\documentclass[12pt]{article}


%\renewcommand{\theequation}{\thesection.\arabic{equation}}


\oddsidemargin=0in
\textwidth=6.25in
\topmargin=0in
\textheight=609pt
\parskip=14pt
\setlength{\unitlength}{0.5cm}
\def\adj{\mathop{\rm adj}}
\def\diag{\mathop{\rm diag}}
\def\rank{\mathop{\rm rank}}
\def\span{\mathop{\rm span}}
\def\rdots{\mathinner{\mkern1mu\raise1pt\vbox{\kern1pt\hbox{.}}\mkern2mu
    \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}}
%\newcommand{\endproof}{\box}
\newcommand{\Z}{{\rm Z\kern-.35em Z}}
\newcommand{\bP}{{\rm I\kern-.15em P}}
\newcommand{\Q}{\kern.3em\rule{.07em}{.65em}\kern-.3em{\rm Q}}
\newcommand{\R}{{\rm I\kern-.15em R}}
\newcommand{\h}{{\rm I\kern-.15em H}}
\newcommand{\C}{\kern.3em\rule{.07em}{.65em}\kern-.3em{\rm C}}
\newcommand{\T}{{\rm T\kern-.35em T}}
\newcommand{\goth}[1]{#1}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bb}{\bf b}
\newcommand{\ba}{\bf a}
\newcommand{\ve}{\varepsilon}
\newcommand{\la}{\lambda}
\newcommand{\ga}{\gamma}
\newcommand{\om}{\omega}
\newcommand{\pa}{\partial}
\newcommand{\cd}{\cdot}
\newcommand{\ra}{\rightarrow}
\newcommand{\na}{\nabla}
\newcommand{\Om}{\Omega}
\newcommand{\de}{\delta}
\newcommand{\Del}{\Delta}
\newcommand{\La}{\Lambda}
\newcommand{\vp}{\varphi}
\newcommand{\al}{\alpha}
\newcommand{\nn}{\nonumber}
\newtheorem{theorem}{Theorem}
\newtheorem{example}{Example}
\newtheorem{rem}{Remark}
\newtheorem{lem}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\newtheorem{definition}{Definition}


\begin{document}
%\font\twelverm=cmr12
%\font\cs=CMSSI12



\openup 1.5\jot
\centerline{A random walk on the permutation group, some formal long- 
time asymptotic relations}

\vspace{1in}
\centerline{Paul Federbush}
\centerline{Department of Mathematics}
\centerline{University of Michigan}
\centerline{Ann Arbor, MI 48109-1109}
\centerline{(pfed@umich.edu)}

\vspace{1in}

\centerline{\underline{Abstract}}

We consider the group of permutations of the vertices of a lattice.   
A random walk is generated by unit steps that each interchange two  
nearest neighbor vertices of the lattice.  We study the heat equation  
on the permutation group, using the Laplacian associated to the  
random walk.  At $t=0$ we take as initial conditions a probability  
distribution concentrated at the identity.  A natural conjecture for  
the probability distribution at long times is that it is  
'approximately' a product of Gaussian distributions for each vertex.   
That is, each vertex diffuses independently of the others.  We obtain  
some formal asymptotic results in this direction.  The problem arises  
in certain ways of treating the Heisenberg model in statistical  
mechanics.

\vfill\eject

This paper represents work in progress, and is written in a casual  
manner, as notes for a seminar or perhaps a physics article.  But the  
work is interesting and non-trivial, and perhaps will inspire  
research, many natural questions will appear.  We of course will be  
clear about what is proved and what is conjectured, some proofs will  
be sketched.  As will be seen this work is somewhat discouraging  
about the application sought to the Heinsenberg model problem, but  
opens some pleasant mathematical vistas.

