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\title  {
        Lattice trees and super-Brownian motion\thanks{To appear in
	{\it Canad.\ Math.\ Bull.}}
        }

\author {Eric Derbez and Gordon Slade \\ 
        Department of Mathematics and Statistics \\
        McMaster University \\ 
        Hamilton, ON, Canada L8S 4K1 \\
%       {\tt derbez@icarus.math.mcmaster.ca} \\ 
%       {\tt derbez@mcmaster.ca} \\ 
	{\tt slade@mcmaster.ca}
        }

\begin{document}

\maketitle

\begin{abstract}
This article discusses our recent proof that above eight dimensions 
the scaling limit of sufficiently spread-out
lattice trees is the variant of super-Brownian motion called
integrated super-Brownian excursion (ISE), 
as conjectured by Aldous.  
The same is true for nearest-neighbour lattice trees in sufficiently
high dimensions.
The proof, whose details will appear elsewhere, 
uses the lace expansion.  Here, a related but simpler analysis is applied to
show that the scaling limit of a mean-field theory is ISE, in all dimensions.  
A connection is drawn between ISE and 
certain generating functions and 
critical exponents, which may be useful for the
study of high-dimensional percolation models at the critical point.
\end{abstract}

\section{Introduction}

Lattice trees arise in polymer physics as a model
of branched polymers and in statistical mechanics as an example exhibiting
the general features of critical phenomena.  A lattice tree in the
$d$-dimensional integer lattice ${\Bbb Z}^d$ is a finite connected 
set of lattice bonds containing no cycles.  Thus any two sites in a lattice
tree are connected by a unique path in the tree.  For the nearest-neighbour
model, the bonds are nearest-neighbour bonds
$\{x,y\}$, $x,y \in {\Bbb Z}^d$, $|x-y|=1$ (Euclidean distance), but
we will also consider ``spread-out'' lattice trees constructed from
bonds $\{x,y\}$ with $0< \| x-y\| \leq L$.  Here $L$ is a parameter
which will later be taken large, and the norm is given by
$\|x\| = \max \{x^{(1)},\ldots, x^{(d)}\}$ for 
$x = (x^{(1)},\ldots,x^{(d)}) \in {\Bbb Z}^d$.
We associate the uniform probability measure 
to the set of all $n$-bond lattice trees which contain the origin.

We are interested in the existence of a scaling limit for lattice
trees.  This involves taking a continuum limit of lattice trees, in 
which the size of the trees increases simultaneously with a shrinking
of the lattice spacing, in such a way as to produce a random fractal.  
The nature of the scaling limit is believed to depend
in an essential way on the spatial dimension, but 
the existence of the limit has not been
proven in low spatial dimensions.  The corresponding problem for simple random
walk has the well-known solution that when space is scaled down by
a factor $n^{1/2}$, as the length $n$ of the walk goes to infinity,
there is convergence to Brownian motion in any dimension.
For self-avoiding walks, it has been shown using the lace
expansion that the scaling limit
is also Brownian motion in dimensions $d \geq 5$ \cite{BS85,HS92a,HS92b}.
The same is believed to be true for $d=4$ with a logarithmic adjustment
to the spatial scaling, but in dimensions 2 and 3 a different limit, currently
not understood, is expected.

\begin{figure}
\vspace{7cm}
\caption{\lbfg{tree5000}
A 2-dimensional lattice tree with 5000 vertices, created with the
algorithm of \protect\cite{JM92}.} 
\end{figure} 


Here we give an overview of recent work on high-dimensional lattice trees 
which proves that under certain assumptions
the scaling limit is ISE (integrated super-Brownian excursion) for $d>8$.
To be precise about the assumptions, the scaling limit has been shown
to be ISE for the spread-out model if $d>8$ and $L$ is sufficiently large,
and for the nearest-neighbour model if $d$ is sufficiently large.
Detailed proofs will appear elsewhere \cite{Derb96,DS96}. 
The hypothesis of universality implies that the scaling limit should be
the same for spread-out and nearest-neighbour lattice trees, and assuming
this, our results provide evidence that the scaling limit of nearest-neighbour
lattice trees is ISE for $d>8$.
That the scaling limit of lattice trees
should be ISE for $d>8$ was conjectured by Aldous,
who has emphasized the role of ISE as a model for the
random distribution of mass \cite{Aldo93}.  In particular, Aldous has
shown that ISE arises in various situations where random trees
are randomly embedded into ${\Bbb R}^d$ \cite{Aldo91a,Aldo91b,Aldo93a}.  
ISE is super-Brownian motion (Brownian motion branching on all time scales)
conditioned to have total mass 1, and is closely
connected to the super-processes intensively studied in the probability
literature.   For our purposes, it will be most convenient to understand ISE
as arising via generating functions.  

It is typical of statistical mechanical models that there is an upper critical
dimension above which a model's scaling properties cease to depend on
the dimension and become identical with those of a simpler so-called
mean-field model.  For the self-avoiding walk, the mean-field model is
simple random walk and the upper critical dimension is 4.  For lattice
trees, the fact that ISE occurs as the scaling limit for $d>8$ adds
to the already considerable evidence that the upper critical dimension is 8 
\cite{LI79,BFG86,TH87,HS90b,HS92c}.
The proof of convergence to ISE for $d>8$ is based on the lace expansion,
and involves the treatment of high-dimensional
lattice trees as a small perturbation of a corresponding mean-field model.

This paper is organized as follows.  In Section~\ref{sec-ise} we
introduce a generating function approach to ISE; 
no previous knowledge of ISE is assumed.  A connection is pointed out
between ISE and the critical exponents of statistical mechanics, which
may be relevant for the study of high-dimensional percolation 
models at the critical point.
Section~\ref{sec-sl} contains precise statements of results showing that
the scaling limit of high-dimensional lattice trees is ISE.
Proofs of these results, deferred to \cite{Derb96,DS96}, use the lace
expansion to perturb around a corresponding argument for a mean-field model.
The mean-field model and its connection with ISE is discussed in 
Section~\ref{sec-mf}.

\section{Integrated super-Brownian excursion (ISE)}
\setcounter{equation}{0}
\label{sec-ise}

\subsection{ISE probability densities}

ISE can be considered as an abstract continuous random tree embedded
in $\rd$, rooted at the origin
and having total mass 1 \cite{Aldo93}.  It is designed in such a way that if
$0,x_1, \ldots, x_{m-1}$ are points in $\Rd$ contained in ISE then
there is an underlying tree structure with branch points
$b_1,\ldots,b_{m-2} \in {\Bbb R}^d$ 
and Brownian motion paths connecting the branch
points and the points $0,x_1, \ldots, x_{m-1}$ according to an abstract 
skeleton (minimal spanning subtree); see Figure~\reffg{absskel}.  
There are $(2m-5)!!$ distinct ``shapes''
for the skeleton.  See \cite[(5.96)]{Grim89} for a proof of
this elementary fact; here $N!!$ is defined recursively for $N=-1,1,3,5,7,9,
\ldots$ by $(-1)!!=1$ and $N!!=N(N-2)!!$, $N \geq 1$.  The shapes for
$m=2,3,4$ are illustrated in Figure~\reffg{shapes}.
The joint probability density function for the skeleton shape, 
the durations $t_1,\ldots, t_{2m-3}$ of each
of the Brownian motion paths and the positions of points and branch points
is given by the explicit formula
\eq
\lbeq{ise}
	\left( \sum_{i=1}^{2m-3}t_i \right) e^{-(\sum_{i=1}^{2m-3}t_i)^2/2} 
	\prod_{i=1}^{2m-3} p_{t_i}(y_i),
\en
where the $y_i$ are the vector displacements along the skeleton paths
and $p_{t}(y)$ is the Brownian transition function
\eq
	p_{t}(y) = \frac{1}{(2\pi t)^{d/2}}e^{-y^2/2t}.
\en
In Figure~\reffg{absskel}, the vector displacements 
(in ${\Bbb R}^d$) along the
skeleton paths are $y_1=b_1$, $y_2=b_2-b_1$, $y_3=x_1-b_2$, $y_4=x_2-b_2$,
and so on.   The ordering of the 
labelling of the displacements is fixed according to some convention,
for each skeleton shape $\sigma$.
The density \refeq{ise} is discussed in \cite{Aldo91b,Aldo93a,Aldo93}; see
also \cite{LeGa93}.

