
(1.8) are respectively
$$
C_3=0.0301\qquad\frac12c_3=0.0112\qquad\log2\cdot C_3=0.0209.
\tag 1.9
$$
The proof is based on a Feynman-Kac formula applied to the
partition function of lattice boson gas (section 2.), the so
called random stirring representation of the simple exclusion
process (section 3.), a coupling argument and a simple
probabilistic estimate on the recurrence probability of the
simple symmetric random walk on $\Bbb Z^d$ (section 4.).
[Terminology: by random walk we mean continuous time simple
symmetric random walk of jump rate $2d$ throughout this paper.]
In section 5. formulas are given for the spontaneous
magnetization and the long range order in terms of
random stirring expectations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf 2. Bose Gas Formulation and Feynman-Kac Formulas}
\medskip
The local raising, lowering and excitation (i.e. magnon) number
operators are defined in the usual way:
$$
S^\pm(x)=\sx(x)\pm i\sy(x),\qquad
m(x)=\sz(x)+\frac12.
\tag 2.1
$$
They satisfy the commutation relations
$$
\align
[S^\pm(x),S^\pm(y)]=0,\qquad
[\sm(x)&,\sp(y)]=\delta_{x,y}\big(1-2m(x)\big)
\tag 2.2
\\
m(x)=\sp(x)&\sm(x)=m^2(x).
\tag 2.3
\endalign
$$
It is well known that these are exactly the commutation
relations of the local creation, annihilation and occupation
number operators of a {\it hard core\/} Bose lattice gas (see
e.g. [CS1] or [T1]).
On the other hand the {\it canonical\/} bosonic creation,
annihilation and particle number operators satisfy the
relations:
$$
\align
[a^+(x),a^+(y)]=[a(x),&a(y)]=0,\quad
[a(x),a^+(y)]=\delta_{x,y}
\tag 2.4
\\
n(x)=&a^+(x)a(x).
\tag 2.5
\endalign
$$
Thus the Heisenberg Hamiltonian written in terms of these
operators,
$$
H_\Lambda=-\sum_{x,y\in\Lambda}\sp(x)\Delta_{x,y}\sm(y)-
\sum_{|x-y|=1}m(x)m(y),
\tag 2.6
$$
is identified with a Bose lattice gas Hamiltonian
$$
H^{\text{BLG}}_\Lambda=
-\sum_{x,y\in\Lambda}a^+(x)\Delta_{x,y}a(y)
+\frac12\sum_{x,y\in\Lambda}V(x,y)n(x)n(y)
\tag 2.7
$$
with one-particle kinetic energy operator $-\Delta$ given by the
discrete La\-pla\-cian on the lattice:
$$
\Delta_{x,y}=\left\{
\matrix\format \c & \l \\
-2d\quad &\text{ if }\quad |x-y|=0 \\
1 \quad &\text{ if }\quad |x-y|=1 \\
0 \quad &\text{ if }\quad |x-y|>1
\endmatrix\right.
\tag 2.8
$$
and pair interaction
$$
V(x,y)=\left\{
\matrix\format \c & \l \\
\infty\quad &\text{ if }\quad |x-y|=0 \\
-2\quad &\text{ if }\quad |x-y|=1 \\
0 \quad &\text{ if }\quad |x-y|>1
\endmatrix\right.
\tag 2.9
$$
By a standard Feynman-Kac argument we express the N-particle
canonical partition function of a Bose lattice gas with {\it
arbitrary} pair interaction $V$, as follows:
$$
\align
&Q_\Lambda(\beta,N)=\frac{1}{N!}\sum_{\pi\in\Cal P_N}
\sum_{x^1\!,\dots\!,x^N\in\Lambda}
\\
&\quad\bold E\left(
\text{e}^{-\int_0^\beta\sum_{i\beta]
\Bbb I[\{X^1_\beta,\dots,X^N_\beta\}\!=\!A]
\!\cond\!\{X^1_0,\dots,X^N_0\}\!=\!A\!\right)
\tag 3.2
\endalign
$$
where for $A\subset\Lambda$
$$
\Cal B(A)=|\{\,(x,y)\in A\times A\,:\,|x-y|=1\,\}|
\tag 3.3
$$
(Each neighbouring pair is counted twice!)
