Critical points for spread-out self-avoiding walk, percolation\\ and the contact process above the upper critical dimensions
Remco van der Hofstad\footnote{Department of Mathematics and Computer Science
5600 MB Eindhoven
Akira Sakai\footnote{EURANDOM
P.O. Box 513
We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p\,D(\,\cdot\,)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p\,D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\pc$. We investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$, where the universal term 1 is the mean-field critical value, and the model-dependent term $C(D)=O(L^{-d})$ is written explicitly in terms of the random walk transition probability $D$. We also use this result to prove that $\pc=1+cL^{-d}+O(L^{-d-1})$, where $c$ is a model-dependent constant. Our proof is based on the lace expansion for each of these models.
critptPTRFrev.tex
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% December 1, 2003, RvdH: Final version, all put into a single file.
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% Jan 16, 2004, AS: Revised version.
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\title{Critical points for spread-out self-avoiding walk, percolation\\
and the contact process above the upper critical dimensions}

\author{
    Remco van der Hofstad\footnote{Department of Mathematics and
        Computer Science, Eindhoven University of Technology,
        5600 MB Eindhoven, The Netherlands.
        {\tt rhofstad@win.tue.nl}}\\
    Akira Sakai\footnote{EURANDOM, P.O. Box 513, 5600 MB Eindhoven,
        The Netherlands.  {\tt sakai@eurandom.tue.nl}}
    }

\date{December 1, 2003\footnote{Revised August 3, 2004}}

\begin{document}
\setlength{\baselineskip}{16pt}
\maketitle

%\begin{center}
%{\large \bf IN PROGRESS: NOT FOR DISTRIBUTION}
%\end{center}

\begin{abstract}
We consider self-avoiding walk and percolation in $\Zd$, oriented percolation
in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p\,D(\,\cdot\,)$
being the coupling function whose range is denoted by $L<\infty$.  For
percolation, for example, each bond $\{x,y\}$ is occupied with probability
$p\,D(y-x)$.  The above models are known to exhibit a phase transition when
the parameter $p$ varies around a model-dependent critical point $\pc$.  We
investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the
other models, and $L\gg1$.  We prove in a unified way that
$\pc=1+C(D)+O(L^{-2d})$, where the universal term 1 is the mean-field
critical value, and the model-dependent term $C(D)=O(L^{-d})$ is written
explicitly in terms of the random walk transition probability $D$.  We also
use this result to prove that $\pc=1+cL^{-d}+O(L^{-d-1})$, where $c$ is a
model-dependent constant.  Our proof is based on the lace expansion for each
of these models.
\end{abstract}



\section{Introduction and main results}\label{s:results}
Self-avoiding walk, percolation, and the contact process are well-known
models that exhibit critical phenomena.  For percolation in two or higher
dimensions, for example, there exists a percolation threshold $\pcperc$
such that there is almost surely no infinite cluster for $p<\pcperc$,
while for $p>\pcperc$ there is almost surely a unique infinite cluster.
As $p\uaw\pcperc$, the average cluster size and the correlation length
diverge.  The precise value of $\pcperc$ depends on the details of the
model, and is only explicitly known in a few cases, such as for
two-dimensional nearest-neighbor bond percolation \cite{k80}.

In this paper, we will consider self-avoiding walk, percolation, oriented
percolation and the contact process, where the interaction range $L$ is
taken to be large.  When $L\gg1$, the interaction in the considered models
is relatively weak, and therefore the critical values can be expected to be
close to the critical value 1 of the respective mean-field models, i.e.,
random walk and branching random walk.  We study the difference of the
critical values and 1 for the above four models as $L\to\infty$.  It turns
out that, above the respective upper critical dimensions, we can write this
difference to leading order as simple functions of the underlying random
walk.

\subsection{Models}\label{ss:models}
First, we define the models.  A self-avoiding walk is a path $\omega$ in
the $d$-dimensional integer lattice $\Zd$ with $\omega(i)\ne\omega(j)$
for every distinct $i,j\in\{0,1,\dots,|\omega|\}$.  We also take the 
zero-step walk into account.  We define the weight of a non-zero path 
$\omega$ by
\begin{align}\lbeq{weight-def}
W_p(\omega)=p^{|\omega|}\prod_{i=1}^{|\omega|}D\big(\omega(i)-\omega
 (i-1)\big),
\end{align}
where $D$ is a probability distribution on $\Zd$, and let $W_p(\omega)=1$ 
if $|\omega|=0$.  We suppose that $D$ is symmetric with respect to the 
lattice symmetries and that $D(o)=0,$ where $o$ is the origin in $\Z^d$.  
A more detailed definition will be given below.
%We define the self-avoiding
%constraint by
%\begin{align}
%K(\omega)=\prod_{0\leq i<j\leq|\omega|}
% \big[1-\indic{\omega(i)=\omega(j)}\big],
%\end{align}
%and define $K(\omega)=1$ for the zero-step walk.
The self-avoiding walk two-point function is defined by
\begin{align}\lbeq{saw-two}
\tausaw_p(x)=\sum_{\substack{\omega:o\too x\\ {\rm saw}}}W_p(\omega),
\end{align}
where the sum is over all self-avoiding paths from $o$ to $x$.  It is known
(see, e.g., \cite{ms93}) that there is a critical value $\pcsaw$
such that
\begin{align}\lbeq{chisaw}
\chisaw_p=\sum_{x\in\Zd}\tausaw_p(x)
\end{align}
is finite if and only if $p<\pcsaw$ and diverges as $p\uaw\pcsaw$.

For percolation, each bond $\{x,y\}$ is occupied with probability
$p\,D(y-x)$ and vacant with probability $1-p\,D(y-x)$, independently of
the other bonds, where $p\in[0,\|D\|_\infty^{-1}]$.  Since $\sum_xD(x)=1$,
the percolation parameter $p$ is the expected number of occupied bonds per
site.  We denote by $\mP_p$ the probability distribution of the bond
variables.  We say that $x$ is connected to $y$, and write $x\cntd y$,
if either $x=y$ or there is a path of occupied bonds between $x$ and $y$.
The percolation two-point function and its sum over $\Zd$ are denoted by
\begin{align}\lbeq{perc-two}
\tauperc_p(x)=\mP_p(o\longleftrightarrow x),&&
\chiperc_p=\sum_{x\in\Zd}\tauperc_p(x).
\end{align}
Similarly to self-avoiding walk, there is a critical value $\pcperc$ such
that $\chiperc_p$ is finite if and only if $p<\pcperc$ and diverges as
$p\uaw\pcperc$ (see, e.g., \cite{g99}).

Oriented percolation is a time-directed version of percolation.  Each bond
$((x,t),(y,t+1))$ is an ordered pair of sites in $\Zd\times\Zp$,
and is occupied with probability $p\,D(y-x)$ and vacant with probability
$1-p\,D(y-x)$, independently of the other bonds, where
$p\in[0,\|D\|_\infty^{-1}]$.  We say that $(x,s)$ is connected to $(y,t)$,
and write $(x,s)\too(y,t)$, if either $(x,s)=(y,t)$ or there is an oriented
path of occupied bonds from $(x,s)$ to $(y,t)$.  Let $\mP_p$ be the
probability distribution of the bond variables.  The oriented percolation
two-point function and its sum over $\Zd\times\Zp$ are denoted by
\begin{align}\lbeq{op-two}
\tauop_p(x,t)=\mP_p((o,0)\too (x,t)),&&
\chiop_p=\sum_{t\in\Zp}\,\sum_{x\in\Zd}\tauop_p(x,t).
\end{align}
Also oriented percolation exhibits a phase transition such that
$\chiop_p<\infty$ if and only if $p$ is less than the critical value $\pcop$,
and that $\chiop_p\uaw\infty$ as $p\uaw\pcop$ (see, e.g., \cite{gh01}).

The contact process is a model of the spread of an infection in $\Zd$,
and is a continuous-time version of oriented percolation in $\Zd\times\Rp$. We
now describe a graphical representation for the contact process.  Along each time line
$\{x\}\times\Rp$, where $x\in \Z^d$, we place points according to a Poisson process with
intensity 1, independently of the other time lines.  For each ordered pair
of distinct time lines from $\{x\}\times\Rp$ to $\{y\}\times\Rp$, we place
oriented bonds $((x,t),(y,t))$, $t\geq0$, according to a Poisson process with
intensity $p\,D(y-x)$, independently of the other Poisson processes, where
the parameter $p\geq0$ is the infection rate.  We say that $(x,s)$ is connected to $(y,t)$,
and write $(x,s)\too(y,t)$, if either $(x,s)=(y,t)$ or there is an oriented
path in $\Zd\times\Rp$ from $(x,s)$ to $(y,t)$ using the Poisson bonds and
time-line segments traversed in the increasing-time direction without
traversing the Poisson points.  Let $\mP_p$ be the corresponding probability
distribution.  We denote the contact process two-point function and its
integro-sum over $\Zd\times\Rp$ by
\begin{align}\lbeq{cp-two}
\taucp_p(x,t)=\mP_p((o,0)\too(x,t)),&&
\chicp_p=\int_0^\infty\!\!dt~\sum_{x\in\Zd}\taucp_p(x,t).
\end{align}
Again there is a critical value $\pccp$ such that
$\chicp_p$ is finite if and only if $p<\pccp$ and diverges as $p\uaw\pccp$
(see, e.g., \cite{Ligg99}).

We will omit the superscript referring to the precise model, and write $\pc$
when referring to the critical values in all models simultaneously.  The goal
in this paper is to study $\pc$ when the range $L$ of $D$ is sufficiently
large.  In the proofs, we will have versions of $D$ in mind which are such
that $L^d D(Lx)$ is a discrete approximation of a function on $\R^d$.  We
will formalize this assumption on $D$ in the following definition:

\begin{defn}\label{defn:D}
Let $h$ be a probability distribution over $\Rd\setminus\{o\}$, which is
invariant under rotations by $\pi/2$ and reflections in the coordination
hyperplanes.  We suppose that $h$ is piecewise continuous, so that
$\int_{\mR^d}h(x)\,d^dx\equiv1$ can be approximated by the Riemann sum
$L^{-d}\sum_{x\in\Zd}h(x/L)$.  Then, we define
\begin{align}
D(x)=\frac{h(x/L)}{\sum_{y\in\Zd}h(y/L)}.
\end{align}
\end{defn}

We will make heavy use of results proved elsewhere for the models under
consideration.  For these results, some further assumptions are made on $D$,
of which we now list the most important ones.  We require that there exist
$c>0$, $C<\infty$, $\eta\in(0,1)$ such that
\begin{align}\lbeq{Dprop}
\sup_{x\in\Zd}D(x)\leq CL^{-d},&&
\eta\wedge(cL^2|k|^2)\leq1-\hat D(k)\leq2-\eta,
\end{align}
where $\hat D(k)=\sum_{x\in\Zd}D(x)\,e^{ik\cdot x}$ and
$|k|^2=\sum_{j=1}^dk_j^2$.  There are a few more minor requirements
that depend on the precise model under investigation.  For details, see
\cite{HS90a} for percolation and \cite{hsa04,hs02,hs01,hs03} for the other
three models, for which the requirements are virtually identical.  A simple
example of $D$, where all the above assumptions are satisfied, is
\begin{align}\lbeq{dex}
D(x)=\frac{\indic{0<\|x\|_\infty\leq L}}{(2L+1)^d-1},
\end{align}
for which $h(x)=2^{-d}$ if $0<\|x\|_\infty\leq1$ and $h(x)=0$ otherwise.