\begin{figure}
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\put(345,705){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$b_5$}}}
\end{picture}
\end{center}
\caption{\lbfg{absskel}
The branch points $b_1,\ldots,b_5$ and abstract skeleton 
for a realization of ISE containing 
the sites $0,x_1,\ldots,x_6$.} 
\end{figure} 

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\put(480,765){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$x_2$}}}
\end{picture}
\end{center}
\caption{\lbfg{shapes}
The unique shapes for $m=2,3$ and the three shapes for $m=4$,
joining points $0,x_1, \ldots, x_{m-1}$.} 
\end{figure} 

The joint probability density function for the skeleton shape and the
positions of the points and branch points, with the time variables 
integrated out, is given by
\eq
\lbeq{Amdef}
	A^{(m)}(\sigma; y_1, \ldots, y_{2m-3})
	= \int_0^\infty dt_1 \cdots \int_0^\infty dt_{2m-3} \,
	\left( \sum_{i=1}^{2m-3}t_i \right) e^{-(\sum_{i=1}^{2m-3}t_i)^2/2} 
	\prod_{i=1}^{2m-3} p_{t_i}(y_i).
\en
The right side is independent of the shape $\sigma$ and 
depends only on the displacements $y_i$.  
This is indeed a probability measure, because integrating over the $y_i$'s
simply removes the product over Brownian transition functions, and the
remaining integral over the $t_i$'s
equals $1/(2m-5)!!$, the reciprocal of the number of shapes.
If we leave the $x_i$'s fixed and integrate out the positions of the 
branch points and sum over
all the $(2m-5)!!$ possible shapes, the result is a measure $P^{(m)}$ on
${\Bbb R}^{d(m-1)}$.  The measures $P^{(m)}$, for $m=2,3,4,\ldots$, 
represent the joint probability densities for ISE to contain the sites
$0,x_1,\ldots,x_{m-1}$, and hence form a consistent family.  For example,
\eqarray
	\int_{{\Bbb R}^d} P^{(3)}(x_1,x_2) d^d x_2 \,
	& = & \int_0^\infty dt_1 \int_0^\infty dt_2 \int d^d b \,
	p_{t_1}(b) p_{t_2}(x_1-b)
	\nonumber \\ \nonumber
	&& \times  \int_0^\infty dt_3 \, (t_1+t_2+t_3) e^{- (t_1+t_2+t_3)^2/2} 
	\int d^d x_2 \, p_{t_3}(x_2-b) 
	\\ & = & P^{(2)}(x_1),
\enarray
as can be seen by performing the integrals from right to left, using
the semi-group property of $p_t(x)$ 
for the $b$-integral.  

In the simplest case $m=2$, 
$P^{(2)}(x)$ represents the probability density function for a point chosen
randomly from the distribution of ISE.
Explicitly, for $m=2$,
\eq
	A^{(2)}(x) = P^{(2)}(x) 
	= (2\pi)^{-d/2}\int_0^\infty t^{1-d/2}e^{-t^2/2} e^{-x^2/2t} dt
\en
and
\eq
\lbeq{A2k}
	\hat{A}^{(2)}( k) = \int_0^\infty t e^{-t^2/2} e^{-k^2 t/2} \, dt ,
%	= \frac{1}{\sqrt{\pi}}  \int_0^\infty
%	\frac{\sqrt{x}e^{-x}}{x+k^4/8}dx,
\en
where our convention for the Fourier transform of a function 
$f:{\Bbb R}^{dn} \to \CBbb$ is
\eq
\lbeq{RdFT}
	\hat{f}(k_1, \ldots, k_n)  = 
	\int_{{\Bbb R}^{dn}} f(y_1,\ldots,y_n )
	e^{ik_1\cdot 
	y_1 + \cdots + ik_{n}\cdot y_{n}}\, 
	d^d y_1\cdots d^d y_n , \quad k_i \in {\Bbb R}^d.
\en
The integral \refeq{A2k} can be written in terms of 
the parabolic cylinder function 
$D_{-2}$ as $\hat{A}^{(2)}( k) = e^{k^4/16}D_{-2}(k^2/2)$
\cite[3.462.1]{GR65}. 
% and 3.383.6
For general $m \geq 2$,
\eq
\lbeq{Amhat}
	\hat{A}^{(m)}(\sigma; k_1,\ldots,k_{2m-3}) 
	= \int_0^\infty dt_1 \cdots \int_0^\infty dt_{2m-3}
	\left( \sum_{i=1}^{2m-3} t_i \right) 
	e^{-(\sum_{i=1}^{2m-3}t_i)^2/2} e^{-\sum_{i=1}^{2m-3}k_i^2 t_i/2}  .
\en

\subsection{ISE and generating functions}
\label{sec-isegf}

In this section, we indicate that the family of probability distributions
$A^{(m)}$, $m=2,3,4,\ldots$, can be encoded simply in terms of generating
functions.  As an analogy, consider the generating function
\eq
\lbeq{Bzk}
	B_z(k) = \frac{1}{k^2+1-z}, \quad k \in {\Bbb R}^d .
\en
Expanding in a power series, we can write 
$B_z(k) = \sum_{n=0}^\infty b_n(k)z^n$, where $b_n(k) = (1+k^2)^{-n-1}$.  
The generating
function $B_z(k)$ thus gives rise to the 
(unit-time) Brownian transition function via
\eq
	e^{-k^2/2} = \lim_{n \to \infty} \frac{b_n(k(2n)^{-1/2})}{b_n(0)} .
\en

For ISE, beginning with $m=2$, we define
\eq
\lbeq{Cdef}
	C_z(k) = \frac{1}{k^2+\sqrt{1-z}}
\en
where the square root is 
defined to be positive for real $z<1$ and has branch cut $[1,\infty)$ 
in the $z$-plane.  
This definition was motivated by the considerations of Section~\ref{sec-ISElt}
below.
Define coefficients $c_n(k)$ by
\eq
	C_z(k) = \sum_{n=0}^\infty c_n(k) z^n, \quad |z|<1,
\en
so that 
\eq
\lbeq{cnkint}
	c_n(k) = \frac{1}{2\pi i} \oint_\Gamma C_z(k) \frac{dz}{z^{n+1}}
\en
where $\Gamma$ is a small circle centred at the origin.  The following lemma
provides a link with ISE.  

\begin{lemma}
\label{lem-cnk}
For any $k \in {\Bbb R}^d$, 
\eq
\lbeq{cnkn}
	c_n(kn^{-1/4}) \sim \frac{1}{\sqrt{\pi n}} 
	\int_0^\infty t e^{-t^2/2} e^{- \sqrt{2} k^2 t}  \, dt 
	= \frac{1}{\sqrt{\pi n}} \hat{A}^{(2)}( 2^{3/4} k)
	\quad \quad \mbox{as} \; \; n \to \infty .
\en
In particular,
\eq
\lbeq{cnklim}
	\lim_{n \to \infty}  \frac{c_n(kn^{-1/4})}{c_n(0)}
	= \hat{A}^{(2)}( 2^{3/4} k) .
\en
\end{lemma}