We study a d-dimensional periodic lattice cube, $\Lambda$ of edge  
size $L$.  The number of its vertices, $\cal V$, is given as
\be	|\Lambda| \equiv N \equiv L^d \equiv \# \{ {\cal V} \}.	\ee
We set ${\cal G}$ to be the permutation group on $\cal V$.  $H$ is  
the element in the group algebra given as
\be	H = - \sum_{i \sim j } (I_{ij} - I).	\ee
here $i\sim j$ indicates that $i$ and $j$ are neighboring vertices in  
the lattice. $I$ is the identity element in $\cal G$, and $I_{ij}$ is  
the group element that interchanges vertices $i$ and $j$ leaving the  
other vertices alone.  $H$ is thus the Heisenberg model Hamiltonian  
``promoted" from an operator in the Hilbert space to an element of  
the group algebra.  (One need not be familiar with the Heisenberg  
model for purposes of this paper.)

We then consider the group algebra element $e^{-Ht}$.  This can be  
expanded as a linear combination of group elements, the $g_p$
\be     e^{-Ht} = \sum_p \ f(g_p, t)g_p \ee
where $p$ labels the group elements.  Each $g_p$ represents a mapping  
of the vertices,
\be		g_p : i \rightarrow p(i)	\ee
so $p$ is specified by
\be   \Big( p(1), p(2), ..... \Big)	\ee
Equation (5) is a point in $(Z^d)^N$, in fact in $(\Lambda)^N$.  The  
cardinality of the set of such points is $N!$.  Such points in $ 
(\Lambda)^N$ are restricted by the condition that all the $p(i)$ in  
(5) are distinct.  This subset we label $(\Lambda)^{N*}$.  The  
respective cardinalities of $(\Lambda)^{N*}$ and $(\Lambda)^{N}$ are  
$N!$ and $N^N$.  The latter space is simpler, being a periodic  
lattice cube.

We have a time dependent probability function $f(g,t)$ on the  
permutation group (or on $(\Lambda)^{N*}$).  We now extend $f(g,t)$  
to $f^e(\vec{x}, t), \vec{x}$ in $(\Lambda)^{N}$.  Of course such  
extensions are not unique.  The motivation for extending $f$ will  
soon be clear, largely depending on the simplicity of $(\Lambda)^{N}$  
over $(\Lambda)^{N*}$.  The extended function will no longer be a  
probability function.

The relation between the Heisenberg model of ferromagnetism and our  
random walk on the permutation group was beautifully developed by R.  
Powers in [1].  Inspired by this work, the author presented a  
possible avenue towards proving the phase transition of the  
Heisenberg model in [2], a development of ideas in [1].  The central  
relation needed in the proof envisioned in [2] is of the form
\be    f^e(\vec{x}, t) \cong C_N \prod_{i\in {\cal V}} \ (e^{\Delta  
t})_{i,x_i} \ \ {\rm for} \ t \ {\rm large}.     \ee
$\Delta$ is the lattice Laplacian on $\Lambda$.  (Of course, equation  
(6) need only hold on $f(g,t)$, but we presume the extension  
satisfies (6).)  We do not now make explicit the degree of  
approximation implied by $\cong$.  The right side of (6) is a product  
of gaussians (associated to independent random walks).  It also is a  
solution of the heat equation on $(\Lambda)^{N}$ with its natural  
lattice Laplacian!

At the very least we would want (6) to imply
\be \lim_{t \ra \infty}  f^e(\vec{x}, t) = C_N \lim_{t \ra \infty}  
\prod_{i\in {\cal V}} \ (e^{\Delta t})_{i,x_i}. \ee
We note
\be   \lim_{t \ra \infty} f(g,t) = \frac 1{N!}    \ee
and
\be   \lim_{t \ra \infty} (e^{\Delta t})_{i,j} = \frac 1{N}.   \ee
Restricting to points $\vec x$ in $(\La)^{N*}$ equation (7) becomes
\be	\frac 1{N!} = C_N\left(\frac 1 N\right)^N	\ee
 From (10) and Stirling's formula we have determined the $C_N$ in (6)  
satisfy
\be 	\lim_{N \ra \infty} (C_N)^{\frac 1 N} = e	\ee

We will find an extension $f^e$ of $f$ satisfying a differential  
equation
\be		\frac {\pa f^e}{\pa t} = \Delta \ f^e + V \ f^e	\ee
$\Delta$ is the usual lattice Laplacian on $(\La)^N$ a periodic  
version of the lattice Laplacian on $Z^{dN}$.  Here we will only  
consider $V$ built up with ``two-particle" interactions.  Thus
\be	V = \sum \ V_{i,j}	\ee
where $V_{i,j}$ describes the interaction of the two vertices that at  
time $t=0$ were at $i$ and $j$ respectively.  The $V$ must be such  
that the solution of (12) (with initial conditions, the identity at  
$t=0$) restricted to $(\La)^{N*}$ agrees with $f$ as defined in  
equation (3).  It indeed is possible to find such $V$ that achieve this.