We denote by $D*G$ the convolution of $D$ and a function $G$ in $\mZ^d$,
and by $D^{*n}$ the $n$-fold convolution of $D$ in $\Zd$, where we define
$D^{*0}(x)=\delta_{o,x}$.  We will frequently use
    \begin{align}\lbeq{d-gauss}
    D^{*n}(x)\leq\delta_{0,n}\delta_{o,x}+\frac{O(\beta)}{(1\vee n)^{d/2}},
    \end{align}
where
    \begin{align}
    \beta=L^{-d}.\lbeq{beta-def}
    \end{align}
The inequality \refeq{d-gauss} is a consequence of \refeq{Dprop},
as we will show in Appendix~\ref{s:lclt}.



\subsection{Main results}
Let $\dc$ denote the respective upper critical dimensions, i.e., $\dc=6$ for
percolation and $\dc=4$ for the other three models.  In this paper, we
investigate the respective critical values when $d>\dc$ and $L\gg1$, in a
unified fashion.

\begin{thm}\label{thm:main}
For each model with $d>\dc$, as $L\rightarrow \infty$,
\begin{align}
\pcsaw,~\pccp
&=1+\sum_{n=2}^\infty D^{*n}(o)+O(\beta^2),\lbeq{sawcp-critpt}\\
\pcop &=1+\frac12\sum_{n=2}^\infty D^{*2n}(o)+O(\beta^2),
 \lbeq{op-critpt}\\
\pcperc &=1+D^{*2}(o)+\frac12\sum_{n=3}^\infty(n+1)\,D^{*n}(o)
 +O(\beta^2).\lbeq{perc-critpt}
\end{align}
\end{thm}

The universal term 1 is the critical value for the mean-field models
(random walk and branching random walk).  Note that, by \refeq{d-gauss},
the model-dependent terms in \refeq{sawcp-critpt}--\refeq{perc-critpt}
are $O(\beta)$.  In Section~\ref{ss:overview}, we will intuitively
explain why the model-dependent terms have the above respective forms.

%Importantly, the difference between the mean-field model and the
%interacting models can be, to leading order, described by loops.  These
%loops, which have a different origin in the different models, describe
%the main effect, when $L$ is large, of the underlying interaction.  We
%note that the asymptotic formulas for $\pccp$ and $\pcop$ are different.
%This can be heuristically understood by noting that the difference
%between the interacting models and the mean-field model is due to loops,
%and that loops for oriented percolation and the contact process have a
%different origin.  For oriented percolation, loops use an {\it even}
%number of bonds, which explains the sum over even powers of $D$ in
%\refeq{op-critpt}.  For the contact process, in a loop, there can be any
%number of spatial arrows, giving rise to factors of $D$, in the graphical
%representation of a loop, which explains the sum over all $n$ in
%\refeq{sawcp-critpt}.

We next compute the dependence on $\beta$ more explicitly, and compute the
coefficients of $\beta$ in $\pc-1$ explicitly.  For this, we let $U$ be the
uniform probability distribution over $[-1,1]^d\subset\Rd$, i.e., for
$x\in\Rd$,
\begin{align}\lbeq{U-def}
U(x)=2^{-d}\indic{\|x\|_\infty\leq1},
\end{align}
and denote by $U^{\star n}$ the $n$-fold convolution of $U$ in $\Rd$.
Then, the leading order coefficient in $\beta$ for $\pc$ is given in
the following theorem:

\begin{thm}\label{thm:uniform}
Fix $D$ as in \refeq{dex}, and let $d>\dc$.  As $L\rightarrow\infty$,
\begin{align}
\pcsaw,~\pccp&=1+\beta\sum_{n=2}^\infty U^{\star n}(o)+O(\beta L^{-1}),
 \lbeq{sawcp-uniform}\\
\pcop &=1+\frac\beta2\sum_{n=2}^\infty U^{\star2n}(o)+O(\beta L^{-1}),\\
\pcperc &=1+\beta\bigg[U^{\star2}(o)+\frac12\sum_{n=3}^\infty(n+1)\,
 U^{\star n}(o)\bigg]+O(\beta L^{-1}).
\end{align}
\end{thm}

We now comment on the relation between the asymptotics in
Theorems~\ref{thm:main}--\ref{thm:uniform}.  The advantage of
Theorem~\ref{thm:uniform} is that it is more concrete, and the continuum
limit of the critical points appears explicitly.  However, the error term
in Theorem~\ref{thm:main} is $O(\beta^2)$, while in Theorem~\ref{thm:uniform}
it is equal to $O(\beta L^{-1})$, which is much larger.  In order to compute
the critical value more precisely, Theorem~\ref{thm:main} gives a much more
powerful result, at the expense of having to compute the random walk terms
appearing in its statement.  In principle, it should be possible to compute
the coefficients of $\beta L^{-1}, \beta L^{-2}, \dots, \beta L^{-d+1}$, but
this requires a substantial amount of work.  Finally, it should be possible
to compute the random walk sums in Theorem~\ref{thm:main} for other examples
than the one in \refeq{dex}, but we refrain from doing so.

We now summarize previous results on the critical values.  We start with
self-avoiding walk.  Penrose's result in \cite{p94} implies that the
critical value for self-avoiding walk defined by
\refeq{dex} with $L\gg1$ satisfies
    \begin{align}\lbeq{p94}
    1+c\,\beta^{2/7}\log\beta^{-1}\geq\pcsaw\geq\begin{cases}
    1,&\text{if }~d\geq3,\\
    1+c'\beta\log\beta^{-1},&\text{if }~d=2,\\
    1+c''\beta^{4/5},&\text{if }~d=1,
    \end{cases}
    \end{align}
for some $\beta$-independent constants $c,c',c''$.  For {\it spread-out
lattice trees}, a related result with a different leading term,
namely $=e^{-1}$, was also obtained in \cite{p94}.  For $d>4$, Madras and Slade
\cite[Corollary 6.2.7]{ms93} improved \refeq{p94} to $\pcsaw=1+O(\beta)$.
In \cite{hs02,hs03}, this result was extended to more general $D$ as
defined in Definition~\ref{defn:D}.  We will rely on the results in
\cite{hs02,hs03}, whose proof is based on the {\it lace expansion} and a
generalized inductive approach.  We will also use the lace expansion to
derive the expression of the $O(\beta)$ term in \refeq{sawcp-critpt}.

For percolation, the best previous result is
$\pcperc=1+O(\beta^{2/d-\epsilon})$ for $d>6$ and $L\gg1$, where
$\epsilon>0$ is an arbitrarily small number \cite{HHS01a}.  However,
if we combine Lemma~\ref{lem-T} proved below and the estimates for
the lace expansion in \cite{HS90a}, then we obtain the better estimate
$\pcperc=1+O(\beta)$.  The result in \refeq{perc-critpt}, which is also
obtained by an application of the lace expansion, identifies the
expression of this $O(\beta)$ term.

When $d>4$ and $L\gg1$, both $\pcop$ and $\pccp$ were proved to be
$1+O(\beta)$ \cite{hsa04,hs02,hs01}.  Similarly to self-avoiding walk,
the proofs of these results rely on the lace expansion and an adaptation
of the inductive approach.  The contact process defined in terms of $D$
of \refeq{dex} was first considered by Bramson, Durrett and Swindle
\cite{bds89}, and they proved that, as $L\to\infty$,
    \begin{align}\lbeq{bds}
    \pccp-1\asymp f(\beta)\equiv\begin{cases}
    \beta,&\text{if }d\geq3,\\
    \beta\log\beta^{-1},&\text{if }d=2,\\
    \beta^{2/3},&\text{if }d=1,
    \end{cases}
    \end{align}
where $\pccp-1\asymp f(\beta)$ means that the ratio $(\pccp-1)/f(\beta)$ is
bounded away from zero and infinity.  Later, Durrett and Perkins \cite{dp99}
proved that
    \begin{align}\lbeq{ed}
    \lim_{L\to\infty}\frac{\pccp-1}{f(\beta)}=\begin{cases}
    \sum_{n=2}^\infty U^{\star n}(o),&\text{if }d\geq3,\\
    3/(2\pi),&\text{if }d=2.
    \end{cases}
    \end{align}
Our result \refeq{sawcp-uniform} in Theorem~\ref{thm:uniform} is stronger
when $d>4$ in the sense that not only the coefficient of $\beta$, but also
the speed of convergence in \refeq{ed} is identified.  In \cite{hsa04}, we
also obtained certain lace expansion results for a local mean-field limit,
where the range and time grow large simultaneously, for the contact process
in $d\leq4$, and we expect that these results could be used to prove
a stronger version of \refeq{ed} for $d=3,4$, as well as for oriented
percolation.  However, this will need serious work using block constructions
as used in \cite{dp99}.

We expect that \refeq{sawcp-critpt}--\refeq{perc-critpt} remain valid for
$d=\dc-1$ and $\dc$ when we change $O(\beta^2)$ to $o(\beta)$.  As mentioned
above, this is the case for the contact process \cite{dp99}.  When
$d\leq\dc-2$, the second terms in \refeq{sawcp-critpt}--\refeq{perc-critpt}
diverge, so that Theorem~\ref{thm:main} cannot hold.  However, we
expect that the asymptotics of the critical point will, as for the contact
process, again be described by the divergence of the sums in
\refeq{sawcp-critpt}--\refeq{perc-critpt}.

When $d>\dc$, we expect that the $O(\beta^2)$ terms could be identified in
terms of $D$ as well, using a similar method as in this paper, but to
do so will require a serious amount of work.

A related problem is to obtain the asymptotics of the critical points for the
nearest-neighbor models, when $D(x)=(2d)^{-1}\indic{|x|=1}$ and $d\to\infty$.
In \cite{hs95}, $\pcsaw$ was proved to have an asymptotic expansion into
powers of $(2d)^{-1}$, and the first six coefficients were obtained.  For
unoriented percolation, the first three coefficients were computed in
\cite{hs95} and \cite{HS04a}, but the proof of the asymptotic expansion only
appeared in \cite{HS04b}.  The proofs of these results are again based on the lace
expansion.  For nearest-neighbour oriented percolation and the
nearest-neighbour contact process, it is proved that $\pcop=1+O(d^{-2})$
(see \cite{cd83}) and $\pccp=1+O(d^{-1})$ (see, e.g., \cite{l97}), using
different methods.

\subsection{Overview of the proof}\label{ss:overview}
To prove Theorem~\ref{thm:main}, we will apply the lace expansion (see, e.g.,
\cite{HS90a,hsa04,hs03,ms93,ny93}).  For example, the lace expansion for
self-avoiding walk gives the recurrence relation
    \begin{align}\lbeq{saw-renewal}
    \tausaw_p(x)=\delta_{o,x}+\sum_{v}[p\,D(v)+\Pisaw_p(v)]\,
    \tausaw_p(x-v),
    \end{align}
where $\Pisaw_p(x)$ is a certain expansion coefficient.  It was proved
in \cite{hs02,hs03} that $\hPisaw_p\equiv\sum_x\Pisaw_p(x)=O(\beta)$ for
$p\leq\pcsaw$, if $d>4$ and $L\gg1$ (see Section~\ref{s:saw}).  Summing
both sides of \refeq{saw-renewal} over $x\in\Zd$ and solving the resulting
equation in terms of $\chisaw_p$, we obtain
    \begin{align}\lbeq{saw-chi}
    \chisaw_p=(1-p-\hPisaw_p)^{-1},
    \end{align}
and thus
    \begin{align}\lbeq{saw-pc}
    \pcsaw=1-\hPisaw_{\pcsaw}.
    \end{align}
To estimate $\pcsaw$, we thus need to investigate $\hPisaw_{\pcsaw}$.
We will prove that, since $\pcsaw=1+O(\beta)$, we can replace
$\hPisaw_{\pcsaw}$ by $\hPisaw_1$ up to an error of order $O(\beta^2)$.
When $p=1$, the exponentially growing factor $p^{|\omega|}$ in
\refeq{weight-def} does not play any role, and $\hPisaw_1$ can be
investigated in terms of random walks.  This is the key ingredient for
the proof of Theorem~\ref{thm:main}.