\proof 
For $k =0$, $c_n(0)$ is given by a binomial coefficient and is asymptotic
to $(\pi n)^{-1/2}$, in agreement with \refeq{cnkn}.  Suppose henceforth
that $k \neq 0$.
Beginning with \refeq{cnkint}, we deform the contour of integration to
the branch cut and make the change of variables $w=n(z-1)$.  
This gives
\eq
	c_n(kn^{-1/4}) = \frac{1}{\sqrt{n}} \frac{1}{2\pi i} \int_{\Gamma'}
	\frac{1}{k^2+\sqrt{-w}} \frac{dw}{(1+w/n)^{n+1}},
\en
where the contour
$\Gamma'$ runs around the branch cut $[0,\infty)$ in the $w$-plane,
oriented from right to left below the
cut and from left to right above the cut.  Then we use
\eq
\lbeq{exprep}
	\frac{1}{k^2  + \sqrt{-w}} 
	=  \sqrt{2} \int_0^\infty dt \, 
	\exp [ -\sqrt{2}\, t(k^2  + \sqrt{-w}) ] .
\en
Taking into account the correct branches of the square root on either side
of the branch cut, and applying Fubini's theorem, gives
\eq
	c_n(kn^{-1/4}) = \frac{\sqrt{2}}{\pi \sqrt{n}} \int_0^\infty dt 
	\, e^{-\sqrt{2}k^2 t} \int_0^\infty  
	\frac{dw}{(1+w/n)^{n+1}}  \sin (t\sqrt{2w}).
\en
Since $(1+\frac{w}{n})^{n+1} \geq 1 + \frac{(n+1)n}{2}(\frac{w}{n})^2
\geq 1+\frac{w^2}{2}$ for all $n \geq 1$, the dominated convergence theorem
can be applied to give
\eq
\lbeq{cnlim}
	c_n(kn^{-1/4}) \sim \frac{\sqrt{2}}{\pi \sqrt{n}} \int_0^\infty dt 
	\, e^{-\sqrt{2}k^2 t} \int_0^\infty dw \, e^{-w} \sin (t\sqrt{2w}).
\en
The desired result then follows, since 
$\int_0^\infty dw \, e^{-w} \sin (t\sqrt{2w}) = (\pi/2)^{1/2}t e^{-t^2/2}$.
\qed

For any shape $\sigma$ and any $m \geq 3$, recalling the definition of
$C_z(k)$ in \refeq{Cdef}, let
\eq
\lbeq{Cmdef}
	C^{(m)}_z(\sigma; k_1,\ldots , k_{2m-3}) = \prod_{j=1}^{2m-3} C_z(k_j).
\en
We write the Maclaurin series of \refeq{Cmdef} as
\eq
	C^{(m)}_z(\sigma; k_1,\ldots , k_{2m-3}) 
	= \sum_{n=0}^\infty c_n^{(m)}(\sigma; k_1,\ldots,k_{2m-3}) z^n, 
	\quad |z|<1.
\en
A calculation similar to that used in the proof of Lemma~\ref{lem-cnk},
using \refeq{exprep} for each of the $2m-3$ factors in \refeq{Cmdef}
and a limiting argument if any $k_j=0$, 
then gives
\eq
\lbeq{cnmasy}
	c_n^{(m)}(\sigma; k_1n^{-1/4}, \ldots , k_{2m-3}n^{-1/4})
	\sim \frac{2^{m-2}n^{m-5/2}}{\sqrt{\pi}}
	\hat{A}^{(m)}(\sigma; 2^{3/4}k_1, \ldots, 2^{3/4}k_{2m-3}).
\en
Since $\hat{A}^{(m)}(\sigma; 0,\ldots, 0) = 1/(2m-5)!!$ is the reciprocal
of the number of shapes, this gives
\eq
\lbeq{cmnlim}
	\lim_{n \to \infty} 
	\frac{c_n^{(m)}(\sigma; k_1 n^{-1/4},\ldots,k_{2m-3} n^{-1/4})}
	{\sum_\sigma c_n^{(m)}(\sigma;0,\ldots,0)}
	= \hat{A}^{(m)}(\sigma; 2^{3/4}k_1, \ldots, 2^{3/4}k_{2m-3}).
\en

Thus the distributions $A^{(m)}$ arise as the scaling limits of the
coefficients of the generating functions $C^{(m)}_z$, $m \geq 2$.  
In particular, this essential 
aspect of ISE follows solely from \refeq{Cdef} and \refeq{Cmdef}.
Moreover, small perturbations of \refeq{Cdef} and \refeq{Cmdef} will not
affect the scaling limit; see Section~\ref{sec-mfa}.  
%For example, adding a term proportional to
%$k^{2+\epsilon}$ or $(1-z)^{\epsilon +1/2}$ to the denominator of the
%right side of \refeq{Cdef} will not change the scaling limit.
In Sections~\ref{sec-sl} and \ref{sec-mf} below, we indicate how 
generating functions
can be related directly to \refeq{ise} itself, rather than to its 
integral \refeq{Amdef}.
%The occurrence of the product of generating functions in \refeq{Cmdef}
%corresponds to the independence of the branches of 
%super-Brownian motion, i.e., of ISE in the absence of any
%restriction that the total mass equal 1.

\subsection{ISE and critical exponents}
\label{sec-ISEce}

This section shows that the
generating function approach to ISE outlined in Section~\ref{sec-isegf}
provides a link between ISE and the critical exponents of statistical 
mechanics.  For lattice trees, it is the exponents $\eta$ and $\gamma$
which are relevant, while for percolation it is $\eta$ and $\delta$.

\subsubsection{Lattice trees}
\label{sec-ISElt}

\noindent
A lattice tree containing the points $0,x_1,\ldots,x_{m-1}$ 
has a unique skeleton (the minimal spanning subtree for
$0,x_1,\ldots,x_{m-1}$), with $m-2$ branch points $b_1,\ldots,
b_{m-2}$ and $2m-3$ paths.  Let $y_1,\ldots,y_{2m-3}$ denote the
vector displacements of the skeleton paths, as in Figure~\reffg{skel},
and let $t_n^{(m)}(\sigma; y_1,\ldots,y_{2m-3})$ denote the number of $n$-bond
trees having skeleton of shape $\sigma$ and skeleton path displacements
$y_1,\ldots,y_{2m-3}$.  
Equivalently, $t_n^{(m)}(\sigma; y_1,\ldots,y_{2m-3})$ is the number of
$n$-bond lattice trees containing the branch points $b_1,\ldots,b_{m-2}$
and sites $0,x_1,\ldots,x_{m-1}$ consistent with the displacements
$y_1,\ldots,y_{2m-3}$ and joined together by a skeleton of shape $\sigma$.
Define
\eq
	G^{(m)}_z(\sigma; y_1,\ldots,y_{2m-3}) = \sum_{n=0}^\infty
	t_n^{(m)}(\sigma; y_1,\ldots,y_{2m-3}) z^n.
\en
It can be shown via a subadditivity argument that 
summing the above expression over $y_1,\ldots,y_{2m-3}$ results in 
a power series having a radius of convergence $z_c \in (0,\infty)$, independent
of $m$.
The two principal ingredients involved in the 
proof of convergence of lattice trees to ISE in high dimensions
are to show that the functions $G_z^{(m)}$ obey, to leading order,
\refeq{Cdef} and \refeq{Cmdef}.

\begin{figure} 
\begin{center}
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\put(185,660){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$0$}}}
\put(225,640){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$b_1$}}}
\put(235,590){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$x_1$}}}
\put(305,625){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$b_2$}}}
\put(305,585){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$b_3$}}}
\put(295,550){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$x_2$}}}
\put(365,660){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$x_3$}}}
\put(385,600){\makebox(0,0)[lb]{\raisebox{0pt}[0pt][0pt]{$x_4$}}}
\end{picture}
\end{center}
\caption{\lbfg{skel}
A lattice tree containing the sites $0,x_1,x_2,x_3,x_4$ with its corresponding
skeleton and branch points $b_1,b_2,b_3$.  The vector displacements along the
skeleton paths are $y_1=b_1$, $y_2=x_1-b_1$, $y_3=b_2-b_1$, $y_4=x_3-b_2$,
and so on.  The ordering of the 
labelling of the displacements is fixed according to some convention,
for each skeleton shape $\sigma$.} 
\end{figure} 