A form for $V_{i,j}$ that works is given as follows.  We apply this $V 
$ to a product function
\be	\phi(\vec x) = \prod_i \ \phi_i(x_i)	\ee
($V$ applied to product functions determines $V$ uniquely.)
\[	\left(V_{i,j} \phi \right) (\vec x) =  - \; \prod_{k\not= i,j}  
\phi_k(x_k) \cdot \sum_{y \in {\cal V}} \cdot \sum_{{\vec i}} \cdot \]
\[	\cdot \left[ \de_{x_i,y} \; \de_{x_j,y+{\vec i}} + \de_{x_j,y} \de_ 
{x_i,y+{\vec i}} + r\; \de_{x_i,y}\de_{x_j,y} + r \;
\de_{x_i,y+{\vec i}} \de_{x_j,y+{\vec i}} \right] \cdot \]
\be \cdot \left[\phi_i(y) - \phi_i(y+{\vec i})\right] \cd \left 
[ \phi_j(y) - \phi_j(y + \vec i)\right]  \ee
$r$ is arbitrary.  There are $d$ orthonormal unit vectors, $\vec i$.   
So the paths interact only when the vertices are in nearest neighbor  
position.  The expression (15) certainly is uniquely determined by  
the conditions above.  This is straightforward to show....though I  
labored weeks on it.  Herein we work with $r$ set equal to zero.   
(But we believe working with an $r \not = 0$ at least if $|r| < 1$,  
leads to no essential changes in the form of our calculations and  
results.  We have studied this $r\not= 0$ situation a little, one  
need consider more diagrams than in the $r=0$ case.  We will briefly  
comment on this again later.)

We return to equation (6) for some further deliberations.  We  
consider summing $\vec {x}$ over $(\La)^N \equiv A$ and over $(\La)^ 
{N*} \equiv B$.
\be
\sum_{\vec {x} \in A} f^e(\vec{x}, t) \cong C_N \sum_{\vec {x} \in A}  
\ \prod_{i\in {\cal V}}  (e^{\Delta t})_{i,x_i} = C_N  \ {\rm for} \  
t \ {\rm large}
\ee
\be
\sum_{\vec {x} \in B} f^e(\vec{x}, t) = \sum_{\vec {x} \in B} f(\vec 
{x}, t) = 1 \cong C_N \sum_{\vec {x} \in B} \ \prod_{i\in {\cal V}}   
(e^{\Delta t})_{i,x_i}  \ {\rm for} \ t \ {\rm large}
\ee
We will want the content of (16) and (17) to be given by the  
following conjectures.
\bigskip
\centerline{--------------------}
\underline{Conjecture 1}
\be   \lim_{t\ra \infty} \left( \sum_{\vec{x} \in A} f^e(\vec{x}, t) 
\right)^{1/N} = \left(C_N\right)^{1/N}  \ee
where $C_N$ is given by (10) and the limit is uniform in $N$.

\bigskip
\centerline{--------------------}

It is the uniformity requirement that makes the conjecture most  
difficult.
\bigskip
\centerline{--------------------}

\underline{Conjecture 2}

\be   \lim_{t\ra \infty} \left( \sum_{\vec{x} \in B} \ \prod_{i\in  
{\cal V}}  (e^{\Delta t})_{i,x_i}\right)^{1/N} = \left(C_N\right)^{- 
\; \frac1 N}  \ee
where $C_N$ is given by (10) and the limit is uniform in $N$.

\centerline{--------------------}

We believe Conjecture 2 is not very difficult to prove, and we plan  
to turn to it soon.