%For nearest-neighbor self-avoiding walk, this strategy was carried out in
%\cite{hs95} and an asymptotic expansion of $\pcsaw$ when $d\to\infty$ was
%derived.  For percolation models, there is a similar expression to
%\refeq{saw-pc} (see Section~\ref{s:perc}), and an expansion of
%$\pc$ for nearest-neighbor percolation was also discussed in \cite{hs95}.

The strategy for percolation models is the same as above.  There is
a similar recursion relation to \refeq{saw-renewal}, with some
model-dependent expansion coefficient $\Pi_p(x)$.  Therefore, to obtain
the formulae in Theorem~\ref{thm:main}, we will have to investigate
$\hat\Pi_1=\sum_x\Pi_1(x)$, again in terms of random walks.

As we will explain in Sections~\ref{s:saw}--\ref{s:perc}, $\hPisaw_1$
and $\hat\Pi_1$ can be described by an alternating sum of a model-dependent
sequence $\hat\pi_1^{\sss(N)}$ for $N\geq0$, where $\hat\pi_1^{\sss(N)}$
for $N\geq1$ decays as $\beta^N$ for all models.  For self-avoiding walk,
$\hat\pi_1^{\sss(0)}$ equals zero, while $\hat\pi_1^{\sss(0)}$ for
percolation models is nearly a half of $\hat\pi_1^{\sss(1)}$.  (This is why
we have the factor $\frac12$ in \refeq{op-critpt}--\refeq{perc-critpt}.)
Therefore,
roughly speaking, we only need to investigate $\hat\pi_1^{\sss(1)}$ to
obtain \refeq{sawcp-critpt}--\refeq{perc-critpt}.  We will show later that
the diagrammatic interpretation of $\hat\pi_1^{\sss(1)}$ for self-avoiding
walk is a single random walk taking more than one step and going back
to the starting point (cf., \refeq{sawcp-critpt}), while the diagrammatic
interpretation of $\hat\pi_1^{\sss(1)}$ for percolation models is that two
random walks, at least one of which is non-vanishing, meet at some point.
Therefore, the correction to the mean-field value 1 are related to
{\it random walk loops}.

For loops in the time-oriented percolation models, the lengths in
the time-increasing direction of these two walks have to be equal
(which explains the sum over even convolution powers in \refeq{op-critpt}),
while for unoriented percolation this is not the case (which explain the
sum over all powers and the factor $n+1$ in \refeq{perc-critpt}).

For the contact process, the two paths are {\it continuous time} random walk paths,
for which the number of convolution powers of $D$ is equal to the number of
spatial steps made by the random walk up to a given time, which has
a Poisson distribution. Therefore, the sum over the
convolution powers of $D$ is not restricted to even powers, and we see that
the correction to the mean-field value for the contact process and
oriented percolation are different. For the contact process, it will turn
out that also the factor $\frac12$ in \refeq{op-critpt} disappears, which
is due to the fact that the two walks are in fact {\it avoiding} each other,
and which will be explained in more detail in Section \ref{ss:op}. This is an
intuitive explanation of the model-dependent terms in
\refeq{sawcp-critpt}--\refeq{perc-critpt}.

We organize the rest of this paper as follows.  We begin with self-avoiding
walk in Section~\ref{s:saw}, and explain the key steps to estimate
$\pcsaw$.  Following the same steps, we estimate $\pcop$ and $\pccp$ in
Section~\ref{ss:op}, and $\pcperc$ in Section~\ref{ss:perc}.  Finally,
we prove an extension of \refeq{d-gauss} in Appendix~\ref{s:lclt}, and
Theorem~\ref{thm:uniform} in Appendix~\ref{s:uniform}.



\section{Critical point for self-avoiding walk}\label{s:saw}
In this section, we prove \refeq{sawcp-critpt}, using \refeq{saw-pc}.
Throughout this section, we will omit the superscript ``sa'' and write,
e.g., $\pc=\pcsaw$ and $\hat\Pi_p=\hPisaw_p$.

Before computing the asymptotics of $\hat\Pi_{\pc}$ in \refeq{saw-pc},
we first note that $\pc\geq1$.  This is because the removal of the self-avoidance
constraint in \refeq{saw-two} results in $\sum_{\omega:o\too x}W_p(\omega)$,
whose sum over $x\in\Zd$ equals $(1-p)^{-1}$ for any $p\leq1$.
For self-avoiding walk,
    \begin{align}\lbeq{saw-lace}
    \Pi_p(x)=\sum_{N=1}^\infty(-1)^N\pi_p^{\scscst(N)}(x),
    \end{align}
where, e.g., $\pi_p^{\scscst(1)}(x)$ is a ``1-loop diagram'' at the origin
\cite{ms93}:
    \begin{align}\lbeq{saw-1stdiag}
    \pi_p^{\scscst(1)}(x)=\delta_{o,x}~(pD*\tau_p)(o)=\delta_{o,x}\sum_{
    \substack{\omega:o\too o\\ |\omega|\geq1}}W_p(\omega)\,I(\omega),
    \end{align}
where $I(\omega)=1$ if there are no self-intersection points except
for $\omega(0)=\omega(|\omega|)$, otherwise $I(\omega)=0$.

For $d>4$ and $L\gg1$, it was proved in \cite{hs03} that, for
$\hat\pi_p^{\scscst(N)}=\sum_x\pi_p^{\scscst(N)}(x)$, we have
    \begin{align}\lbeq{saw-piNbd}
    \hat\pi_p^{\scscst(N)}\leq O(\beta)^N,&& p\,\partial_p\hat\Pi_p\leq O(\beta),
    \end{align}
for all $p\leq\pc$ and $N\geq1$.  Together with \refeq{saw-pc} and
\refeq{saw-lace}, we immediately obtain that $\pc=1+O(\beta)$.  Moreover, by the
mean-value theorem, there is a $p\in(1,\pc)$ such that
    \begin{align}\lbeq{saw-pcrewrite}
    \pc=1-\hat\Pi_1-(\hat\Pi_{\pc}-\hat\Pi_1)&=1-\hat\Pi_1-(\pc-1)~\partial_p
    \hat\Pi_p=1+\hat\pi_1^{\scscst(1)}+O(\beta^2),
    \end{align}
where
    \begin{align}
    \hat\pi_1^{\scscst(1)}=\sum_{\substack{\omega:o\too o\\ |\omega|\geq1}}
    W_1(\omega)\,I(\omega)
    %&=\sum_{\substack{\omega:o\too o\\ |\omega|\geq1}}W_1
    % (\omega)-\sum_{\substack{\omega:o\too o\\ |\omega|\geq1}}W_1(\omega)~[1-
    % I(\omega)]\nnmb\\&
    =\sum_{n=2}^\infty D^{*n}(o)-\sum_{\substack{\omega:o\too o\\ |\omega|
    \geq1}}W_1(\omega)~[1-I(\omega)].\lbeq{saw-pi1exp}
    \end{align}

To complete the proof of \refeq{sawcp-critpt}, it thus suffices to prove
that the second term in the right-hand side of \refeq{saw-pi1exp} is
$O(\beta^2)$ if $d>4$.  We first note that $I(\omega)$ is an indicator function.
If $I(\omega)=0$, so that $1-I(\omega)=1$, then there must be a pair
$\{s,t\}\ne\{0,|\omega|\}$ with $0\leq s<t\leq|\omega|$ such that
$\omega(s)=\omega(t)$.  Denoting the parts of $\omega$ corresponding to
these three time intervals by $\omega_i$, $i=1,2,3$, respectively, we obtain
    \begin{align}\lbeq{saw-error}
    \sum_{\substack{\omega:o\too o\\ |\omega|\geq1}}W_1(\omega)~[1-I(\omega)]
    \leq\sum_{x\in\Zd}\,\sum_{\substack{\omega_1,\omega_3:o\too x\\ |\omega_1|
    +|\omega_3|\geq1}}\,\sum_{\substack{\omega_2:x\too x\\ |\omega_2|\geq1}}\,
    \prod_{i=1}^3W_1(\omega_i)=(D*G^{*2})(o)~(D^{*2}\!*G)(o),
    \end{align}
where $G(x)=\sum_{n=0}^\infty D^{*n}(x)$, and
$(D^{*2}*G)(o)=\sum_{n=2}^\infty D^{*n}(o)=O(\beta)$ if $d>2$.  Moreover,
by \refeq{d-gauss},
    \eq
    (D*G^{*2})(o)=\sum_{n=1}^\infty nD^{*n}(o)=O(\beta)
    \en
if $d>4$.  This completes the proof of \refeq{sawcp-critpt}
for self-avoiding walk.
\qed
%
%\bigskip
%
%Finally, we discuss an extension of the above result to higher order terms,
%e.g., to identifying the expression of the $O(\beta^2)$ term.  Let us
%consider $(\pc-1)\,\partial_p\hat\pi_p^{\sss(1)}$ that is included in
%$(\pc-1)\,\partial_p\hat\Pi_p$ in \refeq{saw-pcrewrite}.  By definition and
%the mean-value theorem, there is an $r\in(1,p)$ such that
%\begin{align}
%p\,\partial_p\hat\pi_p^{\sss(1)}&=\sum_{\substack{\omega:o\too o\\ |\omega|
% \geq1}}|\omega|\,W_p(\omega)\,I(\omega)=\sum_{\substack{\omega:o\too o\\
% |\omega|\geq1}}|\omega|\,\big[W_1(\omega)+(p-1)\,\partial_rW_r(\omega)\big]
% \,I(\omega)\nnmb\\
%&=\sum_{\substack{\omega:o\too o\\|\omega|\geq1}}|\omega|\,W_1(\omega)+
% \sum_{\substack{\omega:o\too o\\|\omega|\geq1}}|\omega|\,W_1(\omega)\,
% [1-I(\omega)]+\frac{p-1}r\sum_{\substack{\omega:o\too o\\|\omega|\geq1}}
% |\omega|^2\,W_r(\omega)\,I(\omega).\lbeq{pi1der-reexpr}
%\end{align}
%By the random walk representation as in \refeq{saw-error}, the first
%sum equals $(D*G^{*2})(o)=O(\beta)$.
%
%The remaining two sums turn out be $O(\beta^2)$ when $d>6$.  Here, we only
%show that the second sum is $O(\beta^{(d\wedge6)/2-1-\epsilon})$ for $d>4$,
%where $\epsilon>0$ is an arbitrarily small number if $d=6$, and $\epsilon=0$
%otherwise.  By \refeq{d-gauss}, we can easily bound the sum due to
%$|\omega|>\beta^{-1}$ by
%\[ \sum_{m>\beta^{-1}}m\,\|D^{*m/2}\|_1\,\|D^{*m/2}\|_\infty\leq O(\beta)\!
% \sum_{m>\beta^{-1}}m^{1-d/2}\leq O(\beta^{d/2-1}). \]
%The remaining sum is bounded as
%\begin{align*}
%&\sum_{x\in\Zd}\;\sum_{\substack{\omega:o\too x\too o\\ 1\leq|\omega|\leq
% \beta^{-1}}}W_1(\omega)\,[1-I(\omega)]\\
%&\leq\sum_{x\in\Zd}\!\sum_{\substack{\omega_1,\omega_2:o\too x\\ 1\leq
% |\omega_1|+|\omega_2|\leq\beta^{-1}}}\shift\!W_1(\omega_1)\,W_1(\omega_2)
% \,\big[1-K(\omega_1)\,K(\omega_2)+K(\omega_1)\,K(\omega_2)\,[1-J(\omega_1,
% \omega_2)]\big],
%\end{align*}
%where $J(\omega_1,\omega_2)=1$ if there are no intersection points between
%$\omega_1$ and $\omega_2$ except for their common endpoints, otherwise
%$J(\omega_1,\omega_2)=0$.  Using $K(\omega_1)\,K(\omega_2)\leq1$ and
%following the same argument as in \refeq{saw-error}, we can bound the
%contribution due to $1-J(\omega_1,\omega_2)$ by $(D*G^{*2})(o)^2=O(\beta^2)$.
%By symmetry, the remainder is bounded by
%\[ 2\sum_{x\in\Zd}\!\sum_{\substack{\omega_1,\omega_2:o\too x\\ 1\leq
% |\omega_1|+|\omega_2|\leq\beta^{-1}}}\shift\!\!W_1(\omega_1)\,W_1(\omega_2)
% \,[1-K(\omega_1)]\leq2\sum_{x,y\in\Zd}\!\!\sum_{\substack{\omega'_1:o\too
% y\too x\\ \omega_2:o\too x\\ 1\leq|\omega'_1|+|\omega_2|\leq\beta^{-1}}}
% \shift\!\!W_1(\omega'_1)\,W_1(\omega_2)\!\sum_{\substack{\omega''_1:y\too
% y\\ |\omega''_1|\geq1}}\!\!W_1(\omega''_1). \]
%Recall that the sum over $\omega''_1$ is $O(\beta)$.  Letting $n$ and $l$
%be the lengths of the first and second halves of $\omega'_1$, and then using
%\refeq{d-gauss}, we can bound the right-hand side by
%\[ \sum_{m=1}^{\beta^{-1}}\,\sum_{n,l=0}^m\frac{O(\beta^2)}{[n\vee l\vee
% (m-n-l)]^{d/2}}\leq\begin{cases}
%  O(\beta^{(d\wedge6)/2-1}),&\text{if }d\ne6,\\
%  O(\beta^2\log\beta^{-1}),&\text{if }d=6.
% \end{cases} \]
%
%Summarizing the above computation gives, for $d>6$,
%\[ \pc=1+(D^{*2}\!*G)(o)+\big[(D^{*2}\!*G)(o)\,(D*G^{*2})(o)+\cdots\big]
% +O(\beta^3). \]
%The other $O(\beta^2)$ terms can be derived in the same way.  We remark
%that the ``critical dimension'' for the above expansion of $\pc$ becomes
%higher, as we determine higher order terms of the expansion.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%Section 3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\input{critpt4}
% cript4.tex
% Sep  4, 2003, AS
% Nov 11, 2003, AS completed
% December 1, 2003, RvdH: Final version completed