In terms of critical exponents, the Fourier transform of the two-point
function is believed to behave asymptotically as 
\eq
\lbeq{eta-gamma}
	\hat{G}_{z_c}^{(2)}(k) \sim \frac{c_1}{k^{2-\eta}}
	\;\; \mbox{as} \; k \to 0 , \quad  
	\hat{G}_z^{(2)}(0) \sim
	\frac{c_2}{(1-z/z_c)^\gamma}
	\;\; \mbox{as} \;  z \to z_c,
\en
with the mean-field values $\eta=0$ and $\gamma=\frac{1}{2}$ for $d>8$.
Here the Fourier transform is the discrete one, given for $f : {\Bbb Z}^d \to
{\Bbb C}$ by
\eq
	\hat{f}(k) = \sum_{x \in {\Bbb Z}^d} f(x) e^{ik\cdot x},
	\quad k \in [-\pi,\pi]^d.
\en
For $d>8$, the simplest possible combination of the two asymptotic relations
in \refeq{eta-gamma} is
\eq
\lbeq{etagamma}
	\hat{G}_z^{(2)}(k) =
	\frac{C_1}{D_1^2k^{2} + 2^{3/2}(1-z/z_c)^{1/2}}
	+ \mbox{error term},
\en
where $C_1$ and $D_1$ are positive constants depending on $d$ and $L$, and
the factor $2^{3/2}$ has been inserted for later convenience.
The error term is meant to be of lower order than the main term, 
in some suitable sense, as $k \to 0$ and $z \to z_c$. 
The first step in the proof of convergence to ISE
is to show that \refeq{etagamma} does
hold in high dimensions, with a controlled error,
so that much as \refeq{Cdef} leads to \refeq{cnklim}, 
\eq
	\lim_{n \to \infty} 
	\frac{\hat{t}_n^{(2)}(kD_1^{-1}n^{-1/4})}{\hat{t}_n^{(2)}(0)}
	= \hat{A}^{(2)}(k).
\en
This can be interpreted as asserting that in the scaling limit the distribution
of a site $x D_1 n^{1/4}$ in an $n$-bond lattice tree is the 
distribution of a point from ISE.  It was the anticipation of this
conclusion which motivated Section~\ref{sec-isegf}.

The second step in the proof of convergence to ISE
involves showing that in high dimensions there is an approximate
independence of the form 
\eq
\lbeq{Hzasymp}
	\hat{G}_z^{(m)}(\sigma;k_1,\ldots,k_{2m-3}) 
	= v^{m-2} \prod_{j=1}^{2m-3} \hat{G}_z(k_j)
	+ \mbox{error term},
\en
where $v$ is a finite positive constant which translates the
self-avoidance interactions of lattice trees into a renormalized
vertex factor.
Then, with sufficient control on the error term in \refeq{Hzasymp},
the finite-dimensional distributions can be shown to have
scaling limit $\hat{A}^{(m)}(\sigma; k_1,\ldots,k_{2m-3})$,
just as \refeq{Cmdef} leads to \refeq{cmnlim}.

We believe that the above discussion should apply also to lattice
animals for $d>8$, yielding ISE for their scaling limit for $d>8$ and
consistent with the general belief that lattice trees and 
lattice animals have the same
scaling properties in all dimensions.


\subsubsection{Percolation}
\label{sec-perc}

Reasoning of the above type has led 
Hara and Slade to conjecture
that for $d>6$ the scaling limit of large percolation clusters at the
critical point is ISE.  
The remainder of this section discusses the basis for
the conjecture.
Further discussion of the scaling limit can be found in \cite{Aize96}.

Consider independent Bernoulli
bond percolation on ${\Bbb Z}^d$ with $p$ fixed and
equal to its critical value $p_c$ \cite{Grim89}.  
Let $C(0)$ denote the random set of sites connected to $0$, let
\eq
	\tau_n^{(2)}(x) = P_{p_c}\{C(0) \ni x, |C(0)|=n \}
\en
denote the probability at the critical point that the origin is connected
to $x$ via a cluster containing $n$ sites, and let 
\eq
\lbeq{Mzdef}
	T_{z}^{(2)}(x) = \sum_{n=1}^\infty \tau_n^{(2)}(x) z^{n},
	\quad |z| \leq 1.
\en
The generating function \refeq{Mzdef} converges absolutely if $|z|\leq 1$.
Let $\tau(p;0,x)$ denote the probability that $0$ is connected to $x$.  Then
$T_1^{(2)}(x) = \tau(p_c ; 0,x)$ (assuming no infinite cluster at $p_c$).
The conventional definitions 
\cite[Section~7.1]{Grim89} of the critical exponents $\eta$ and $\delta$
lead to
\eq
	\hat{T}^{(2)}_1(k) \sim \frac{c_1}{k^{2-\eta}}, \;\; \mbox{as} \;
	k \to 0, \quad
	\hat{T}_z^{(2)}(0) \sim \frac{c_2}{(1-z)^{1-1/\delta}}, \;\; 
	\mbox{as} \; z \to 1.
\en
Using the mean-field values $\eta =0$ and $\delta = 2$
above six dimensions, the simplest combination of the above asymptotic
relations for $d>6$, analogous to \refeq{etagamma}, is
\eq
\lbeq{Mapprox}
	\hat{T}_z^{(2)}(k) 
	= \frac{C_2}{D_2^2k^{2} + 2^{3/2}(1-z)^{1/2}}
	+ \mbox{error term},
\en
for some constants $C_2$, $D_2$.

Proving \refeq{Mapprox} 
would provide an analogue of \refeq{Cdef}.  With sufficient control of the
error in \refeq{Mapprox}, 
contour integration with respect to $z$ may then lead to 
\eq
	\lim_{n \to \infty} 
	\frac{\hat{\tau}_n^{(2)}(kD_2^{-1}n^{-1/4})}{\hat{\tau}_n^{(2)}(0)}
	= \hat{A}^{(2)}(k).  
\en
The above equation can be interpreted as asserting that
in the scaling limit the distribution of a site $x D_2 n^{1/4}$ in the cluster
of the origin, conditional on the cluster being of size $n$,
is the distribution of a point in ISE.

The study of percolation clusters containing $m \geq 3$ sites
is more difficult than for lattice trees because for percolation
there is not a unique skeleton nor therefore unique branch points 
and corresponding displacements for a cluster
containing $m$ specified points (the same is true for lattice animals).  
Nevertheless, for $d>6$ this lack of
uniqueness should be a ``local'' effect whose role is unimportant in the 
scaling limit, and we expect that a relation of the form
\eq
\lbeq{Mlapprox}
	\hat{T}^{(m)}_z(\sigma; k_1,\ldots, k_{2m-3}) = v^{m-2}
	\prod_{i=1}^{2m-3} \hat{T}^{(2)}_z(k_i) + 
	\mbox{error term}
\en
should hold for a suitably defined generating function 
$\hat{T}^{(m)}_z(\sigma; k_1,\ldots, k_{2m-3})$.
Such a statement would provide a relation
analogous to \refeq{Hzasymp}, and could lead to the ISE
correlation $\hat{A}^{(m)}(\sigma;
k_1, \ldots, k_{2m-3})$ in the scaling limit.  
An asymptotic relation in the spirit of \refeq{Mlapprox} was conjectured
for $d>6$ already in \cite{AN84}.  There it was argued that the
sum, over sites $x_1,\ldots,x_{m-1}$ in the lattice, of the 
probability that the cluster of the origin contains $x_1,\ldots,x_{m-1}$,
should behave asymptotically
as $v^{m-2}\hat{\tau}(p;0)^{2m-3}$
in the limit $ p \to p_c$, where $v$ is a positive constant. 

Hara and Slade are currently investigating whether 
the method of \cite{DS96} can be combined with the method of \cite{HS90a} 
to prove the conjecture.    
The methods of \cite{NY93,NY95}
could possibly serve as a starting point to study related questions
for oriented percolation.


\section{Lattice trees in high dimensions}
\setcounter{equation}{0}
\label{sec-sl}

In this section, we state precise results for the
scaling limit of high-dimensional lattice trees.  We begin by
introducing some notation and recalling some previous results.