We now consider the solution of (12), treating $V$ as a perturbation  
in the form of a Rayleigh-Schrodinger expansion.
\be
f^e(\vec x, t) = \sum^\infty_{n=0} \int^t_0 dt_n \int^{t_n}_0 dt_ 
{n-1} \cdots \int^{t_2}_0 dt_1 \ e^{\Del(t-t_n)} Ve^{\Del(t_n-t_ 
{n-1})} \cdots Ve^{\Del t_1}.	\ee
The right side of (20) is naturally represented as a sum of  
contributions of diagrams.  In some more detail as a sum of products  
of the contributions of connected diagrams.  All the computations of  
this paper deal with results for the sum over final states (in (20)  
over $\vec x$ in $(\La)^N)$ for connected tree-graph diagrams.  If in  
equation (6) we sum over final states, over $\vec x$, on both sides  
we get
\be	\sum_{{\vec x}} f^e(\vec x, t) \cong C_N \ \ \ {\rm for} \ \ t \  
\ {\rm large}.	\ee
(See Conjecture 1.)

Yes, in this paper we study the truth of (21), a much weakened form  
of (6).  But we expect that if we can get on top of (21) we are well  
on our way to treating (6).  Studying the decomposition of (20) into  
connected diagram contributions puts us in the ``cluster expansion"  
framework familiar in statistical mechanics.  A very complete  
treatment is in [3], but the level of sophistication of an  
undergraduate course in statistical mechanics is more than adequate.

\bigskip

\centerline{--------------------}


\begin{theorem}
We consider a two-particle (vertex) connected tree-graph contribution  
to (20).  The diagram is an $H$ shaped figure.  The bottom legs end  
at vertices $i$ and $j$ and the upper arms end at $x_i$ and ${x_j}$.   
The cross segment represents an interaction at time $t_1$.  The  
contribution of this diagram, with $x_i$ and $x_j$ summed over $ 
{\La^2}$, but before integrating over $t_1$ is
\be \sum_y \frac d{dt} \left( \phi_1(y,t) \ \phi_2(y,t)\right)\bigg|_ 
{t=t_1}   \ee
where $\phi_1$ satisfies
\begin{eqnarray}
\frac \pa{\pa t} \phi_1 &=& \Del \phi_1 \\
\phi_1(x,0) &=& \de_{x,i}
\end{eqnarray}
with $\Del$ in (23) the Laplacian in $\La$.  Similarly for $\phi_2$.
\end{theorem}

\centerline{--------------------}

This is a simple computation patterned on the continuum or lattice  
version of
\be
-2 \int \vec \nabla \phi_1 \vec \nabla \phi_2 = \int \left( \phi_1  
\Del \phi_2 + \phi_2 \Del \phi_1 \right) =
\int \left( \dot \phi_1 \phi_2 + \phi_2 \dot \phi_1 \right) = \int  
\frac d{dt} (\phi_1 \phi_2)
\ee

We will always be dealing with tree-graph diagrams.  Non tree-graph  
diagrams have contributions that fall off with $t$ (for $r = 0)$, and  
we are considering the $t \ra \infty$ limit.  We do not control the  
uniformity of this limit for the sum over all such diagrams, one  
reason for the formal nature of our computations.  Our final comment  
on the $r \not= 0$ case is that corresponding to Theorem 1, in this  
case, one must consider all ``ladder" diagrams to get the same formal  
estimate, and not just the single tree graph.  Choosing $r \not= 0$  
leads to much more work and no gain.  To do better than our  
"results", if that is possible, one must consider potentials $V$ with  
other than two-body forces.
\bigskip
\centerline{-----------------------}

Theorem 2 will be the analog of Theorem 1 for connected diagrams  
involving $n$ particles, $n$ not necessarily 2.