\section{Critical points for percolation models}\label{s:perc}
In this section, we compute the asymptotics of the critical values
for the other three models, and thus complete the proof of
Theorem~\ref{thm:main}.

To discuss oriented percolation and the contact process simultaneously,
it is convenient to introduce the following oriented percolation on
$\Zd\times\vep\Zp$, which is the time-discretized contact process with
a discretization parameter $\vep\in(0,1]$.  A bond is a directed pair
$((x,t),(y,t+\vep))$ of sites in $\Zd\times\vep\Zp$.  Each bond
is either occupied or vacant, independently of the other bonds, and a bond
$((x,t),(y,t+\vep))$ is occupied with probability
\begin{align}\lbeq{bprob}
q_p(y-x)=\begin{cases}
 1-\vep,&\text{if }x=y,\\
 p\vep D(y-x),&\text{if }x\ne y,
 \end{cases}
\end{align}
provided that $\sup_x q_p(x)\leq1$.  In this notation, the model with
$\vep=1$ is the usual oriented percolation model as defined in
Section \ref{ss:models}, and the weak limit as
$\vep\daw0$ is the contact process \cite{bg91}.  Similarly to
oriented percolation with $\vep=1$, for each $\vep \in (0,1]$,
there is a critical value $\pc^{\sss(\vep)}$
for every $\vep\in(0,1],$ such that $\pc^{\sss(1)}=\pcop$ and
$\lim_{\vep\downarrow 0}\pc^{\sss(\vep)}=\pccp$ \cite{s01}. We will call the
model with $\vep\in (0,1]$ the {\it time-discretized contact process}.

To summarise notation for percolation and the time-discretized contact
process, we will write $\Lambda=\Zd$ for percolation and $\Lambda=
\Zd\times\vep\Zp$ for oriented percolation. For notational convenience,
we will take $\vep=1$ for percolation. We will also use bold letters
to represent elements of $\Lambda$.  For example, $\ovec=o$, $\xvec=x$
for percolation, and $\ovec=(o,0)$, $\xvec=(x,t)$ for the time-discretized
contact process. For a bond $b=(\uvec,\vvec)$, we write $\bb=\uvec$ and
$\tb=\vvec$.  We also omit the superscripts $\vep$, pe, op and cp, if no
confusion can arise.

As mentioned in Section~\ref{s:results}, the lace expansion for percolation
models takes a similar form as in \refeq{saw-renewal}, and reads (see, e.g.,
\cite{HS90a, hsa04})
    \begin{align}\lbeq{perc-renewal}
    \tau_p(\xvec)=[\delta_{\ovec,\xvec}+\Pi_p(\xvec)]+\sum_{\uvec,\vvec\in\Lambda}
    [\delta_{\ovec,\uvec}+\Pi_p(\uvec)]\,q_p(\vvec-\uvec)\,\tau_p(\xvec-\vvec).
    \end{align}
In particular, $q_p(\vvec-\uvec)=p\,D(v-u)$ for percolation and oriented
percolation for which $\vep=1$.  (To unify notation, we recall that we regard
unoriented percolation as a model with $\vep=1$.)  The lace expansion
coefficient $\Pi_p(\xvec)$ equals
    \begin{align}\lbeq{perc-lace}
    \Pi_p(\xvec)=\sum_{N=0}^\infty(-1)^N\pi_p^{\scscst(N)}(\xvec),
    \end{align}
where $\pi_p^{\scscst(N)}(\xvec)$, $N\geq0$, are model-dependent diagram
functions.  The result of the lace expansion will be explained in
Sections~\ref{ss:op}--\ref{ss:perc}.  For the time-discretized contact process with
$\vep\in(0,1]$, $d>\dc$ and $L\gg1$, it has been proved \cite{hsa04,hs01}
that $\hat\Pi_p\equiv\vep\sum_{\xvec\in\Lambda}\Pi_p(\xvec)$ is
$O(\beta)\,\vep^2$ for all $p\leq\pc$.  The same estimate is proved to hold
for unoriented percolation (with $\vep=1$), using the lace expansion in
\cite{HS90a} and Lemma~\ref{lem-T} proved below in Section~\ref{ss:perc}.

As in the derivation of \refeq{saw-chi}, solving \refeq{perc-renewal} in
terms of $\chi_p=\vep\sum_{\xvec\in\Lambda}\tau_p(\xvec)$ gives
    \begin{align}
    \chi_p=\frac{1+\frac1\vep\hat\Pi_p}{1-p-(1-\vep+p\vep)\frac1{\vep^2}
    \hat\Pi_p},
    \end{align}
and thus, equating the denominator to zero,
    \begin{align}\lbeq{perc-pc}
    \pc=1-\frac1{\vep^2}\hat\Pi_{\pc}-(\pc-1)\frac1\vep\hat\Pi_{\pc}.
    \end{align}
This expression holds uniformly in $\vep$. We will use it to compute $\pcop$
and $\pcperc$ by taking $\vep=1$ and $\pccp$ by taking the limit when
$\vep\daw0$ \cite{s01}, respectively.  In particular, the third term is
$O(\beta^2)$ when $\vep=1$, and it has no contribution in the limit
$\vep\daw0$. Therefore, we are left to prove that, apart from an error term
of order $O(\beta^2)$, the second term in \refeq{perc-pc} equals the second
term in \refeq{sawcp-critpt} when $\vep\daw0$, and equals the second term in
\refeq{op-critpt} for oriented percolation and that in \refeq{perc-critpt}
for (unoriented) percolation when $\vep=1$.  We again note that
$\pc^{\smallsup{\vep}}\geq1$, since
$\chi_p\leq\vep\sum_{n=0}^\infty\sum_xq_p^{*n}(x)=(1-p)^{-1}$ for $p\leq1$.
In addition, similarly to \refeq{d-gauss}, when $p=1$ and $\vep<1$, we have
    \begin{align}\lbeq{q-gauss}
    q^{*n}(x)\equiv q_1^{*n}(x)\leq(1-\vep)^n\,\delta_{o,x}+\frac{O(\beta)}
    {[1\vee(n\vep)]^{d/2}}.
    \end{align}
We will prove \refeq{q-gauss} in Appendix~\ref{s:lclt}.  Note that when $\vep=1$,
\refeq{q-gauss} reduces to \refeq{d-gauss}.

To complete the proof of Theorem~\ref{thm:main}, we investigate
$\hat\Pi_{\pc}$ for oriented percolation and the contact process
in Section~\ref{ss:op}, and for unoriented percolation in
Section~\ref{ss:perc}.

\subsection{Asymptotics of $\pcop$ and $\pccp$}
\label{ss:op}
In this section, we investigate $\hat\Pi_{\pc}$ for the discretized contact
process, and derive \refeq{op-critpt} for oriented percolation (i.e.,
$\vep=1$) and \refeq{sawcp-critpt} for the contact process (i.e., $\vep\daw0$).

To describe the diagram functions $\pi_p^{\sss(N)}(\xvec)$, $N\geq0$, we
need some definitions.  We say that $\xvec$ is {\it doubly connected to}
$\yvec$, if either $\xvec=\yvec$ or there are at least two nonzero
bond-disjoint occupied paths from $\xvec$ to $\yvec$.  Following the
notation in \cite{hs01} as closely as possible, we denote this event by
$\xvec\db\yvec$, and define
    \begin{align}\lbeq{op-pi0}
    \hat \pi_p^{\scscst(0)}=\vep \sum_{\xvec \in \Lambda} \pi_p^{\scscst(0)}(\xvec),
    \qquad \text{where}\qquad \pi_p^{\scscst(0)}(\xvec)=\mP_p(\ovec\db\xvec)-\delta_{\ovec,\xvec}.
    \end{align}
If $\ovec$ is connected but not doubly connected to $\xvec$, there is
a {\it pivotal bond} $b=(\bb,\tb)$ for $\ovec\too\xvec$ such that both
$\ovec\too\bb$ and $\tb\too\xvec$ occur, and that $\ovec\too\xvec$ occurs
if and only if $b$ is set occupied.  For $A\subseteq\Lambda$, we say that
$\yvec$ {\it is connected to $\xvec$ through} $A$ when every occupied path
from $\yvec\too\xvec$ has at least one bond with an endpoint in $A$.  We define
$E(b,\xvec;A)$ to be the event that $b$ is occupied, that $\tb\too\xvec$
through $A$, and that there are no pivotal bonds $b'$ for $\tb\too\xvec$
such that $\tb\too\bb'$ through $A$.  Let $\bC^b(\ovec)$ be the set of
vertices in $\Lambda$ connected from $\ovec$ without using $b$.  Then,
    \begin{align}
    \hat \pi_p^{\scscst(1)}=\vep \sum_{\xvec \in \Lambda} \pi_p^{\scscst(1)}(\xvec),
    \qquad \text{where}\qquad\pi_p^{\scscst(1)}(\xvec)=\sum_b\mP_p\big(\ovec\db\bb;\,E(b,\xvec;\bC^b
    (\ovec))\big).\lbeq{op-pi1}
    \end{align}
The higher order diagram functions $\pi_p^{\scscst(N)}(\xvec)$,
$N\geq2$, are defined in a similar way, but are irrelevant in this
paper (see \cite[Section~3]{hsa04} for a complete definition,
with slightly different notation).