At this stage we have to introduce the {\it simple symmetric
exclusion process\/} (SSEP) defined in Spitzer's classic [Sp]
and extensively studied since then. (For a survey see also
[L].) The SSEP of $N$ particles on $\Lambda$ is a Markov process
$\eta_t$ on $\{\,A\,:\,A\subset\Lambda,\,\,\,|A|=N\,\}$.
It is an evolution of configurations of $N$ indistinguishable
particles on $\Lambda$ with at most one particle per site. A
particle at $x\in\Lambda$ waits an exponentially distributed
time with mean ${(2d)}^{-1}$, then chooses a neighbouring site
$y$ with probability ${(2d)}^{-1}$. If $y$ is vacant at that
time, the particle jumps from $x$ to $y$, otherwise it stays at
$x$. All the waiting times and choices of neighbouring sites are
independent.
A SSEP on $\Lambda$ can be expressed in terms of the so called
{\it random stirring process\/} (RSP) on $\Lambda$, defined
originally by Harris [H] (see also [G]). The RSP is a Markov
process $\sigma_t$ on the set of permutations of $\Lambda$,
$\{\,\pi:\Lambda\to\Lambda\,:\,
[x\not=y]\Rightarrow[\pi(x)\not=\pi(y)]\,\}$. The labels
$\sigma_t(x)$ and $\sigma_t(y)$ are interchanged at rate $1$,
with independent Poisson flow of event times for each pair of
neighbouring sites $(x,y)$ in $\Lambda$. In other words, the
random permutation $\sigma_t$ starts from the identity and a
transposition $(x,y)$ is appended to it after exponentially
distributed times with mean $1$, independently for each pair of
nearest neighbours.
Given a RSP $\sigma_t$ on $\Lambda$ and a subset
$A\subset\Lambda,\,\,\,|A|=N$
$$
\eta_t=\{\,\sigma_t(x)\,:\,x\in A\,\}
\tag 3.4
$$
is clearly a SSEP of $N$ particles on $\Lambda$, starting
initially from the set $A$. (The particles of the SSEP are
indistinguishable!)
We are going now to express $Q_{\Lambda}$ in terms of
expectations over SSEP trajectories and eventually $\Xi_\Lambda$
in terms of expectations over RSP trajectories. The point is
that until the first collision time the trajectories of $N$
independent random walkers at one hand and $N$ particles
performing SSE random walks on the other hand can be identified.
More precisely: given a SSEP $\eta_t$, enlarge the probability
space by an extra random event (``killing'') which occurs at a
random time $\tilde\tau$ with instantaneous rate $\Cal
B(\eta_t)$. Clearly
$\big(\tau,\,\,\{X^1_s,\dots,X^N_s\}:\,s<\tau\big)$ and
$\big(\tilde\tau,\,\,\eta_s:\,s<\tilde\tau\big)$ have the same
joint distribution, given $\{X^1_0,\dots,X^N_0\}=\eta_0$.
Consequently
$$
\allowdisplaybreaks
\align
&\bold E\left(
\text{e}^{\int_0^\beta \Cal
B\left(\{X^1_s,\dots,X^N_s\}\right)\text{d}s}
\Bbb I[\tau>\beta]
\Bbb I[\{X^1_\beta,\dots,X^N_\beta\}\!=\!A]
\cond\{X^1_0,\dots,X^N_0\}\!=\!A\,\right)
\\
&\qquad=\bold E\left(
\text{e}^{\int_0^\beta \Cal B\left(\eta_s\right) \text{d}s}
\Bbb I[\tilde\tau>\beta]
\Bbb I[\eta_\beta\!=\!A]
\cond\eta_0\!=\!A\,\right)
\\
&\qquad=\bold E\left(
\bold P\bigg(\tilde\tau>\beta\big|\!\big|\eta_s:
\,\,0\le s\le\beta\bigg)
\text{e}^{\int_0^\beta \Cal B\left(\eta_s\right) \text{d}s}
\Bbb I[\eta_\beta\!=\!A]
\cond\eta_0\!=\!A\right)
\\
&\qquad=\bold P\left(\eta_\beta\!=\!A\cond\eta_0\!=\!A\right)
\tag 3.5
\endalign
$$
In the last step we used the equality
$$
\bold P\bigg(\tilde\tau>\beta\big|\!\big|\eta_s:
\,\,0\le s\le\beta\bigg)=
\text{e}^{-\int_0^\beta \Cal B\left(\eta_s\right) \text{d}s}
\tag 3.6
$$
which holds by definition of $\tilde\tau$.