\bigskip
\centerline{-----------------------}

\begin{theorem}
We consider all $n$-particle (vertex) connected tree-graph  
contributions to (20) involving vertices $z_1,z_2,...,z_n$ at $t=0$.   
We sum over final positions, over $\La^n$.  We do not integrate over  
$t_1$, and over the other times in the order $t_n, t_{n-1}, ..., t_2 
$. Using Theorem 1, and Theorem 2 inductively on $n$, each of these  
integrals will be of an explicit time derivative.  In evaluating  
these integrals \underline{we keep only the lower limit}, as if in
\be	\int^b_a f'(t)dt = f(b) - f(a)	\ee
we keep only the $-f(a)$ term.  We will collect contributions of the  
upper limits later.  The ``contribution of lower limits" is
\be  (-1)^N (n-1)! \frac d{dt} \sum_y \ \prod^n_{i=1} \ \phi_i(y)  
\bigg|_{t=t_1}	\ee
where
\begin{eqnarray}
\frac \pa{\pa t} \; \phi_i &=& \Delta \phi_i	\\
\phi_i(y,0) &=& \delta_{y,z_i}
\end{eqnarray}
\end{theorem}
\bigskip
\centerline{---------------------}
\bigskip
\underline{Proof.}  The proof is by induction on $n$.  Let the  
earliest interaction corresponding to $t_1$ be between vertices $n-1$  
and $n$.  (That is, one of these vertices at $z_n$ at $t=0$ and the  
other at $z_{n-1}$.)  Upon integrating over the later times and  
keeping only lower limits we have terms with vertices $1,2,...,j$ at  
$y$, and vertices $j+1, ..., n-2$ at $y + \vec i$, where vertices $n$  
and $n-1$ are at these two points at $t_1$.  One sums over the value  
of $j$, the points $y$, the unit vectors $\vec i$, and the  
permutation of different possibilities for the vertices attached to $y 
$ and $y+\vec i$.  We let $S$ stand for the sum of the $(n-1)!$  
permutations of vertices $1,...,n-1$.  The following telescopic sum  
relation is the heart of the proof.
\[	\sum_y \sum_{\vec i} S \left\{ \left[ \sum^{n-2}_{j=0} \phi_1 
(y) .... \phi_j(y) \phi_{j+1}(y+\vec i) ... \phi_{n-2}(y+ \vec i) 
\right] \cd \left(\phi_{n-1}(y) - \phi_{n-1}(y+\vec i)\right) \right 
\} \left(\phi_n(y) - \phi_n(y+\vec i)\right)  \]
\[  = \sum_y \sum_{\vec i} S \left[ \phi_1(y) ... \phi_{n-1}(y) -  
\phi_1(y+\vec i) ... \phi_{n-1}(y + \vec i)\right]
\left(\phi_n(y) - \phi_n(y+\vec i)\right)  \]
\be = \sum_y \sum_{\vec i} (n-1)! \left[ \phi_1(y) ... \phi_{n-1}(y)  
- \phi_1(y+\vec i) ... \phi_{n-1}(y + \vec i)\right]
\left(\phi_n(y) - \phi_n(y+\vec i)\right)
\ee
This will represent the contribution from terms where the first  
interaction involves vertex $n$.  One then sums over the $n$  
possibilities for this first vertex coupled.  (There is a factor of 2  
in the first, $t=t_1$, interaction we have absorbed against the fact  
that we are counting double since either end of the interaction at  
$t=t_1$, could have been called $n$.)  Thus the proof is short.  Even  
writing out the details which we have raced over.  But the proof is  
tricky enough, so that it's hard to be sure you're right.  Counting  
is hard.

\centerline{--------------}

Theorem 1 may be included in the statement of Theorem 2 as the $n=2$  
case.  In the next theorem we take the same contributions as in  
Theorem 2 but in addition integrate over $t_1$ from $0$ to $t$, and  
sum over $z_2,...,z_n$ but requiring that $z_1,...,z_n$ be a point in  
$(\La)^{n*}$.  We first define
\be   a_i \equiv - (-1)^i.	\ee

\centerline{--------------}

\begin{theorem} Let $T_n(t)$ be all the contributions considered in  
Theorem 2 for given $n$ in addition integrated over $t_1$ from 0 to $t 
$, and summed over $z_2,...,z_n$ with the requirement that  
$z_1,...,z_n$ be a point in $(\La)^{n*}$.
\be	\lim_{t\ra \infty} T_n(t) = a_{n-1}	\ee
\end{theorem}

\centerline{--------------}

\underline{Proof.}  We detail the proof for $n=2$ which contains all  
the essential points.  From (27) to (29) we have upon integrating  
(27) from 0 to $t$ and summing over $z_2$
\be \sum_{z_2 \not= z_1} \left( \sum_y \ \phi_1(y,t) \phi_2(y,t) -  
\sum_y \ \phi_1(0,t) \phi_2(0,t) \right)  \ee
with $\phi_1(y,0) = \de_{y,z_1}, \ \ \phi_2(y,0) = \de_{y,z_2}$.