For $d>4$ and $L\gg1$, it was proved in \cite{hsa04} that, for
$\hat\pi_p^{\scscst(N)}=\vep\sum_{\xvec\in\Lambda}\pi_p^{\scscst(N)}(\xvec)$,
we have
    \begin{align}\lbeq{op-piNbd}
    \hat\pi_p^{\scscst(N)}\leq O(\beta)^{N\vee1}\,\vep^2,&&
    p\,\partial_p\hat\Pi_p\leq O(\beta)\,\vep^2,
    \end{align}
for all $p\leq\pc$ and $N\geq0$.  Together with \refeq{perc-lace} and
\refeq{perc-pc}, we obtain $\pc=1+O(\beta)$.  Moreover, by the mean-value
theorem, there is a $p\in(1,\pc)$ such that
    \begin{align}\lbeq{op-pcrewrite}
    \pc=1-\frac1{\vep^2}\hat\Pi_{\pc}-(\pc-1)\frac1\vep\hat\Pi_{\pc}
    &=1-\frac1{\vep^2}\hat\Pi_1-(\pc-1)\frac1{\vep^2}\partial_p\hat\Pi_p
    +O(\beta^2)\,\vep\nnmb\\
    &=1-\frac1{\vep^2}\hat\pi_1^{\scscst(0)}+\frac1{\vep^2}\hat\pi_1^{\scscst(1)}
    +O(\beta^2).
    \end{align}
To prove \refeq{sawcp-critpt}--\refeq{op-critpt}, it thus suffices to
investigate $\hat\pi_1^{\scscst(0)}$ and $\hat\pi_1^{\scscst(1)}$.

\begin{proof}[Analysis of $\hat\pi_1^{\scscst(0)}$]
We prove
    \begin{align}\lbeq{pi0-main}
    \frac1{\vep^2}\hat\pi_1^{\scscst(0)}\begin{cases}
    =\frac12\sum_{n=2}^\infty D^{*2n}(o)+O(\beta^2),&\text{for }~\vep=1,\\
    \to\sum_{n=2}^\infty D^{*n}(o)+O(\beta^2),&\text{when }~\vep\daw0.
    \end{cases}
    \end{align}

Recall \refeq{op-pi0}.  To describe a double connection by a pair of two
random walk paths, we order the support of $D$ in an arbitrary but fixed
manner.  For $x,y$ in the support of $D$, we write $x\prec y$ if $x$ is
{\it lower} than $y$ in that order.  For a pair of paths consisting of
bonds in $\Lambda$, $\omega=(b_1,\dots,b_{\sN})$ and $\omega'=(b'_1,\dots,b'_{\sN})$
with $\bb_1=\bb'_1$ and $\tb_{\sN}=\tbp_{\sN}$, we say that $\omega$ is {\it lower}
than $\omega'$, denoted by $\omega\prec\omega'$, if at the first time
$n\in\{1,\dots,N\}$ when $\omega$ is incompatible with $\omega'$
(therefore $b_i=b'_i$ for all $i<n$) we have $\tb_n\prec\tbp_n$.
We also say that $\omega_2$ is {\it higher} than $\omega_1$.

A path $\omega=(b_1,\dots,b_{|\omega|})$ is said to be {\it occupied} if
all bonds along $\omega$ are occupied.  We define $\Elow(\omega)$ to be
the event that $\omega$ is the lowest occupied path among all occupied
paths from $\bb_1$ to $\tb_{|\omega|}$, and that there is another occupied
path $\omega'$ from $\bb_1$ to $\tb_{|\omega|}$ which is bond-disjoint
from $\omega$ (denoted by $\omega\cap\omega'=\varnothing$).  Given a path
$\omega$, we also define $\Ehigh(\omega';\omega)$ to be the event that
$\omega'$ is the highest occupied path among all occupied paths from $\bb_1$
to $\tb_{|\omega|}$ that are bond-disjoint from $\omega$.  Such an occupied
path $\omega'$ exists on $\{\bb_1\db\tb_{|\omega|}\}\cap\Elow(\omega)$ by
definition.

Using the above notation, we have, for $\xvec\ne\ovec$,
    \begin{align}\lbeq{pi0-eventreexpr}
    \{\ovec\db\xvec\}=\BDcup{\substack{\omega_1,\omega_2:\ovec\too\xvec\\
    \omega_1\cap\omega_2=\varnothing\\ \omega_1\prec\omega_2}}\big\{
    \omega_1,\omega_2\text{ occupied};\,\Elow(\omega_1)\cap\Ehigh
    (\omega_2;\omega_1)\big\}.
    \end{align}
We define the right-hand side to be empty if $\xvec=\ovec$.  Then,
    \begin{align}\lbeq{pi-reexpr}
    \hat\pi_1^{\scscst(0)}=\vep\sum_{\xvec\in\Lambda}~\sum_{\substack{\omega_1,
    \omega_2:\ovec\too\xvec\\ \omega_1\cap\omega_2=\varnothing\\ \omega_1\prec
    \omega_2}}\mP_1\big(\omega_1,\omega_2\text{ occupied};\,\Elow(\omega_1)\cap
    \Ehigh(\omega_2;\omega_1)\big).
    \end{align}
Since $\mP_1$ is a product measure, if we ignore
$\Elow(\omega_1)\cap\Ehigh(\omega_2;\omega_1)$, then we obtain
    \begin{align}
    &\sum_{\substack{\omega_1,\omega_2:\ovec\too\xvec\\ \omega_1\cap\omega_2=
    \varnothing\\ \omega_1\prec\omega_2}}\mP_1(\omega_1,\omega_2\text{ occupied})
    \nnmb\\
    &\quad=\sum_{\substack{u,v:u\prec v\\ y,z:y\ne z}}q(u)\,q(v)\,q(x-y)\,q(x-z)
    \sum_{\substack{\omega_1:\uvec\too\yvec\\ \omega_2:\vvec\too\zvec\\ \omega_1
    \cap\omega_2=\varnothing}}\mP_1(\omega_1\text{ occupied})~\mP_1(\omega_2
    \text{ occupied}),\lbeq{pi-premain}
    \end{align}
where $\uvec=(u,\vep)$, $\vvec=(v,\vep)$, $\yvec=(y,t-\vep)$,
$\zvec=(z,t-\vep)$, and $q(x)=q_1(x)$ (cf., \refeq{q-gauss}).  By an
inclusion-exclusion relation, %monotonicity
%$\Ehigh(\omega_2;\omega_1)\supset\Ehigh(\omega_2)$ (because the former
%event allows existence of an occupied path higher than $\omega_2$ that
%uses a bond in $\omega_1$), and symmetry between $\Elow$ and $\Ehigh$,
the correction is bounded by
    \[ \sum_{\substack{\omega_1,\omega_2:\ovec\too\xvec\\ \omega_1\cap\omega_2=
    \varnothing\\ \omega_1\prec\omega_2}}\big[\mP_1\big(\omega_1,\omega_2
    \text{ occupied};\,\Elow(\omega_1)^{\rm c}\big)+\mP_1\big(\omega_1,\omega_2
    \text{ occupied};\,\Ehigh(\omega_2;\omega_1)^{\rm c}\big)\big]. \]
We will prove below that, for $E$ equal to $\Elow(\omega_1)$ or
$\Ehigh(\omega_2;\omega_1)$,
    \begin{align}\lbeq{intersect1}
    \vep\sum_{\xvec\in\Lambda}~\sum_{\substack{\omega_1,\omega_2:\ovec\too\xvec\\
    \omega_1\cap\omega_2=\varnothing}}\mP_1(\omega_1,\omega_2\text{ occupied};\,
    E^{\rm c})=O(\beta^2)\,\vep^2.
    \end{align}

We investigate \refeq{pi-premain} to obtain the expression of
$O(\beta)$ from \refeq{pi-reexpr}.  If we ignore the restriction
$\omega_1\cap\omega_2=\varnothing$, then we obtain
    \begin{align}\lbeq{pi-main}
    \sum_{\substack{u,v:u\prec v\\ y,z:y\ne z}}q(u)\,q(v)\,q(x-y)\,
    q(x-z)~q^{*(t/\vep-2)}(y-u)~q^{*(t/\vep-2)}(z-v),
    \end{align}
where $t/\vep\in[2,\infty)\cap\Zp$.  We will prove below that the correction
satisfies
    \begin{align}\lbeq{intersect2}
    \vep\sum_{\xvec\in\Lambda}\,\sum_{\substack{u,v:u\prec v\\ y,z:y\ne z}}\!
    \!q(u)\,q(v)\,q(x-y)\,q(x-z)\!\!\sum_{\substack{\omega_1:\uvec\too\yvec\\
    \omega_2:\vvec\too\zvec\\ \omega_1\cap\omega_2\ne\varnothing}}\!\!\mP_1(
    \omega_1\text{ occupied})~\mP_1(\omega_2\text{ occupied})=O(\beta^2)\,\vep^2.
    \end{align}
Therefore, we only need to consider the contribution to \refeq{pi-reexpr}
from \refeq{pi-main}.
By changing variables as $y'=x-y$ and $z'=x-z$ and using the symmetry between
$u\prec v$ and $u\succ v$, the sum of \refeq{pi-main} over $x\in\Zd$ equals
    \begin{gather}
    \sum_{\substack{u,v:u\prec v\\ y',z':y'\ne z'}}q(u)\,q(v)\,q(y')\,q(z')\sum_x
    q^{*(t/\vep-2)}(x-y'-u)~q^{*(t/\vep-2)}(x-z'-v)\nnmb\\
    =\frac12\sum_{\substack{u,v:u\ne v\\ y,z:y\ne z}}q(u)\,q(v)\,q(y)\,q(z)~
    q^{*(2t/\vep-4)}(v+z-y-u).\lbeq{decomp}
    \end{gather}
Recall \refeq{bprob}.  Since there is at most one temporal (or vertical)
bond growing out of every site in $\Lambda$, we must have
$q(u)=\vep D(u)$ or $q(v)=\vep D(v)$, so that we obtain at
least one factor of $\vep$.  By the same reason, we should have
$q(y)=\vep D(y)$ or $q(z)=\vep D(z)$, so that we obtain
a second factor of $\vep$.  Therefore, the number of
combinations for the product of four factors of $q$ in \refeq{decomp} is
nine: one combination is proportional to $\vep^4$, four others are
proportional to $(1-\vep)\,\vep^3$, and the remaining four are proportional
to $(1-\vep)^2\,\vep^2$.  Only the first case arises for oriented percolation
for which $\vep=1$ , while only the third case arises for the contact process
for which $\vep\daw0$, respectively.

We first complete the proof of \refeq{pi0-main} for oriented percolation.
When $\vep=1$, and using inclusion-exclusion on the restrictions
$u\neq v$ and $y\neq z$, the sum of \refeq{decomp} over $t\geq2$ equals
    \begin{gather}\lbeq{oppi-main}
    \frac12\sum_{u,v,y,z}D(u)\,D(v)\,D(y)\,D(z)\sum_{t=2}^\infty D^{*(2t-4)}
    (v+z-y-u)+O(\beta^2)=\frac12\sum_{t=2}^\infty D^{*2t}(o)+O(\beta^2),
    \end{gather}
where we use \refeq{Dprop} to obtain an error of order $O(\beta^2)$ that
comes from contributions where $u=v$ or $y=z$.