Inserting (3.5) into (2.10) we get the following form
of the grand canonical partition function
$$
\Xi_\Lambda(\beta,\mu)=
\sum_{A\subset\Lambda}\text{e}^{\beta\mu|A|}
\bold P\left(\eta_\beta\!=\!A\,\big|\!\big|\,\eta_0\!=\!A\right)
\tag 3.7
$$
This form is of course equivalent to the one given in [CS1],
where similar arguments are applied to the QHF of any spin.
Our forthcoming arguments (the random stirring representation
and its consequences) work in the $S=1/2$ case only.
We are going to exploit now the random stirring representation
of the simple exclusion process. Let us denote by $\frak
m_s(l)$, $l\ge1$ the number of cycles of length $l$ in the
random permutation $\sigma_s$.
\proclaim{Theorem 2}
$$
\Xi_\Lambda(\beta,\mu)=\bold E\left(\prod_{l\ge1}
{\left(1+\text{e}^{\beta\mu l}\right)}^{\frak m_\beta(l)}\right)
\tag 3.8
$$
\endproclaim
\demo{Proof}
According to (3.7) and (3.4)
$$
\Xi_\Lambda(\beta,\mu)=
\sum_{A\subset\Lambda}\text{e}^{\beta\mu|A|}
\bold P\left(\sigma_\beta(A)=A\right)
\tag 3.9
$$
Denote
$$
p=\frac{\text{e}^{\beta\mu}}{1+\text{e}^{\beta\mu}}.
\tag 3.10
$$
Then
$$
\Xi_\Lambda(\beta,\mu)=
{\left(1+\text{e}^{\beta\mu}\right)}^{|\Lambda|}
\sum_{A\subset\Lambda}p^{|A|}(1-p)^{|\Lambda\setminus A|}
\bold P\big(\sigma_\beta(A)=A\big).
\tag 3.11
$$
But the sum on the right hand side has a straightforward
probabilistic meaning: imagine that at time zero we put a red or
a blue marble with probability $p$ respectively $1-p$ on each
site of $\Lambda$, independently of one another. After this we
perform the random stirring of the marbles up to time $\beta$,
independently of the initial choice of colours. The sum on the
right hand side of (3.11) is equal to the probability of the
event ``the initial and final configuration of coluored marbles
are the same''. Which is the same as ``all cycles of the
permutation $\sigma_\beta$ are monocolour''. That is
$$
\Xi_\Lambda(\beta,\mu)=
{\left(1+\text{e}^{\beta\mu}\right)}^{|\Lambda|}
\bold E\left(\prod_{l\ge1}{\left(p^l+(1-p)^l\right)}^
{\frak m_\beta(l)}\right)
\tag 3.12
$$
Let us denote by $\lambda_s(x),\,\,
x\!\in\!\Lambda,\,\,s\!\ge\!0$, the length of the cycle
containing $x\!\in\!\Lambda$ in the random permutation
$\sigma_s$. The following identity is straightforward:
$$
\sum_{l\ge1}F(l)l\frak m_s(l)=\sum_{x\in\Lambda}F(\lambda_s(x))
\tag 3.13
$$
for any function $F:\Bbb N\to \Bbb R$. Using (3.13) with
$F\equiv1$, from (3.12) we get exactly (3.8).
\enddemo
\noindent
{\sl Remark:\/} Random permutations have been used recently in
different contexts of 1-d quantum spin chains [AN] and of
interacting Bose gas [S\"u].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf 4. Proof of Theorem 1.}
\medskip
By Jensen's inequality we have
$$
\beta p_\Lambda(\beta,0)=
{|\Lambda|}^{-1}\log\bold E\left(2^{\sum_{l\ge1}\frak
m_\beta(l)}\right) \ge
{|\Lambda|}^{-1}\log2\cdot\bold E\left(\sum_{l\ge1}\frak
m_\beta(l)\right)
\tag 4.1
$$
We apply (3.13) with the choice $F(l)=1/l$ and use translation
invariance of the distribution of $\lambda_\beta(x)$ to get
$$
\beta p_\Lambda(\beta,0)\ge \log2\cdot\bold
E\left(\frac{1}{\lambda_\beta(0)}\right).