\noindent
The second term in (33) therefore vanishes, leaving
\be	 \sum_{z_2 \not= z_1}  \sum_y \phi_1(y,t) \phi_2(y,t)  \ee
which equals
\be	 \sum_{z_2}  \sum_y \phi_1(y,t) \phi_2(y,t)  - \sum_y \ \phi^2_1 
(y,t)  .  \ee
The first term is 1 by
\begin{eqnarray}
\sum_{z_2}   \phi_2(y,t) &=& 1 \\
\sum_y   \phi_1(y,t)  &=& 1.
\end{eqnarray}
and the second term in (35) gets to zero with $t$.  This concludes  
the proof for $n=2$.  For $n=3$, say, the initial points $z_1, z_2,  
z_3$ and $z_1, z_3, z_2$ lead to the same set of diagram  
contributions explaining the lost $(n-1)!$ factors when pursued.

We now define quantities $A_i$ defined recursively from the $a_i$ of  
(31).  We set
\be P \equiv \sum_{j=0} \ A_j \ t^j  \ee
a formal power series in $t$.  Then the $A_i$
are defined by
\be	A_0 = 1 \ee
\be A_i = a_i + \sum^{i-1}_{k=1} a_k \ {\rm coef} \ \left(P^{2k+1}, t^ 
{i-k} \right), \ i > 0    \ee
where using Maple notation, coef $(f,t^s)$ picks out the coefficient  
of $t^s$ in the formal power series $f$.  This is the procedure by  
which we found the $A_i$.  Actually, with $A_1 = 1, A_2 = 2, A_3 =5,  
A_4 =14....$, the $A_i$ are the Catalan numbers, given as
\be	A_i = \frac 1{i+1} \left( \begin{array}{c}
2i \\
\ \\
i
\end{array} \right). \ee

\centerline{--------------}

\begin{theorem} Let $\tilde{T} _ n(t)$ be the analog of $T_n(t)$ of  
Theorem 3 but now \underline{including upper limits}, the whole  
megillah.  Again integrating over all times, summing final states  
over $(\La)^n$, keeping one initial vertex fixed and summing the  
other vertices at $t=0$ over points lying in $(\La)^{n*}$.  Then
\be \lim_{t\ra \infty} \tilde{T} _ n(t) = A_{n-1} \ee
\end{theorem}

\centerline{--------------}

\underline{On the Proof}

The computation of the right side of (42), arising as a solution of  
(40), was perhaps the most difficulty and tricky business I have ever  
been associated with.  Also I would find it extremely difficult to  
write a presentable proof.  Perhaps someone can come up with a  
reasonable proof.  (Skeptics may prefer to call Theorem 4 a  
conjecture, but it is certainly true.)  I content myself here with  
some points on the computation of $A_2$.

Equation (40), for $i=2$ becomes
\be	A_2 + a_2 + 3a_1.	\ee
The contributions of all contributing diagrams,when only the lower  
limit of the $t_2$ integration is kept, is $a_2 = -1$, the first term  
in (43).  This by Theorem 3.

Keeping the upper limit on the $t_2$ integration involves us with  
three cases.

1)  Case 1, associated to $t_1$ is $V_{z_1,z_2}$, associated to $t_2$  
is $V_{z_1, z_3}$.

2)  Case 2, associated to $t_1$ is $V_{z_1,z_2}$, associated to $t_2$  
is $V_{z_2, z_3}$.

3)  Case 3, associated to $t_1$ is $V_{z_2,z_3}$, associated to $t_2$  
is $V_{z_1, z_2}$.

With this notation there is a sum over $z_2$ and $z_3$ with the  
restriction $z_2 \not= z_3 \not= z_1 \not= z_2$.  Here the  
contribution of $z_1, z_2, z_3$ does not equal the contribution of  
$z_1, z_3, z_2$.