Let $t^{(1)}_n$ denote the number of $n$-bond lattice trees containing
the origin, with $t^{(1)}_0 =1$.  
By a subadditivity argument \cite{Klei81}, the limit 
$z_c^{-1} = \lim_{n \to \infty} (t^{(1)}_n)^{1/n}$ exists and is positive
and finite.
For $m \geq 2$, let $t^{(m)}_{n}(\sigma; \vec{y},\vec{s})$ 
be the number of $n$-bond lattice trees with 
skeleton shape $\sigma$ and skeleton displacements 
$y_1,\ldots, y_{2m-3}$ as in Figure~\reffg{skel}, with the 
skeleton path corresponding to $y_i$ consisting of $s_i$ 
steps ($i=1,\ldots,2m-3)$.  We also define
\eqarray
	t^{(m)}_{n}(\sigma; \vec{y}) & = & 
	\sum_{\vec{s}} t^{(m)}_{n}(\sigma; \vec{y},\vec{s}) , \\
	t^{(m)}_{n}(\vec{y}) & = & 
	\sum_{\sigma} t^{(m)}_{n}(\sigma; \vec{y}).
\enarray
We will make use of Fourier transforms with respect to the $\vec{y}$ variables,
for example,
\eq
	\hat{t}_n^{(m)}(\sigma; \vec{k}) 
	= \sum_{\vec{y}} t_n^{(m)}(\sigma; \vec{y}) 
	e^{i(k_1 \cdot y_1 + \cdots + k_{2m-3}\cdot y_{2m-3})},
	\quad k_i \in [-\pi,\pi]^d .
\en
Note that for $m \geq 2$,
\eq
\lbeq{tnm0}
	\hat{t}_n^{(m)}(\vec{0}) 
	= \sum_\sigma \sum_{\vec{y}} t_n^{(m)}(\sigma; \vec{y})
	= (n+1)^{m-1} {t}_n^{(1)}.
\en
To see this, perform the sums over $\sigma$ and $\vec{y}$ by first fixing
the values of $x_1,\ldots,x_{m-1}$ and then summing over all shapes and
branch points compatible with $x_1,\ldots,x_{m-1}$ as in Figure~\reffg{skel}.
This leaves the sum over $x_1,\ldots,x_{m-1}$ of the number of $n$-bond
lattice trees containing the origin and $x_1,\ldots,x_{m-1}$.  Then \refeq{tnm0}
follows from the fact that an $n$-bond lattice tree contains $n+1$ sites.

In \cite{HS90b,HS92c}, some critical exponents for lattice trees were
proven to exist and to assume their mean-field values 
when $d>8$.  More precisely, the results were
obtained for the nearest-neighbour model when $d \geq d_0$ for some 
undetermined 
dimension $d_0 > 8$, and for spread-out trees when $d >8$ and $L$ is
sufficiently large depending on $d$.  We will refer to 
either of these restrictions
on the dimension and $L$ as the ``high-dimension condition.''

In particular, it was shown in \cite{HS92c} that under the high-dimension
condition there is a positive constant $A$ (depending on $d$ and $L$)
such that
\eq
\lbeq{theta}
	{t}_n^{(1)} \sim A z_c^{-n} n^{-3/2}, \quad \mbox{as}\;\; n \to \infty.
\en
In terms of the critical exponent $\theta$ occurring 
in the conjectured relation 
${t}_n^{(1)} \sim A z_c^{-n} n^{1-\theta}$, this says that 
$\theta = \frac{5}{2}$ under the high-dimension condition.
The bounds 
$c_1 n^{-c_2 \log n}z_c^{-n} \leq t_n^{(1)} \leq c_3 n^{1/d}z_c^{-n}$,
proved respectively in \cite{Jans92} and \cite{Madr95}
and believed not to be sharp, are the best general
bounds known at present for $t_n^{(1)}$.  
The critical exponent $\theta$ is formally 
related to the exponent $\gamma$, discussed in Section~\ref{sec-ISElt} and 
defined by $\hat{G}_z^{(2)}(0) \sim \mbox{const.}(1-z/z_c)^{-\gamma}$
as $z \to z_c$, by $\theta = 3-\gamma$.  It had been proved earlier,
in \cite{HS90b}, that $\gamma = \frac{1}{2}$ under the high-dimension
condition.  With \refeq{theta}, \refeq{tnm0} gives
\eq
	\hat{t}_n^{(m)}(\vec{0}) \sim A z_c^{-n} n^{m-5/2}.
\en

Another critical exponent involves $R_n$, 
the average radius of gyration of $n$-bond trees.  The squared average radius
of gyration is defined by
\eq
	R_n^2 = \frac{1}{t_n^{(1)}} \sum_{T : |T|=n, T \ni 0} R(T)^2,
\en
where
\eq
	R(T)^2 = \frac{1}{|T|+1}\sum_{x \in T} |x-\bar{x}_T|^2
\en
is the squared radius of gyration of $T$.  Here we write $|T|$ to denote the
number of bonds in a lattice tree $T$, 
$\bar{x}_T = (|T|+1)^{-1}\sum_{x \in T} x$ 
to denote the centre
of mass of $T$ (considered as a set of unit masses at the {\em sites}\/ of
$T$), and we say that $x \in T$ if $x$ is an element of a bond
in $T$.  
Equivalently,
\eq
	R_n^2 = \frac{1}{2\hat{t}_n^{(2)}(0)} \sum_x |x|^2 t_n^{(2)}(x).
\en
It is believed that there is a critical exponent $\nu$
such that $R_n \sim Dn^{\nu}$ as $n \to \infty$, but very little has been
proved rigorously about this in general dimensions.

Under the high-dimension condition, 
it is proved in \cite{HS92c} that
\eq
\lbeq{nu}
	R_n \sim Dn^{1/4} ,
\en
so that $\nu = \frac{1}{4}$.
The amplitude $D$ of \refeq{nu} is a
positive constant which depends on $d$, and for the spread-out
model, also on $L$.  Asymptotically, for fixed $d$,
$D$ behaves like a multiple of $L$ as $L \to \infty$.  For later use,
we define
\eq
\lbeq{D1def}
	D_1 = 2^{3/4} d^{-1/2} \pi^{-1/4} D.
\en
The fact that $\nu = \frac{1}{4}$ under the high-dimension condition
can be interpreted as saying
that the mass $n$ of a tree grows on average like the fourth power of
its radius, suggesting a 4-dimensional nature for 
lattice trees in high dimensions.  This compares well with the 
fact that ISE has Hausdorff dimension 4 (as can be shown to follow
from \cite[Theorem~1.4]{DIP89}), 
and also permits the upper critical dimension
8 to be interpreted as the dimension above which two 4-dimensional
objects generically do not intersect.  

Define
\eq
	p_n^{(m)}(\sigma;\vec{y}) = \frac{t_n^{(m)}(\sigma; \vec{y})}
	{\hat{t}_n^{(m)}(\vec{0})},
\en
which is the probability that an $n$-bond lattice tree containing the origin
has a skeleton of shape $\sigma$ mediating displacements $y_1,\ldots,y_{2m-3}$.
The following theorem \cite{Derb96,DS96}
shows that this distribution has the corresponding
ISE distribution as its scaling limit, under the high-dimension condition.

\begin{theorem}
\label{thm-fdd}
Let $m \geq 2$ and $k_i \in \Rd$ ($i=1,\ldots,2m-3$).
For nearest-neighbour trees in sufficiently high dimensions $d \geq d_0$, 
and for sufficiently spread-out trees above eight dimensions, 
\[
	\lim_{n \to \infty} \hat{p}_n^{(m)}(\sigma; \vec{k}D_1^{-1}n^{-1/4} )
	= \hat{A}^{(m)}(\sigma; \vec{k}),
\]
where $D_1$ is given by \refeq{D1def}.
\end{theorem}

For a more refined statement than Theorem~\ref{thm-fdd}, 
we wish to see the integrand 
\eq
\lbeq{Amhatint}
	\hat{a}^{(m)}(\sigma; \vec{k}, \vec{t}) \equiv
	\left( \sum_{i=1}^{2m-3}t_i \right) e^{-(\sum_{i=1}^{2m-3}t_i)^2/2} 
	e^{-\sum_{i=1}^{2m-3} k_i^2 t_i /2}
\en
of the integral representation \refeq{Amhat} of $\hat{A}^{(m)}(\sigma; \vec{k})$
as corresponding to Brownian motion
paths arising from the scaling limit of the skeleton.  For this, 
we denote by
\eq
	p_n^{(m)} (\sigma; \vec{y},\vec{s}) =
	\frac{t^{(m)}_{n}(\sigma; \vec{y},\vec{s})}{\hat{t}_n^{(m)}(\vec{0})}
\en
the probability that an $n$-bond lattice tree 
containing the origin
has a skeleton of shape $\sigma$ mediating displacements $y_1,\ldots,y_{2m-3}$
with skeleton paths of respective lengths 
$s_1,\ldots, s_{2m-3}$.  The following theorem
\cite{DS96} shows that, for $m=2,3$, the skeleton paths converge 
to Brownian motions,
with the weight factor appearing in \refeq{Amhatint}.