For the contact process, for which $\vep\daw0$, the leading contribution is
due to the four combinations of order $(1-\vep)^2\,\vep^2$ mentioned above,
where either $u$ or $v$ is $o$, and either $y$ or $z$ is $o$.  Therefore,
the coefficient of $(1-\vep)^2\,\vep^2$ in \refeq{decomp} is
    \begin{align*}
    &\frac12\bigg[\sum_{u,y}D(u)\,D(y)~q^{*(2t/\vep-4)}(-y-u)+\sum_{u,z}D(u)
    \,D(z)~q^{*(2t/\vep-4)}(z-u)\\
    &\quad+\sum_{v,y}D(v)\,D(y)~q^{*(2t/\vep-4)}(v-y)+\sum_{v,z}D(v)\,D(z)
    ~q^{*(2t/\vep-4)}(v+z)\bigg]=2\big(D^{*2}\!*q^{*(2t/\vep-4)}\big)(o).
    \end{align*}
Summing this expression (multiplied by $\vep$) over
$t/\vep\in[2,\infty)\cap\Zp$ gives
    \begin{align}
    2\int_{\Box_\pi}\frac{d^dk}{(2\pi)^d}~\hat D(k)^2~\vep\sum_{n=0}^\infty
    \big[1-\vep+\vep\hat D(k)\big]^{2n}&=\int_{\Box_\pi}\frac{d^dk}{(2
    \pi)^d}~\frac{2\hat D(k)^2}{[1-\hat D(k)][2-\vep+\vep\hat D(k)]}\nnmb\\
    &\stackrel{\vep\daw0}{\too}\int_{\Box_\pi}\frac{d^dk}{(2\pi)^d}~\frac{
    \hat D(k)^2}{1-\hat D(k)}=\sum_{n=2}^\infty D^{*n}(o),\lbeq{cppi-main}
    \end{align}
where $\Box_\pi=[-\pi,\pi]^d$.  This completes the proof of \refeq{pi0-main}.
\end{proof}

%\begin{remnum}
%\label{rem-pairs} For future references, we note that as $\vep\downarrow 0$,
%for any pair of bonds $b_1$ and $b_2$ with $b_1=((u_1, s), (v_1, s+\vep))$
%and $b_2=((u_2,s), (v_2,s+\vep))$ with equal starting point ($u_1=u_2$)
%and unequal endpoint $(v_1\neq v_2)$, the probability at $\pc=1$ that
%the bonds $b_1, b_2$ are occupied is to leading order a factor
%$\vep D(v_1-u_1)\delta_{u_1,u_2}\delta_{v_2,u_2}+\vep D(v_2-u_2)\delta_{u_1,u_2}\delta_{v_1,u_1}$.
%Similarly, when the endpoints of the bond agree, but the starting points are different,
%then the probability at $\pc=1$ that
%the bonds $b_1, b_2$ are occupied is to leading order a factor
%$\vep D(v_1-u_1)\delta_{v_1,v_2}\delta_{v_2,u_2}+\vep D(v_2-u_2)\delta_{v_1,v_2}\delta_{v_1,u_1}$.
%\end{remnum}

\begin{proof}[Analysis of $\hat\pi_1^{\scscst(1)}$]
We prove that $\frac1{\vep^2}\hat\pi_1^{\scscst(1)}$ is asymptotically
twice as large as the right-hand side of \refeq{pi0-main}:
\begin{align}\lbeq{pi1-main}
\frac1{\vep^2}\hat\pi_1^{\scscst(1)}\begin{cases}
 =\sum_{n=2}^\infty D^{*2n}(o)+O(\beta^2),&\text{for }~\vep=1,\\
 \to2\sum_{n=2}^\infty D^{*n}(o)+O(\beta^2),&\text{when }~\vep\daw0.
 \end{cases}
\end{align}

For a bond $b$, let $\{b\db\xvec\}$ be the event that $b$ is occupied and
$\tb\db\xvec$.  We define $\{\uvec\too b\}$ and a joint event
$\{\uvec\too b\db\xvec\}$ similarly.  For events $E_1$ and $E_2$, we denote
by $E_1\circ E_2$ the event that $E_1$ and $E_2$ occur {\it disjointly},
i.e., using disjoint bond sets of bonds (see e.g., \cite[Section 2.3]{g99}).
Recalling \refeq{op-pi1} and distinguishing between $\bb=\ovec$ and $\bb\ne\ovec$, we
can rewrite $\hat\pi_1^{\scscst(1)}$ as
    \begin{gather}
    \hat\pi_1^{\scscst(1)}=\vep\sum_{\uvec,\xvec\in\Lambda}\mP_1\big(\{(\ovec,
    \uvec)\too\xvec\}\circ\{\ovec\too\xvec\}\big)+\vep\sum_{\xvec\in\Lambda}\,
    \sum_{b:\bb\ne\ovec}\mP_1\big(\ovec\db\bb;\,E(b,\xvec;\bC^b(\ovec))\big)
    \nnmb\\
    -\vep\sum_{\uvec,\xvec\in\Lambda}\mP_1\Big(\big\{\{(\ovec,\uvec)\too\xvec\}
    \circ\{\ovec\too\xvec\}\big\}\setminus E((\ovec,\uvec),\xvec;\bC^{\sss
    (\ovec,\uvec)}(\ovec))\Big).\lbeq{pi1-reexpr1}
    \end{gather}
We will extract the leading contribution from the first term.  Note that
$\{(\ovec,\uvec)\too\xvec\}\circ\{\ovec\too\xvec\}$ is almost identical to
$\{\ovec\db\xvec\}=\{\ovec\too\xvec\}\circ\{\ovec\too\xvec\}$.  However, the
symmetry between the two connections from $\ovec$ to $\xvec$ is
lost in the former event, due to the bond $(\ovec,\uvec)$. We will use this
symmetry breaking in a convenient manner. Recall that below \refeq{pi0-main},
the support of $D$ was ordered in an arbitrary way. Now, instead, we choose
the ordering such that, for $\uvec=(u,\vep)$, the element $u$ in the support of
$D$ is {\it minimal}. This will ensure that the lowest occupied
path $\omega_1$ from $\ovec$ to $\xvec$ will use the bond $(\ovec,\uvec)$.
We also write $\Elow^{\uvec}(\omega_1)$ and $\Ehigh^{\uvec}(\omega_2;\omega_1)$
for $\Elow(\omega_1)$ and $\Ehigh(\omega_2;\omega_1)$ in this $\uvec$-dependent
ordering. Therefore, (cf., \refeq{pi0-eventreexpr}),
    \begin{align}\lbeq{pi1-eventreexpr}
    \{(\ovec,\uvec)\too\xvec\}\circ\{\ovec\too\xvec\}=\BDcup{\substack{\omega_1:
    (\ovec,\uvec)\too\xvec\\ \omega_2:\ovec\too\xvec\\ \omega_1\cap\omega_2=
    \varnothing}}&\big\{\omega_1,\omega_2\text{ occupied};\,\Elow^{\uvec}(\omega_1)
    \cap\Ehigh^{\uvec}(\omega_2;\omega_1)\big\},
    \end{align}
and its contribution to \refeq{pi1-reexpr1} is
    \begin{align}\lbeq{pi1-reexpr2}
    \vep\sum_{\xvec\in\Lambda}~\sum_{\substack{\omega_1:(\ovec,\uvec)\too
    \xvec\\ \omega_2:\ovec\too\xvec\\ \omega_1\cap\omega_2=\varnothing}}
    &\mP_1\big(\omega_1,\omega_2\text{ occupied};\,\Elow^{\uvec}(\omega_1)
    \cap\Ehigh^{\uvec}(\omega_2;\omega_1)\big),
    \end{align}
where $\omega_1:(\ovec,\uvec)\too\xvec$ is a path from $\ovec$ to
$\xvec$ starting by the bond $(\ovec,\uvec)$.  Ignoring the condition
$\Elow^{\uvec}(\omega_1)\cap\Ehigh^{\uvec}(\omega_2;\omega_1)$
as in \refeq{pi-premain} and following the same strategy as
in estimating $\hat\pi_1^{\scscst(0)}$,
we obtain the main contribution to \refeq{pi1-main}.  The leading term
of $\frac1{\vep^2}\hat\pi_1^{\scscst(1)}$ is twice as large as that of
$\frac1{\vep^2}\hat\pi_1^{\scscst(0)}$, because the symmetry is broken and
we do not obtain the factor $\frac 12$ as in \refeq{decomp}
(cf., \refeq{pi-reexpr} and \refeq{pi1-reexpr2}).

To complete the proof of \refeq{pi1-main}, it suffices to show that the
second and third terms in \refeq{pi1-reexpr1} are both $O(\beta^2)\,\vep^2$.
The event in the second term of \refeq{pi1-reexpr1} implies the existence of
$\yvec\in\Lambda$ such that $\{\ovec\too\yvec\too\bb\}\circ\{\ovec\too\bb\}$
and $\{\yvec\too\xvec\}\circ\{b\too\xvec\}$ occur disjointly.  Let $\omega_1$
denote a path from $\ovec$ to $\xvec$ through $\yvec$, $\omega_2$ denote
another path from $\ovec$ to $\xvec$ via the the bond $b$ with $\bb=\zvec$,
and $\omega_3$ denote another path from $\yvec$ to $\zvec$.  Then, the second
term in \refeq{pi1-reexpr1} is bounded by
\begin{align}\lbeq{pi1-error}
\vep\sum_{\substack{\xvec,\yvec,\zvec\in\Lambda\\ \zvec\ne\ovec,\xvec}}\,
\sum_{\substack{\omega_1:\ovec\too\yvec\too\xvec\\ \omega_2:\ovec\too\zvec
\too\xvec\\ \omega_3:\yvec\too\zvec\\ \omega_i\cap\omega_j=\varnothing,~i
\ne j}}\,\prod_{i=1}^3\mP_1(\omega_i\text{ occupied}),
\end{align}
since $\mP_1$ is a product measure.  The third term in \refeq{pi1-reexpr1}
is also bounded by the above expression.  This is because the event in the
third term in \refeq{pi1-reexpr1} implies existence of $\yvec\in\Lambda$
and a pivotal bond $b=(\zvec,\cdot\,)$ for $\uvec\too\xvec$ such that
$\{\ovec\too\yvec\too\xvec\}$, $\{(\ovec,\uvec)\too b\too\xvec\}$ and
$\{\yvec\too\zvec\}$ occur disjointly.  We thus obtain \refeq{pi1-error}
by the same random walk representation.

Therefore, it is sufficient to prove that \refeq{pi1-error} is bounded by
$O(\beta^2)\,\vep^2$.  When $\vep=1$, we simply ignore the restriction
$\omega_i\cap\omega_j=\varnothing$, $i\ne j$, and apply the Gaussian bound
\refeq{d-gauss} to the part of $\omega_1$ from $\yvec$ to $\xvec$ and to
the part of $\omega_2$ from $\ovec$ to $\zvec$.  Since $\yvec\ne\xvec$ and
$\zvec\ne \ovec$, the term $\delta_{\ovec,\xvec}$ in \refeq{d-gauss}
does not contribute, so that \refeq{pi1-error} is bounded by
\begin{align}\lbeq{pi1-errorbd}
\sum_{\substack{t,s,s'\in\Zp\\ 0\leq s\leq s'\leq t}}\frac{O(\beta)}{[1\vee
 (t-s)]^{d/2}}\,\frac{O(\beta)}{(1\vee s')^{d/2}}\leq\sum_{t=0}^\infty
 \frac{O(\beta^2)}{(1\vee t)^{d/2}}\leq O(\beta^2),
\end{align}
where $s,s'$ are the time variables of $\yvec$ and $\zvec$, respectively.
When $\vep<1$, we use the restriction $\omega_i\cap\omega_j=\varnothing$,
$i\ne j$, to extract factors of $q$ with pairwise different arguments,
as in \refeq{pi-premain}, out of the four intersection points
$\ovec,\,\yvec,\,\zvec$ and $\xvec$.  As explained above \refeq{oppi-main},
each pair gives rise to a factor $\vep$, and we obtain a total factor $\vep^4$.
With the help of \refeq{q-gauss}, \refeq{pi1-error} with $\vep<1$ is bounded
by $\vep^{1+4}$ times the left-hand side of \refeq{pi1-errorbd} with the
region of summation being replaced by $\vep\Zp$.  This is further bounded by
$O(\beta^2)\,\vep^2$, since the sum over $t,s,s'\in\vep\Zp$ eats up
a factor $\vep^3$ for the Riemann sum approximation.  This completes the proof of
\refeq{pi1-main}.
\end{proof}

\begin{proof}[Proof of \refeq{intersect1}]
We only consider the case $E^{\rm c}=\Elow^{\uvec}(\omega_1)^{\rm c}$, which is the
event that there is an $\eta\prec\omega_1$ from $\ovec$ to $\xvec$, which must
share at least one step with $\omega_1$, such that $\Elow^{\uvec}(\eta)$ occurs;
the other case $E=\Ehigh^{\uvec}(\omega_2;\omega_1)$ can be estimated in a similar
way.  Let $\omega_3$ be the part of $\eta$ from the point, say $\yvec$,
where $\eta$ starts disagreeing from $\omega_1$ until it hits $\omega_1$
or $\omega_2$ at $\zvec$.  Since $\mP_1$ is a product measure,
\refeq{intersect1} is bounded by
    \[ \vep\sum_{\substack{\xvec,\yvec,\zvec\in\Lambda\\ \zvec\ne\ovec,\xvec}}\,
    \sum_{\substack{\omega_1:\ovec\too\yvec\too\xvec\\ \omega_2:\ovec\too\xvec\\
    \omega_3:\yvec\too\zvec\\ \omega_i\cap\omega_j=\varnothing,~i\ne j}}\big(
    \indic{\zvec\in\omega_1\setminus\{\yvec\}}+\indic{\zvec\in\omega_2}\big)
    \prod_{i=1}^3\mP_1(\omega_i\text{ occupied}). \]