Each of these three cases contributes a factor $a_1 = 1$ to equation  
(43).  Case 3 is the most interesting, and we will deal with this one  
case.

The contribution of Case 3 may be represented as
\be \sum_{x_1,x_2,x_3} \ \sum_{z_2,z_3} \int^t_0 \; dt_1 \; \int^t_ 
{t_1} dt_2 \; K(x_1,x_2,x_3,z_1,z_2,z_3,t_1,t_2) \ee
$x_1,x_2,x_3$ lie in $\La^3$ and $z_1, z_2, z_3$ are restricted to $ 
\La^{3*}$.  $x_1,x_2,x_3$ are positions of the vertices at $t=t$ and  
$z_1, z_2, z_3$, the positions at $t=0$.  Recall we are keeping only  
the upper limit in the integral over $t_2$, getting
\be \sum_{x_1,x_3} \ \sum_{z_2,z_3} \int^t_0 \; dt_1 \;  k 
(x_1,x_3,z_2,z_3,t_1) \left( e^{\Delta t}\right)_{z_1,x_1}  \ee
$k$ is the kernel of $a$ two-vertex diagram with a single interaction  
at $t=t_1$.  We rewrite this as
\be  \sum_{x_1}  \left( e^{\Delta t} \right)_{z_1,x_1} \left( \sum_ 
{x_3} \; \sum_{z_2,z_3} \int^t_0 \; dt_1 \;  k\left 
(x_1,x_3,z_2,z_3,t_1 \right) \right). \ee
We wish to compare this expression to
\be  \sum_{x_1}  \left( e^{\Delta t} \right)_{z_1,x_1} \left( \sum_ 
{\bar x_1, x_3} \; \sum_{z_3} \int^t_0 \; dt_1 \;  k\left(\bar  
x_1,x_3,z_2,z_3,t_1 \right) \right).   \ee
\be \ \ \ \ \ \ \ \ \ \ \cong \ \ 1 \ \ \ \ \cdot \ \ \ \ 1.	\ee
by Theorem 3.  Using translation invariance of the kernel $k$, (46)  
and (47) differ by an error that goes to zero with $t$ (from the  
different restrictions on the $z$'s  in the two expressions).  In  
(47) we neglect the restriction that $z_2$ and $z_3$ may not equal  
$z_1$.

\bigskip

\centerline{--------------------}

\bigskip
We turn to the relation of Conjecture 1, equation (18)
\be   \lim_{t\ra \infty} \left( \sum_{\vec{x} \in A} f^e(\vec{x}, t) 
\right)^{1/N} = \left(C_N\right)^{1/N}.  \ee

We work in the limit $N$ large, and $t \ra \infty$ (before $N \ra  
\infty)$.  We thus want
\be \lim_{t \ra \infty} \left( \sum_{{\vec x} \in A} f^e(x,t) \right)^ 
{1/N} = e  \ee
and eschew considering uniformity of $t$ limit with respect to $N$.   
We view $f^e(x, t)$ expressed a sum of products of connected  
diagrams.  For $N$ large we expect this sum to be dominated by terms  
with some fixed number of connectivity patterns.  That is, in terms  
kept in the product there are $x_1N$ two-connected terms in the  
product, $x_2N$ three-connected terms in the product, $x_3N$ four- 
connected terms in the product, and so on.  The expression for $ 
\displaystyle{\sum_{{\vec x} \in A}} f^e(x,t)$ in this limit is
\vfill\eject

\[	\prod_i \left( \frac{A_i \; i!}{N^i} \right)^{x_iN} \ \cdot \ \frac 
{N\;!}{(\sum_i(i+1)Nx_i)!(N - \sum (i+1)Nx_i)!} \ \cdot \]
\be \cdot \ \frac{(\sum(i+1)Nx_i)!}{\Pi(x_iN)!\;\Pi\left( (i+1)! 
\right)^{x_iN}} .  \ee
The first set of parentheses includes the contributions of the  
diagrams $t \ra \infty$ limit, from Theorem 4.  The next factor, a  
ratio of factorials, sums over which set of initial vertices are  
included in the set that are connected to other vertices.  The final  
ratio of factorials sums over the connectivities of the vertices  
(which vertices are connected with which vertices).