\begin{theorem}
\label{thm-bb}
Let $m = 2$ or $m=3$, 
$k_i \in \Rd$ and $t_i \in [0,\infty)$ ($i=1,\ldots,2m-3$).
For nearest-neighbour trees in sufficiently high dimensions $d \geq d_0$, 
and for sufficiently spread-out trees above eight dimensions, there is
a constant $T_1$ depending on $d$ and $L$ such that
\eq
\lbeq{bbeq}
	\lim_{n \to \infty} (T_1 n^{1/2})^{2m-3}\,
	\hat{p}_n^{(m)} (\sigma; \vec{k}D_1^{-1}n^{-1/4},
	\vec{t}\, T_1 n^{1/2})
	= \hat{a}^{(m)}(\sigma; \vec{k}, \vec{t}) .
\en
(As an argument of $\hat{p}^{(m)}_n$, $t_iT_1n^{1/2}$ is to be interpreted as
its integer part $\lfloor t_i T_1 n^{1/2} \rfloor$.)
\end{theorem}

We believe that Theorem~\ref{thm-bb} is valid for also for $m \geq 4$,
but we encounter technical difficulties in attempting a proof.
It would be of interest to extend Theorem~\ref{thm-bb} to general $m$,
and also to investigate tightness with the aim of obtaining 
a stronger statement of convergence to ISE.

The factor $(T_1n^{1/2})^{2m-3}$ on the left side of \refeq{bbeq}
has a natural interpretation.
In fact, writing $t_i = s_i /(T_1 n^{1/2})$ in the right side of 
\refeq{bbeq}, and
then multiplying by $(T_1n^{1/2})^{-(2m-3)}$ and summing over the 
$s_i$, gives
a Riemann sum approximation to \refeq{Amhat}.
Theorem~\ref{thm-bb} indicates that skeleton paths 
with length of order $\sqrt{n}$
are typical, and that the skeleton paths converge to Brownian motion paths in
the scaling limit.

The proofs of Theorems~\ref{thm-fdd} and \ref{thm-bb} are given in
\cite{Derb96,DS96}.  The proofs
use generating functions and contour integration, with the generating
functions controlled using the lace expansion.
To define the generating functions, we begin with $m=1$ and define
\eq
\lbeq{gdef}
	g(z) = \sum_{n=0}^\infty t_n^{(1)} z^{n}
	= \sum_{T:T\ni 0} z^{|T|}.
\en
For $m \geq 2$, let 
\eq
\lbeq{Gmdef}
	G_z^{(m)}(\sigma; \vec{y})=
	\sum_{n=0}^\infty t^{(m)}_n(\sigma; \vec{y}) z^{n}.
\en
The series in \refeq{Gmdef} and \refeq{gdef} converge if $|z|<z_c$.  
When the high-dimension condition is satisfied, it was shown in \cite{HS90b}
that $g(z_c) < \infty$.

For $m=2$, there is only one shape, and 
the two-point function $G_z^{(2)}(x)$ is the generating
function for lattice trees containing the sites $0$ and $x$.
A lattice tree $T$
containing these two sites contains a unique self-avoiding walk
$\omega$ connecting them (the skeleton), and removing the bonds 
of $\omega$ from $T$ leaves behind trees $R_0, R_1, \ldots, R_{|\omega|}$ 
attached along the skeleton sites, with
the restriction that as sets of sites $R_i \cap R_j = \emptyset$ if $i \neq j$.
Explicitly,
\eq
\lbeq{G2I}
	G_z^{(2)}(x) = \sum_{\omega : 0 \to x} z^{|\omega|}
	\prod_{i=0}^{|\omega|}
	\sum_{R_i: R_i \ni \omega(i)} z^{|R_i|} 
	I[R_j \cap R_k = \emptyset \mbox{ if } j \neq k],
\en
where the sum is over self-avoiding walks $\omega$ from $0$ to $x$
(spread-out or nearest-neighbour as appropriate) and $|\omega|$ denotes
the number of steps of $\omega$.
There is a differential equation relating $\hat{G}_z^{(2)}(0)$ and $g_z$,
since by \refeq{tnm0}
\eq
\lbeq{G2de}
	\hat{G}_z^{(2)}(0) = \sum_{n=0}^\infty \hat{t}^{(2)}_n(0)z^{n}
	= \sum_{n=0}^\infty (n+1) t_n^{(1)} z^{n} 
	= \frac{d}{dz} \left( zg(z) \right).
\en

The lace expansion was used in \cite{HS90b} to show,
under the high-dimension condition, that 
\eq
\lbeq{G2k}
	\hat{G}_z^{(2)}(k) = \frac{g_z+\hat{\Pi}_z(k)}
	{1-z\Omega \hat{D}(k)(g_z+\hat{\Pi}_z(k))},
\en
where $\Omega$ is the number of bonds emanating from the origin ($2d$ for
the nearest-neighbour model and $(2L+1)^d-1$ for the spread-out model), and
\eq
\lbeq{Dkdef}
	\hat{D}(k) = \frac{1}{\Omega}\sum_{x \in \Omega} e^{ik\cdot x}
\en
(letting $\Omega$ denote also the set of sites which together with the
origin form a bond).  The function 
$\hat{\Pi}_z(k)$ is complicated but explicit and is
well-understood    
under the high-dimension condition, where it can be regarded as
a small perturbation of $g(z)$.
It can be shown \cite{Derb96,DS96} that under the high dimension condition
\eq
	\hat{G}_z^{(2)}(k) = 
	\frac{C_1}{D_1^2 k^2 + 2^{3/2}\sqrt{1-z/z_c}} 
	+ \mbox{error term},
\en
with appropriate control on the error term.  This shows
that the critical exponent $\eta$ takes on its mean-field value $\eta =0$.
The bounds on the error term are sufficient
that, using Lemma~\ref{lem-cnk}, and \cite[Lemma 3.3]{HS92a} or 
\cite[Lemma 6.3.3]{MS93} for the error term, the result 
of Theorem~\ref{thm-fdd} for the two-point function can be concluded.
For the $m$-point function, the basic step
involves showing that  
\eq
\lbeq{Gz3}
	\hat{G}_z^{(m)}(\sigma; k_1,\dots,k_{2m-3}) = v^{m-2} \,
	\prod_{j=1}^{2m-3} \hat{G}_z^{(2)}(k_j) 
	+ \mbox{error term},
\en
with $v$ a positive constant.  
With sufficient control on the error term in 
\refeq{Gz3}, this is enough to prove Theorem~\ref{thm-fdd}.

To prove Theorem~\ref{thm-bb}, more refined generating functions are used.
For the two-point function, this involves the insertion of a factor
$\zeta^{|\omega|}$ on the right side of \refeq{G2I}, where $\zeta$ is a complex
fugacity (in the unit disk) for the length of the skeleton.  This
modifies \refeq{G2k} to
\eq
	\hat{G}_{z,\zeta}^{(2)}(k) = \frac{g_z+\hat{\Pi}_{z,\zeta}(k)}
	{1-\zeta z\Omega \hat{D}(k)(g_z+\hat{\Pi}_{z,\zeta}(k))},
\en
where now the interaction term $\hat{\Pi}_{z,\zeta}(k)$ depends also 
on $\zeta$.  It is shown in \cite{DS96} that 
\eq
	\hat{G}_{z,\zeta}^{(2)}(k) 
	= \frac{C_1}{D_1^2 k^2 + 2^{3/2}\sqrt{1-z/z_c} + 2T_1(1-\zeta)}
	+ \mbox{error term},
\en
with appropriate control on the error term.
This permits us to perform a contour integration in the $\zeta$-plane to 
extract the
coefficient of $\zeta^s$, corresponding to trees with an $s$-step skeleton,
and then perform a contour integration in the $z$-plane to extract the
coefficient of $z^n$, corresponding to $n$-bond trees with an $s$-step
skeleton.  
This aspect is discussed in more detail in Section~\ref{sec-mfa}, in the
context of the mean-field model.
The situation is similar for the three-point function, using
also a $\zeta$-dependent analogue of \refeq{Gz3}.