Since the contribution from $\indic{\zvec\in\omega_2}$ is equal to
\refeq{pi1-error}, we only need to investigate the contribution due to the
other indicator $\indic{\zvec\in\omega_1\setminus\{\yvec\}}$.  We again
discuss the case $\vep=1$ first, and then adapt the argument to the case
$\vep<1$, as done below \refeq{pi1-errorbd}.  When $\vep=1$, we ignore
the restriction $\omega_i\cap\omega_j=\varnothing$, $i\ne j$, and apply
\refeq{d-gauss} to the probability of $\omega_2$ and $\omega_3$ being
occupied.  By denoting the time variables of $\yvec$ and
$\zvec$ by $s$ and $s'$ respectively, the contribution from
$\indic{\zvec\in\omega_1\setminus\{\yvec\}}$ is bounded by
\begin{align}
\sum_{\substack{t,s,s'\in\Zp\\ 0\leq s<s'\leq t}}\frac{O(\beta)}{(1\vee t)
 ^{d/2}}\,\frac{O(\beta)}{[1\vee(s'-s)]^{d/2}}\leq\sum_{t=0}^\infty\frac{O
 (\beta^2)}{(1\vee t)^{(d-2)/2}}\leq O(\beta^2).
\end{align}
When $\vep<1$, we use the restriction $\omega_i\cap\omega_j=\varnothing$,
$i\ne j$, along each of the four intersection points and obtain the eight
factors of $q$ with pairwise different arguments.  Following the argument
below \refeq{pi1-errorbd}, we obtain the desired bound $O(\beta^2)\,\vep^2$.
This completes the proof of \refeq{intersect1}.
\end{proof}

\begin{proof}[Proof of \refeq{intersect2}]
Since $\omega_1\cap\omega_2\ne\varnothing$, there is a sequence of bonds
$b_1,\dots,b_n$ such that $\omega_1$ and $\omega_2$ meet for the first time
at $\bb_1$, share $b_1,\dots,b_n$, and split at $\tb_n$ ($\omega_1$ and
$\omega_2$ may share a bond again after $\tb_n$).  This means that, together
with $q(u)\,q(v)\,q(x-y)\,q(x-z)$ in \refeq{intersect2}, the left-hand side
of \refeq{intersect2} is bounded by the convolution of two non-vanishing
bubbles and $\prod_{i=1}^nq(w_i)^2$, where each $w_i$ is the spatial
component of $\tb_i-\bb_i$.  Using \refeq{q-gauss} and
$\sum_wq(w)^2\leq\|q\|_\infty$, we can bound
\refeq{intersect2} by
\begin{align}
\vep\sum_{\substack{t,s,s'\in\vep\Zp\\ \vep<s<s'<t-\vep}}\frac{O(\beta)\,
 \vep^2}{(1\vee s)^{d/2}}~\big[(1-\vep)\vee(\vep\|D\|_\infty)\big]^{\frac
 {s'-s}\vep}\,\frac{O(\beta)\,\vep^2}{[1\vee(t-s')]^{d/2}}\leq O(\beta^2)
 \,\vep^2,
\end{align}
where, as before, $\vep^3$ is used up for the Riemann sum approximation.
The above estimate can be improved to $O(\beta^3)$ for oriented percolation,
using \refeq{Dprop}.  This completes the proof of \refeq{intersect2}.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%Section 3.2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\input{critpt4.2}
% cript4.2.tex

% December 1, 2003, RvdH: Final version completed
% November 12, 2003, RvdH
% November 3, 2003, RvdH
% Oct 13, 2003, RvdH
% Oct 4, 2003, RvdH
% Sep 24, 2003, AS

\subsection{Asymptotics of $\pcperc$}\label{ss:perc}
In this section, we compute the asymptotics of the critical point for
(unoriented) percolation.  We follow the strategy in Section \ref{ss:op} as
closely as possible.  However, there are a number of changes due to the fact
that we have less control of the lace expansion coefficients.  For example,
the bounds on the derivative of $\hat\Pi_p$ with respect to $p$ are not
available in the literature, even though in the unpublished manuscript
\cite{h03}, this derivative is computed.  To make this paper self-contained,
we avoid the use of the derivative, which causes changes in the proof.

We start with some notation. Let
\begin{align}\lbeq{Tdef}
T_p=\sup_{x\in \Z^d} (pD*\tau_p^{*3})(x),&&
T_p'=\sup_{x\in \Z^d} \tau_p^{*3}(x).
\end{align}
We will use the following bounds:

\begin{lemma} Fix $d>6$. For $L$ sufficiently large, and all $p\leq \pc$,
\label{lem-T}
\begin{align}
T_p\leq C\beta,&& T_p'\leq1+C\beta.\lbeq{Tbd}
\end{align}
\end{lemma}
We will defer the proof of Lemma \ref{lem-T} to the end of this section.

To compute the asymptotics of $\hat\Pi_p$, we use \refeq{perc-lace}
and the bound (cf., \cite[Proposition 4.1]{BCHSS04b})
\begin{align}\lbeq{piNbd}
\hat{\pi}^\smallsup{N}_p \leq T_p'(2T_pT_p')^{N\vee 1}.
\end{align}
Note that Lemma \ref{lem-T} together with \refeq{perc-pc} and \refeq{piNbd}
immediately imply
\begin{align}
\pc=1+O(\beta).\lbeq{pcasyone}
\end{align}

We now start the proof to improve \refeq{pcasyone} one term further.
Together with Lemma \ref{lem-T}, \refeq{piNbd} proves that the contribution
to $\sum_{N=2}^{\infty}\hat\pi_{\pc}^{\sss(N)}$ is $O(\beta^2)$.  Thus, we
are left to compute $\hat\pi_{\pc}^{\sss(0)}$ and $\hat\pi_{\pc}^{\sss(1)}$.
The goal of this section is to prove
\begin{align}\lbeq{pi01asy}
\hat{\pi}_{\pc}^{\scscst(0)}=\frac12\sum_{n=3}^{\infty}(n-1)D^{*n}(o)
 +O(\beta^2),&&
\hat{\pi}_{\pc}^{\scscst(1)}=D^{*2}(o)+\sum_{n=3}^{\infty}nD^{*n}(o)
 +O(\beta^2).
\end{align}
Using \refeq{perc-pc} and \refeq{pi01asy}, we arrive at \refeq{perc-critpt}.
Thus, we are left to prove \refeq{pi01asy}.

We again investigate $\hat\pi_{\pc}^{\sss(0)}$ and $\hat\pi_{\pc}^{\sss(1)}$
separately.  First, we compute $\hat\pi_{\pc}^{\sss(0)}$. For percolation,
we denote by $\{w\udb x\}$ the event that $w$ is doubly connected to $x$.
By definition \cite{HS90a}, 
$\hat\pi_p^{\sss(0)}=\sum_{x\in\Zd}\pi_p^{\sss(0)}(x)$, where
\begin{align}
\pi_p^{\sss(0)}(x)=\mP_p(o\udb x)-\delta_{o,x}.
\end{align}
We wish to use {\it Russo's formula} (see, e.g.,
\cite{g99}) to prove that $\hat \pi_{\pc}^{\scscst(0)}=
\hat \pi_1^{\scscst(0)}+O(\beta^2)$. However, Russo's formula
is restricted to events that only depend on a {\it finite} number of bonds,
so that we will first show that Russo's formula may be applied to
$\pi_p^{\scscst(0)}(x)$.

Let $B_\ell=\{x\in\Zd:|x|\leq\ell\}$.  We note that, since
$\hat\pi_p^{\sss(0)}$ is finite for any $p\leq\pc$, there is an $r<\infty$
such that $\sum_{x\notin B_r}\pi_p^{\sss(0)}(x)=O(\beta^2)$ for any
$p\leq\pc$.  In fact, using the {\it BK inequality} (see, e.g., \cite{g99})
and the bound $\tau_{\pc}(x)\leq K|x|^{2-d}$ for $x\ne o$
\cite[Proposition~2.2]{HHS01a}\footnote{In \cite[Proposition~2.2]{HHS01a},
$K$ is of order $O(L^{-2+\epsilon})$ with an arbitrarily small number
$\epsilon>0$, and thus is small when $L$ is large.  Here, we do not care
about the dependence of $K$ on $L$, and will take $K=O(1)$.}\!, we have
    \begin{align}\lbeq{finitesum}
    \sum_{x\notin B_r}\pi_p^{\sss(0)}(x)\leq\sum_{x\notin B_r}\tau_p(x)^2
    \leq\sum_{x\notin B_r}\tau_{\pc}(x)^2\leq c\sum_{\ell>r}\ell^{(d-1)
    +2(2-d)}=O(r^{4-d})=O(\beta^2),
    \end{align}
where we assume $r=O(L^{2d/(d-4)})$.  Let $\{E$ in $B_{\sR}\}$ be the set of
bond configurations whose restriction on bonds $\{u,v\}$ with $u,v\in B_{\sR}$
are in $E$.  Similarly to \refeq{finitesum}, if $R=O(L^{2d/(d-6)})$, then
for any $p\leq\pc$ we have\footnote{The event
$\{o\udb x\}\setminus\{o\udb x\text{ in }B_{\sR}\}$ implies the existence of
$y\notin B_{\sR}$ such that $o\cntd x$, $x\cntd y$ and $y\cntd o$ occur
disjointly.  Therefore, by the BK inequality and Propositions~1.7(i)
and 2.2 of \cite{HHS01a}, we obtain
    \[ \sum_{x\in\Zd}\mP_p\big(\{o\udb x\}\setminus\{o\udb x\text{ in }B_{\sR}\}
    \big)\leq\sum_{\substack{x\in\Zd\\ y\notin B_{\sR}}}\tau_p(x)~\tau_p(y-x)~
    \tau_p(y)\leq c\sum_{y\notin B_{\sR}}|y|^{(4-d)+(2-d)}=O(R^{6-d}). \]
    }
    \begin{align}\lbeq{finitecont}
    \sum_x\mP_p\big(\{o\udb x\}\setminus\{o\udb x\text{ in }B_{\sR}\}\big)
    \leq O(\beta^2).
    \end{align}
By \refeq{finitesum}--\refeq{finitecont} and the mean-value theorem,
there is a $p\in(1,\pc)$ such that
    \begin{align}
    \hat\pi_{\pc}^{\sss(0)}&=\sum_{x\in B_r}\pi_{\pc}^{\sss(0)}(x)+O(\beta^2)=
    \sum_{x\in B_r\setminus\{o\}}\mP_{\pc}(o\udb x\text{ in }B_{\sR})+O(\beta^2)\nn\\
    &=\sum_{x\in B_r\setminus\{o\}}\mP_1(o\udb x\text{ in }B_{\sR})+(\pc-1)\sum_{x\in
    B_r\setminus\{o\}}\partial_p\mP_p(o\udb x\text{ in }B_{\sR})+O(\beta^2)\nn\\
    &=\hat\pi_1^{\sss(0)}+(\pc-1)\sum_{x\in B_r\setminus\{o\}}\partial_p\mP_p(o
    \udb x\text{ in }B_{\sR})+O(\beta^2).\lbeq{mvt}
    \end{align}
We will later identify $\hat\pi_1^{\sss(0)}$, and first show that the second
term is $O(\beta^2)$.  Since the event $\{o\udb x$ in $B_{\sR}\}$ depends only
on finitely many bonds, we are now allowed to apply Russo's formula to obtain
    \begin{align}
    \sum_{x\in B_r\setminus\{o\}}\partial_p\mP_p(o\udb x\text{ in }B_{\sR})=
    \sum_{x\in B_r\setminus\{o\}}\,\sum_{(u,v)}D(v-u)~\mP_p\big((u,v)
    \text{ pivotal for }\{o\udb x\text{ in }B_{\sR}\}\big),\lbeq{russo}
    \end{align}
where the factor $D(v-u)$ arises from the derivative of the bond occupation
probability of $\{u,v\}$ with respect to $p$, and where a bond is pivotal for
$o\udb x$ when $o\udb x$ in the (possibly modified) configuration where
the bond is made occupied, while $o\udb x$ does not occur in the
(possibly modified) configuration where the bond is made occupied.