Maximizing (51) over the choice of $x_i$ one finds
\be	\lim_{t \ra \infty} \ \left(\sum_{{\vec x} \in A} \; f^e(x,t)  
\right)^{1/N} \ \cong e^q \ee
with
\be	q = -1 + \sum_{i=0} \ A_i \frac{p^{i+1}}{i+1} - \ell n \ p  \ee
where
\be		1 = \sum_{i=0} \ A_i p^{i+1} .	\ee
One wants $q=1$ but (53) and (54) do not yield $q=1$.  (It is not  
clear how to define a solution of (54) for $p$, but no reasonable  
definition works.)  At this point we have reached complete frustration!

Sometime after arriving at this impasse, we decided to consider a  
random walk not involving all the lattice vertices, but rather a  
fraction $\rho$ of the vertices, ``uniformly distributed".  This is  
achieved by ``integrating out" a fraction $(1-\rho)$ of the vertices  
in the probability function $f(x,t)$.  This is done before the  
extension to $f^e\;!$.  It is easy to make the corresponding changes  
in all the computations of this paper, a matter of a day or two.

Replacing (52), (53) and (54) one finds
\be	\lim_{t \ra \infty} \left(\Sigma \; f^e(x,t)\right)^{1/N} \cong e^ 
{\tilde q}	\ee
with $f^e$ here depending on $\rho N$ vertices and with
\be  \tilde q = - 1 + \sum_{i=0} \ A_i \; p^{i+1} \ \frac{\rho^i}{i 
+1} - \ell n \; p	\ee
and
\be	1 = \sum_{i=0} \ A_i \; p^{i+1} \rho^i .  \ee
Where before one wanted $q=1$, here we desire
\be  \tilde q = 1 + \frac{1 - \rho}{\rho} \ \ell n (1 - \rho).	\ee
The miracle that happens is as follows.  For $\rho < 1/2$, equation  
(57) is satisfied with
\be	p = 1 - \rho   \ee
and substituting this expression for $p$ into (56) one finds (56) and  
(58) yield the same formal expansion in powers of $\rho$, valid for $ 
\rho < 1/2$.  It is interesting to note that (56), (57) and (58)  
determines both (59) and the $A_i$ (expanding $p$ and $\tilde q$ in  
powers of $\rho$).

If we set
\be    f(\rho\, p) \equiv \sum_{i=1} \ A_i \; p^i \rho^i  \ee
we can solve, from (57) and (59)
\be	p + pf(\rho\, p) = 1 \ee
\be p = 1 - \rho	\ee
to get
\be	f(z) = \frac{1 - \sqrt{1 - 4z}}{1 + \sqrt{1 - 4z}}	\ee
and see the singularity at $z = 1/4$, or $\rho = \frac 1 2$.

So we are led to believe that Equation (6) may still be true in some  
suitable sense, but that a perturbation expansion development as  
undertaken in this paper is not promising.  Presumably as $\rho$  
increases to value $1/2$ one must consider diagrams of arbitrarily  
high connectivity.

Many interesting questions suggest themselves, of which we choose  
two.  For $\rho < 1/2$, control the perturbation expansion, and  
obtain uniformity in $N$ of the $t \ra \infty$ limits.  Find some way  
of treating $\rho > 1/2$.  The first question is likely a problem  
about which to develop several Ph.D. theses.  The second still  
requires some further ideas to gauge its difficulty.


\vfill\eject

\centerline{\underline{References}}

\begin{itemize}

\item[[1]] Robert T. Powers, ``Heisenberg Model and a Random Walk on  
the Permutation Group", {\it Lett. in Math. Phys.} {\bf 1}, 125-130  
(1976).
\item[[2]]  P. Federbush, ``For the Quantum Heisenberg Ferromagnet,  
Tao to the Proof of a Phase Transition", math-ph/0202044.
\item[[3]]  David C. Brydges, ``A Short Course in Cluster Expansions,  
Phenomenes critiques, systems aleatoires, theories de gauge, Part I,  
II" (Les Houches, 1984), 129-183, North Holland, Amsterdam, 1986.


\end{itemize}



\end{document}