\section{Mean-field theory}
\setcounter{equation}{0}
\label{sec-mf}

The proofs of Theorems~\ref{thm-fdd} and \ref{thm-bb} can be interpreted
as perturbations of corresponding calculations for a
mean-field theory.
This section sketches the main ideas of 
an analysis of the mean-field theory, valid
in all dimensions, which serves as a basis for the proofs of
Theorems~\ref{thm-fdd} and \ref{thm-bb} in \cite{Derb96,DS96}.  
The mean-field model studied here can be regarded as a model of 
non-self-interacting lattice trees with no self-avoidance constraint,
and is closely related to the model studied in \cite{BFG86}.  Its scaling limit
is ISE in all dimensions.
For convenience, we will consider throughout this section the case where
$z$ is a site fugacity rather than a bond fugacity,
which has the effect of replacing factors $z^{|T|}$ by $z^{|T|+1}$,
for example in \refeq{gdef}.
This is not a substantive change.  
We also restrict attention to the nearest-neighbour model.
At the end of this section, we
make some remarks of a combinatorial nature.

\subsection{Definition}

Switching to a site fugacity, if we remove the indicator function
containing the interaction in the two-point function \refeq{G2I},
we obtain $\sum_{\omega : 0 \to x} (g(z))^{|\omega|+1}$.
This prompts us to define the two-point function of the mean-field theory 
by
\eq
\lbeq{Fxdef}
	F_z^{(2)}(x) = \sum_{\omega : 0 \to x} 
	(f(z))^{|\omega|+1},
\en
where the function $f(z)$, specified below, can be regarded as the generating
function for mean-field trees containing the origin.
The sum in \refeq{Fxdef} is taken over simple random walks, and
by analogy with \refeq{G2de} (taking into account the switch to site
fugacity), $f(z)$ is required to satisfy the differential equation
\eq
\lbeq{Fhat0}
	\hat{F}_z^{(2)}(0) = z \frac{df(z)}{dz}.
\en
We demand, by analogy with \refeq{gdef}, 
that $f(z) \sim z $ as $z \to 0$.   In the next paragraph, we show
that this uniquely defines $f(z)$ and hence $\hat{F}_z^{(2)}(k)$.
For the $m$-point function of the mean-field model, we define
\eq
\lbeq{Fmdef}
	\hat{F}_z^{(m)}(\sigma; k_1,\ldots,k_{2m-3}) 
	= \prod_{i=1}^{2m-3} \hat{F}_z^{(2)}(k_i),
\en
an exact analogue of \refeq{Cmdef}.  This definition of the $m$-point function
as a product of 2-point functions is consistent with a lack of 
self-interaction for the mean-field model.

Taking the Fourier transform of \refeq{Fxdef} gives
\eq
\lbeq{Fhatk}
	\hat{F}_z^{(2)}(k) = \frac{f(z)}{1-2df(z)\hat{D}(k)},
\en
where, as in \refeq{Dkdef}, $\hat{D}(k) = d^{-1}\sum_{j=1}^d \cos k^{(j)}$.
Combining \refeq{Fhat0} and \refeq{Fhatk} gives
\eq
\lbeq{Fde}
	\frac{f(z)}{1-2df(z)} = z \frac{df(z)}{dz}.
\en
Let $z_0$ be defined by $2df(z_0)=1$.  Solving the separable equation
\refeq{Fde} gives 
\eq
	f(z)e^{-2df(z)} = \frac{z}{2dez_0}.
\en
Comparing the asymptotic behaviour of both sides as $z \to 0$,
using $f(z) \sim z$, gives $z_0 = 1/(2de)$ and hence
\eq
\lbeq{fdef}
	f(z)e^{-2df(z)} = z.
\en

\subsection{Analysis}
\label{sec-mfa}

This section sketches proofs of analogues of Theorems~\ref{thm-fdd}
and \ref{thm-bb} for the mean-field model.

By \refeq{fdef}, $f$ can be written as $f(z)=-(2d)^{-1}W(-2dz)$, where $W$
is the principal branch of the Lambert $W$ function,  
defined by $W(w)e^{W(w)}=w$.  
The principal branch of $W(w)$
is analytic on the $w$-plane with branch cut
$(-\infty,-e^{-1}]$, corresponding to a branch cut $[z_0,\infty)$ for $f(z)$.  
Properties of $f$ can
be derived from known properties of the Lambert function, or derived directly
from \refeq{fdef}.  
In particular, $f$ has a square
root singularity at $z_0$, and 
\eq
\lbeq{fzasymp}
	f(z) = f(z_0) - \frac{1}{d\sqrt{2}}(1-z/z_0)^{1/2} +O(|1-z/z_0|),
\en
with the absolute value of the error term bounded by a constant multiple
of $|1-z/z_0|$ uniformly in the cut plane.
Using also the fact that $\hat{D}(k) = 1 - (2d)^{-1} k^2 +O(k^4)$, this gives
\eq
\lbeq{Fzasy}
	\hat{F}_z^{(2)}(k) 
	= \frac{1 + E_1}
	{k^2 + d 2^{3/2}\sqrt{1-z/z_0}+E_2},
\en
where $E_1= -\sqrt{2}(1-z/z_0)^{1/2} +O(|1-z/z_0|)$ and
$E_2= O(k^4)+O(k^2|1-z/z_0|^{1/2})+O(|1-z/z_0|)$.  
Writing $\hat{F}_z^{(m)}(k) = \sum_{n=0}^\infty \hat{\phi}_n^{(m)}(k) z^n$,
the asymptotic form of the coefficient $\hat{\phi}_n^{(2)}(kn^{-1/4})$ can
then be obtained using contour integration as in the proof of
Lemma~\ref{lem-cnk}.
The result is
\eq
	\hat{\phi}^{(2)}_n(d^{1/2}kn^{-1/4}) \sim
	\frac{1}{2d\sqrt{2n\pi}} \frac{1}{z_0^n} \hat{A}^{(2)}(k),
\en
and hence, as in Theorem~\ref{thm-fdd},
\eq
	\lim_{n \to \infty} 
	\frac{\hat{\phi}^{(2)}_n(d^{1/2}kn^{-1/4})}{\hat{\phi}^{(2)}_n(0)}
	= \hat{A}^{(2)}(k).
\en

For the general $m$-point function, the procedure is similar.  By \refeq{Fmdef}
and \refeq{Fzasy}, $\hat{F}_z^{(m)}(\sigma; \vec{k})$ is approximately equal to
\eq
	\prod_{i=1}^{2m-3} \frac{1}{k_i^2 + d 2^{3/2}\sqrt{1-z/z_0}}.
\en
As in \refeq{cnmasy}, this leads to
\eq
\lbeq{phinmk}
	\hat{\phi}_n^{(m)}(\sigma; d^{1/2} \vec{k} n^{-1/4})
	\sim \frac{n^{m-5/2}}{(2d)^{2m-3} \sqrt{2\pi}} \frac{1}{z_0^n}
	\hat{A}^{(m)}(\sigma ; \vec{k}) 
	\quad \mbox{as}\;\; n \to \infty .
\en
Since $\hat{A}^{(m)}(\sigma ; \vec{0}) = 1/(2m-5)!!$ is the 
reciprocal of the number of shapes, this gives a mean-field version of
Theorem~\ref{thm-fdd}:
\eq
	\lim_{n \to \infty} 
	\frac{\hat{\phi}_n^{(m)}(\sigma; d^{1/2} \vec{k} n^{-1/4})}
	{\sum_\sigma \hat{\phi}_n^{(m)}(\sigma; \vec{0})} 
	= \hat{A}^{(m)}(\sigma ; \vec{k}).
\en