\tag 4.2
$$
Taking the thermodynamic limit we have
$$
\beta p(\beta,0)\ge \log2\cdot\sum_{n\ge1} \frac{1}{n}\bold
P\left(\lambda_\beta=n\right)
\tag 4.3
$$
where now $\lambda_\beta$ is the length of the (possibly
infinite) cycle containing $0\in\Bbb Z^d$, under the random
permutation $\sigma_\beta$ on the infinitely extended lattice
$\Bbb Z^d$. (There is no difficulty in extending the RSP to the
whole hypercubic lattice, see [H].)
We define now a random process $Z^{(\beta)}_t,\,\, t\ge0$ on
$\Bbb Z^d$, which is loosely speaking the trajectory of a
particle starting from the origin, induced by the random
stirring $\sigma_s,\,\,s\in[0,\beta)$ periodically continued in
time. More precisely: we periodically continue the random
permutation process
$$
\bar{\sigma}^{(\beta)}_t=
\sigma_{t-\beta\left[\frac{t}{\beta}\right]}\circ
{(\sigma_\beta)}^{\circ\left[\frac{t}{\beta}\right]}
\qquad t\ge0
\tag 4.4
$$
and define $Z^{(\beta)}_t$ as the trajectory of a particle
starting from the origin, under this permutation process:
$$
Z^{(\beta)}_t=\bar{\sigma}^{(\beta)}_t(0), \qquad t\ge0.
\tag 4.5
$$
In terms of this process the cycle length $\lambda_\beta$ is
$$
\lambda_\beta=\min\{k\ge1\,:\,Z^{(\beta)}_{k\beta}=0\}.
\tag 4.6
$$
The process $Z^{(\beta)}_{\cdot}$ can be easily realized as a
function of the trajectory of a random walk $X_{\cdot}$ in the
following way. Until $t=\beta\lambda_{\beta}$ \phantom{W}
$Z^{(\beta)}$ performs the same jumps as
$X$ except of two deterministic modifications:
Let $t>\beta$, assume $Z^{(\beta)}_{t-0}=x$ and $y$ is a
neighbouring site.\newline
(a) {\sl forced jumps:\/} If at some time $t-k\beta,\,\,
k=1,2,\dots,[t/\beta]$ a jump $y\to x$ of $Z^{(\beta)}$
occured, then, at time $t$, $Z^{(\beta)}$ is forced to
jump backwards from $x$ to $y$. \newline
(b) {\sl erased jumps:\/} If at time $t$ a jump of $X$
occurs, which would force $Z^{(\beta)}$ to jump from $x$
to $y$, but $y$ was occupied by $Z^{(\beta)}$ at some
time $t-k\beta,\,\,k=1,2,\dots,[t/\beta]$, then this jump is not
performed by $Z^{(\beta)}$.\newline
After $t=\beta\lambda_{\beta}$ \phantom{W} $Z^{\beta}_{\cdot}$
is continued periodically, with period $\beta\lambda_{\beta}$.
Denote by $\theta_\beta$ the moment when the trajectories
$Z^{(\beta)}_\cdot$ and $X_\cdot$ come apart for the first
time, if this happens before $\beta\lambda_\beta$, or
$\beta\lambda_\beta$ otherwise. I.e. the time when the first
deterministic modification (forced jump, erased jump or periodic
continuation) of the random walk trajectory occurs.
This is clearly the following stopping time of $X_\cdot$
$$
\align
\theta_\beta&=\inf\left\{t>0:Z^{(\beta)}_t\not=X_t\right\}
\bigwedge \beta\lambda_\beta
\\
&=\inf\{t>\beta\,:\,X_t=X_{t-k\beta}
\text{ for some } k=1,2,\dots,[t/\beta]\}.