To study the scaling behaviour of the skeleton, we introduce fugacities
$\zeta_i$ ($|\zeta_i| \leq 1$) for the bonds in the skeleton paths.  
For the two-point function, we define 
\eq
\lbeq{Fzwdef}
	F_{z,\zeta}^{(2)}(x) = \sum_{\omega : 0 \to x}
	(f(z))^{|\omega|+1} \zeta^{|\omega|} ,
\en
so that
\eq
\lbeq{Fz2hat}
	\hat{F}_{z,\zeta}^{(2)}(k) = \frac{f(z)}{1-2d\zeta f(z)\hat{D}(k)}.
\en
Define, for $|\zeta_i| \leq 1$ and $|z| <z_0$,
\eq
	\hat{F}_{z,\vec{\zeta}}^{(m)}(\sigma; \vec{k}) =
	\prod_{i=1}^{2m-3} \hat{F}_{z,\zeta_i}^{(2)}(k_i) =
	\sum_{n=0}^\infty \sum_{\vec{s}} 
	\hat{\phi}_{n}^{(m)}(\sigma; \vec{k},\vec{s})
	\zeta_1^{s_1} \cdots \zeta_{2m-3}^{s_{2m-3}} z^n ,
\en
where the last equality defines $\hat{\phi}_{n}^{(m)}(\sigma; \vec{k},\vec{s})$.  
To prove a mean-field version of Theorem~\ref{thm-bb}, 
for all $m \geq 2$, we wish to argue that 
\eq
\lbeq{mfbb}
	\lim_{n \to \infty} 
	n^{m-3/2} \frac
	{\hat{\phi}_{n}^{(m)}(\sigma; d^{1/2}\vec{k}n^{-1/4},\vec{t}\sqrt{n})}
	{\sum_\sigma \hat{\phi}_{n}^{(m)}(\sigma; \vec{0})}
	= \hat{a}^{(m)}(\sigma; \vec{k}, \vec{t}) .
\en

Since the right side of \refeq{Fz2hat} is the sum of a geometric series
(in $\zeta$), 
\eq
	\sum_{n=0}^\infty \hat{\phi}_{n}^{(m)}(\sigma; \vec{k},\vec{s}) z^n
	= \prod_{i=1}^{2m-3} \left[
	(2d\hat{D}(k_i))^{s_i}  (f(z))^{s_i+1}
	\right],
\en
and thus
\eq
\lbeq{phisn}
	\hat{\phi}_{n}^{(m)}(\sigma; \vec{k},\vec{s})
	= \left( \prod_{i=1}^{2m-3} \hat{D}(k_i)^{s_i} \right) 
	\frac{1}{(2d)^{2m-3}} \frac{1}{2\pi i}
	\oint_\Gamma  (2d f(z))^{\sum_{i=1}^{2m-3} (s_i+1)} \frac{dz}{z^{n+1}}
\en
where $\Gamma$ is a small circle centred at the origin.  Writing 
$s_i = t_i \sqrt{n}$ and replacing $k_i$ by $d^{1/2}k_i n^{-1/4}$,
the Gaussian factor emerges in the limit $n \to \infty$ from
\eq
	\prod_{i=1}^{2m-3} \hat{D}(d^{1/2}k_i n^{-1/4})^{t_i \sqrt{n}}
	= \prod_{i=1}^{2m-3} 
	\left( 1 - \frac{k_i^2}{2\sqrt{n}} 
	+ O\left( \frac{k_i^4}{n} \right) \right)^{t_i \sqrt{n}}
	\to e^{-\sum_{i=1}^{2m-3} k_i^2 t_i/2}.
\en
Letting $t = \sum_i t_i$ and $M=2m-3$, it remains to determine
the asymptotic form of the integral
\eq
\lbeq{phisnend1}
	\frac{1}{2\pi i}
	\oint_\Gamma (2d f(z))^{t\sqrt{n}+M} \frac{dz}{z^{n+1}}.
\en
This can be done by deforming the contour to 
the branch cut $[z_0,\infty)$, making the change of variables $w=n(z/z_0 -1)$, 
using the fact that $f(z)$ has a square root singularity at $z_0$ 
by \refeq{fzasymp}, and finally applying the dominated convergence theorem,
with the result
\eq
\lbeq{phisnend}
	\frac{1}{nz_0^n}
	\frac{1}{\pi} \int_0^\infty e^{-w} \sin (t\sqrt{2w}) dw
	= \frac{1}{nz_0^n} \frac{1}{\sqrt{2\pi}} t e^{-t^2/2}.
\en
The fact that \refeq{phisnend1} is asymptotic to \refeq{phisnend} is
a special case of a more general fact; see
\cite[(4.2)]{MM90}, with, in their notation, $\alpha=0$ and $\lambda =t$.
Combining \refeq{phisn}--\refeq{phisnend}, and recalling from \refeq{phinmk}
that
\eq
	\sum_\sigma \hat{\phi}_n^{(m)}(\sigma; \vec{0}) \sim 
	\frac{n^{m-5/2}}{(2d)^{2m-3} \sqrt{2\pi}} \frac{1}{z_0^n} ,
\en
gives the desired result \refeq{mfbb} that for the mean-field model
typical skeletons have path lengths proportional to $\sqrt{n}$, and have
a Brownian scaling limit with the ISE weight factor $te^{-t^2/2}$.

\subsection{Combinatorics}

We close with some considerations of a combinatorial nature, relating the
analysis of the mean-field model to the scaling limit of lattice embeddings 
of abstract trees.

The Taylor series at the origin of the Lambert function
is $W(w) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}w^n$, and hence
\eq
\lbeq{fser}
	f(z) = -\frac{1}{2d} W(-2dz)
	= \sum_{n=1}^\infty \frac{(2dn)^{n-1}}{n!}z^n.
\en
The funtion $f(z)$ is thus essentially the exponential generating function
of rooted labelled trees \cite[(1.7.6)]{HP73}, 
and we would like to argue that the
analysis of the mean-field model captures an essential feature
relevant to the scaling limit of embeddings of abstract
trees into the lattice.  

This claim is based on the fact that the coefficient of
$z^n$ in \refeq{fser} is given asymptotically by
\eq
\lbeq{fnasy}
	\frac{(2dn)^{n-1}}{n!} 
	\sim \frac{e}{\sqrt{2\pi}}(2de)^{n-1}\frac{1}{n^{3/2}}.
\en 
The power law $n^{-3/2}$ corresponds to the square root singularity 
of $f(z)$, which in turn led to \refeq{Fzasy} and hence to ISE.
In terms of critical exponents, the power law $n^{-3/2}$ corresponds to 
the critical exponent value $\theta = \frac{5}{2}$, which in turn
corresponds to $\gamma = \frac{1}{2}$.  
This reiterates the theme of Section~\ref{sec-ISEce}, relating
critical exponents and ISE.  

Lattice embeddings of abstract trees typically also give rise to this power law
$n^{-3/2}$.  For example, the number of rooted unlabelled trees with 
$n$ vertices is given \cite[(9.5.29)]{HP73} asymptotically by
$c_1 c_2^n n^{-3/2}$,
where $c_1$ and $c_2$ are positive constants.  
Since each edge of an abstract tree can be embedded in $2d$ ways,
the number of lattice embeddings is therefore asymptotic to 
\eq
\lbeq{ult}
	c_1 (2d)^{n-1}c_2^n \frac{1}{n^{3/2}}.
\en
As another example of 
lattice embeddings of abstract trees that gives rise to the power law
$n^{-3/2}$,
consider the number of embeddings of planted planar trees with $2n$ vertices,
each of degree 1 or 3 (these trees are discussed in \cite[Section 2.7.2]{GJ83}).
The number of such trees is given by the Catalan number
$(n+1)^{-1} (^{2n}_{n})$, and hence the 
number of lattice embeddings is asymptotic to 
\eq
\lbeq{emb}
	\frac{1}{\sqrt{\pi}} 2^{2n} (2d)^{2n-1} \frac{1}{n^{3/2}}.
\en

The difference in the rates of exponential growth of \refeq{fnasy}, 
\refeq{ult} and \refeq{emb}
serves only to give different values of the critical point $z_0$ and 
does not play 
an important role.  However, the common power law $n^{-3/2}$,
which is well-known to combinatorialists \cite{MM90}, 
corresponds to the
square-root nature of the singularity of $f(z)$ which plays an essential role
in the identification of the limit as ISE.



\subsection*{Acknowledgements}
G.S. thanks the Canadian Mathematical Society for its invitation
to present the 1995 Coxeter--James Lecture and to write this article.
We are grateful to Takashi Hara for many conversations and for 
his contribution to Section~\ref{sec-perc}.
We thank David Aldous, Jean-Fran\c{c}ois Le Gall
and Ed Perkins for enlightening correspondence concerning ISE, 
and Buks van Rensburg for providing
Figure~\reffg{tree5000}.  E.D. thanks Sichun Wang for several helpful
discussions.  This work was supported in part by 
NSERC grant OGP0009351.  


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\end{thebibliography}


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