\tag 4.7
\endalign
$$
Consequently
$$
\bold P\bigg(\lambda_\beta=n\bigg)\ge
\bold P\bigg(\theta_\beta=\beta\lambda_\beta=n\beta\bigg)=
\bold P\bigg(X_{n\beta}=0\bigg)-
\bold P\bigg(\big[X_{n\beta}=0\big] \land
\big[\theta_\betaFrom (4.8) and (4.12) we get
$$
\liminf_{\beta\to\infty}\beta^{d/2}\bold
P\left(\lambda_\beta=n\right)
\ge\lim_{\beta\to\infty}\beta^{d/2}\bold
P\left(X_{n\beta}=0\right)
\tag 4.13
$$
for any $n\in\Bbb N$ in three and more dimensions, and for $n=1$
in one and two dimensions. Inserting (4.13) into (4.3), a
comparison with (2.12) yields the proof of Theorem 1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf 5. Further Consequences of Theorem 2.}
\medskip
In this section we give formulas for the spontaneous
magnetization and the long range order parameter of the QHF in
terms of random stirring expectations. These formulas will show
that the expected phase transition of the model is closely
related to the apparence of an infinite cycle in the random
stirring $\sigma_\beta$ of $\Bbb Z^d$, for $\beta$ sufficiently
large.
After straightforward manipulations, from (3.8) we get the
following expression of the magnetization (i.e. density of the
Bose gas minus 1/2)
$$
m_\Lambda(\beta,h)=\frac12\frac
{\bold E\left(\tanh\left(\frac{h}{2}\lambda_\beta(0)\right)
\prod_{l\ge1}
{\left(1+\text{e}^{\beta h l}\right)}^{\frak m_\beta(l)}\right)}
{\bold E\left(\prod_{l\ge1}
{\left(1+\text{e}^{\beta h l}\right)}^{\frak m_\beta(l)}\right)}.
\tag 5.1
$$
Taking the thermodynamic limit we find the following formula
for the spontaneous magnetization
$$
\lim_{h\to0}m(\beta,h)=m(\beta)=
\frac12\lim_{n\to\infty}\lim_{\Lambda\nearrow\Bbb Z^d}
\frac{\bold E\left(\Bbb I[\lambda_\beta(0)>n]
2^{\sum_{l\ge1}\frak m_\beta(l)}\right)}
{\bold E\left(2^{\sum_{l\ge1}\frak m_\beta(l)}\right)}
\tag 5.2
$$
The long range order parameter (as defined e.g. in [DLS]) is
$r(\beta)=\lim_{\Lambda\nearrow\Bbb Z^d}r_\Lambda(\beta)$, where
$$
r_\Lambda(\beta)=\frac1{|\Lambda|^2}\sum_{x,y\in\Lambda}
{\big\langle \bold S(x)\cdot\bold S(y)\big\rangle}_{\Lambda}.
\tag 5.3
$$
(Notice that in the Bose gas formulation $\frac23r(\beta)$ is
exactly the density of the condensate, as defined by [PO].)
Applying similar considerations as in the derivation of Theorem
2, the following expression of the long range order parameter is
found:
$$
r(\beta)=\frac34\lim_{\Lambda\nearrow\Bbb Z^d}
\frac{1}{|\Lambda|}\frac
{\bold E\left(\lambda_\beta(0)
2^{\sum_{l\ge1}\frak m_\beta(l)}\right)}
{\bold E\left(2^{\sum_{l\ge1}\frak m_\beta(l)}\right)}.
\tag 5.4
$$
Formulas (5.2) and (5.4) show striking similarity to
percolation theoretical objects with ``cluster size'' replaced
by ``cycle length''. They could allow nice random-geometrical
speculations about the still open problem of existence of phase
transition in QHF. On a technical level their value is less
clear for two reasons: (1) the random cycles seem to be less
transparent geometric objects than the random clusters and (2)
calculating the averages in (5.2), (5.4) is further complicated
by the weights $2^{\sum_{l\ge1}\frak m_\beta(l)}$ assigned to the
permutations, over the bare random stirring measure.
\bigskip
\noindent
{\bf Acknowledgement:} It is a pleasure to thank L\'aszl\'o
Erd\H os and Jan Philip Solovej their helpful comments on the
first version of this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\noindent
{\bf References}
\parindent=1cm
\medskip
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