Since $\pc=1+O(\beta)$, and since, by the BK inequality, \refeq{russo} is bounded by
    \begin{align}
    &\sum_{x,(u,v)}D(v-u)~\mP_p\big(\{o\cntd u\}\circ\{v
    \cntd x\}\circ\{o\cntd x\}\big)\nn\\
    &\qquad\leq\sum_{x,(u,v)}D(v-u)~\tau_p(u)\,\tau_p(x-v)\,\tau_p
    (x)\leq p^{-1}T_p\leq T_p,
    \end{align}
so that the second term in \refeq{mvt} is $O(\beta^2)$.  We are left to
analyse the first term $\hat\pi_1^{\sss(0)}$.  We follow the strategy
around \refeq{pi0-eventreexpr}, but the details change somewhat.


%A percolation path $\omega$ is said to be {\it occupied} if all bonds
%$(\omega(m),\omega(m+1))$ along $\omega$ are occupied.  For $x\in \Z^d$, we
%define $\Elow(\omega)$ to be the event that $\omega$ is the lowest occupied
%path among all occupied paths starting at $o$ and ending at $x$,
%and which are such that there exists a disjoint paths $\omega'$
%starting and ending at $o$, respectively $x$.
%Given a path $\omega'$, we also define $\Ehigh(\omega; \omega')$ to be the
%event that $\omega$ is the highest occupied path among all occupied paths that
%are bond-disjoint from $\omega'$, denoted by $\omega\cap\omega'=\varnothing$,
%starting and ending at $o$, respectively $x$.  Such a path exists
%on $\{o\udb x\}\cap \Elow(\omega')$ by the definition of $\Elow(\omega')$.
%Again bond-disjoint means that either $\omega(s)\ne\omega'(s)$ or
%$\omega(s+\vep)\ne\omega'(s+\vep)$ holds for every $s\geq0$. We then obtain

Let $\Scal_x$ denote all self-avoiding paths from $o$ to $x$, and order
the elements in $\Scal_x$ in an arbitrary way.  Then we can write
\begin{align}\lbeq{pi-reexprup}
\hat\pi_1^{\sss(0)}=\sum_{x\ne o}~\sum_{\substack{\omega_1,\omega_2\in
 \Scal_x\\ \omega_1\cap\omega_2=\varnothing\\ \omega_1\prec\omega_2}}\mP_1
 \big(\omega_1,\omega_2\text{ occupied};\,\Elow(\omega_1)\cap\Ehigh(\omega_2;
 \omega_1)\big),
\end{align}
where $\Elow(\omega_1)$ and $\Ehigh(\omega_2;\omega_1)$ were defined between
\refeq{pi0-main} and \refeq{pi0-eventreexpr}.  In words, the event
$\Elow(\omega_1)$ holds when $\omega_1$ is the lowest occupied self-avoiding
walk path from $o$ to $x$ such that there is an occupied bond disjoint path
from $o$ to $x$.  The event $\Ehigh(\omega_2;\omega_1)$ holds when $\omega_2$
is the highest occupied self-avoiding walk path from $o$ to $x$ that is bond
disjoint from $\omega_1$.  Since $\mP_1$ is a product measure, if we ignore
$\Elow(\omega_1)\cap\Ehigh(\omega_2;\omega_1)$, we obtain
\begin{align}\lbeq{pi-premainup}
\sum_{\substack{\omega_1,\omega_2\in\Scal_x\\ \omega_1\cap\omega_2
 =\varnothing\\ \omega_1\prec\omega_2}}\mP_1(\omega_1,\omega_2
 \text{ occupied})=\sum_{\substack{\omega_1,\omega_2\in\Scal_x\\
 \omega_1\cap\omega_2=\varnothing\\ \omega_1\prec\omega_2}}\mP_1
 (\omega_1\text{ occupied})~\mP_1(\omega_2\text{ occupied}).
\end{align}
We can then follow the rest of the argument between \refeq{pi-premain}
and \refeq{oppi-main} to arrive at
\begin{align}
\hat\pi_1^{\sss(0)}=\frac12\sum_{x\ne o}\sum_{\substack {\omega_1,\omega_2
 \in\Scal_x\\ |\omega_1|+|\omega_2|\geq3}}W_1(\omega_1)\;W_1(\omega_2)+
 O(\beta^2),\lbeq{percpi0-main}
\end{align}
where we recall the definition of $W_p(\omega)$ in \refeq{weight-def}.  Here
the factor $1/2$ has the same origin as the one in \refeq{decomp}, and the
restriction that $|\omega_1|+|\omega_2|\geq3$ is due to the fact that the
smallest cycle in percolation has length 3.  In \refeq{percpi0-main}, each
$\omega_j$ is a self-avoiding path from $o$ to $x$.  However, as estimated
in Section~\ref{s:saw}, the contribution in which $\omega_1$ or $\omega_2$
has a self-intersection is $O(\beta^2)$.  Therefore, we can remove the
self-avoidance constraint in \refeq{percpi0-main}.  Performing the sum
over $x\ne o$ and writing $\omega=(\omega_1,\omega_2)$, which is a random
walk path starting and ending at $o$ of length at least 3, we obtain
\begin{align}
\hat\pi_1^{\sss(0)}=\frac12\sum_{\substack{\omega:o\too o\\ |\omega|\geq3}}
 (|\omega|-1)\,W_1(\omega)+O(\beta^2)=\frac12\sum_{n=3}^\infty(n-1)\,D^{*n}
 (o)+O(\beta^2),
\end{align}
where $|\omega|-1=n-1$ is the number of vertices along $\omega$, excluding
the starting and ending point of $\omega$.
%where ${\rm ver}(\omega)$ denotes the number of vertices in the path
%$\omega$.  Naturally, ${\rm ver}(\omega)\leq |\omega|+1$, and
%${\rm ver}(\omega)=|\omega|+1$ unless the walk makes at least one
%self-intersection. The contribution from paths containing at least
%one self-intersection is bounded by $O(\beta^2)$, similarly to the
%argument in \refeq{saw-pi1exp}. Therefore, we obtain
%\begin{align}\lbeq{pi0asyconc}
%\hat\pi_1^{\sss(0)}=\frac12\sum_{n=3}^\infty(n-1)D^{*n}(o)+O(\beta^2).
%\end{align}
This completes the computation of the leading asymptotics of
$\hat\pi_{\pc}^{\sss(0)}$.

We next derive the asymptotics of $\hat{\pi}_{\pc}^{\scscst(1)}$, following
the strategy in \cite{HS04a, HS04b}, where the first three coefficients of
the asymptotic expansion into powers of $(2d)^{-1}$ of the critical value
$\pc$ for nearest-neighbour percolation were computed.  The details of the
argument are changed considerably compared to \cite{HS04a, HS04b}.  Indeed,
since we are only interested in the leading order term, while in \cite{HS04a}
the first three coefficients are computed, many terms that need explicit
computation in \cite{HS04a, HS04b} will be error terms for us.  On the other
hand, since in \cite{HS04a, HS04b} the asymptotics in nearest-neighbour
models for large dimensions are considered, long loops lead to error term in
\cite{HS04a, HS04b}, whereas they contribute to the leading asymptotics here.
We follow the proof in \cite[Section 4.2]{HS04a} as closely and as long as
possible, and indicate where the argument diverges.

To define $\hat{\pi}_p^{\smallsup{1}}$, we need the following definitions.
Given a bond configuration and $A\subseteq\Zd$, we recall that $x$ and $y$ are
\emph{connected through} $A$, and write $x\ctx{\;A\;}y$, if every occupied
path connecting $x$ to $y$ has at least one bond with an endpoint in $A$.
As defined below \refeq{op-pi0}, the directed bond $(u,v)$ is said to be
{\em pivotal} for $x\cntd y$, if $x\cntd u$ and $v\cntd y$ occur, and if
$x\cntd y$ occurs only when $\{u,v\}$ is set occupied.  (Note that there is
a distinction between the events $\{(u,v)$ is pivotal for $x\cntd y\}$ and
$\{(v,u)$ is pivotal for $x\cntd y\}=\{(u,v)$ is pivotal for $y\cntd x\}$.)
Let
\begin{align}\lbeq{E'def}
E'(v,x;A)=\{v\ctx{\;A\;}x\}\cap\big\{\nexists(u',v')
 \text{ occupied \& pivotal for }v\cntd x\text{ s.t. }v\ctx{\;A\;}u'\big\}.
\end{align}
Then, by definition \cite{HS90a},
%   \eq
%   \lbeq{pi1def}
%   \Pi^{\smallsup{1}}_p(x) = \sum_{(u,v)}  p  \,
%   \Ebold_{\SSS 0} \left(  I [ 0 \dbc u ]
%   \prob[{\SSS 1}]
%   {E'(v,x; \tilde{C}_{\SSS 0}^{(u, v)}(o))}
%   \right),
%   \en
%so that
\begin{align}\lbeq{Pi1defa}
\hat\pi_p^{\sss(1)}=\sum_x\sum_{(u,v)}p\,D(v-u)~\mE_{\sss0}\Big[\ind{o\udb
 u}~\mP_{\sss1}\big(E'(v,x;\tilde C^{\sss(u,v)}_{\sss0}(o))\big)\Big],
\end{align}
where the sum over $(u,v)$ is a sum over directed bonds.  On the right-hand
side, we use subscripts to identify the different expectations.  Thus, the
subscripts do {\it not} refer to the percolation parameter $p$.  The cluster
$\tilde C^{\sss(u,v)}_{\sss0}(o)$ appearing on the right hand side of
\refeq{Pi1defa} is random with respect to the expectation $\mE_{\sss0}$,
but $\tilde C^{\sss(u,v)}_{\sss0}(o)$ should be regarded as a \emph{fixed} set
inside the probability $\mP_{\sss1}$.  The latter introduces a second
percolation model which depends on the original percolation model via the
set $\tilde C^{\sss(u,v)}_{\sss0}(o)$.  We refer to the bond configuration
corresponding to the $j^{\rm th}$-expectation as the ``level-$j$''
configuration.

By \refeq{piNbd},
\begin{align}\lbeq{Pi1Tbd}
0\leq\hat\pi_p^\smallsup{1}\leq 2T_p'T_p T_p'.
\end{align}
We will use refinements of this bound in the following.

We first claim that the contribution to \refeq{Pi1defa} due to $u\ne o$
is an error term of order $O(\beta^2)$.  Indeed, if $u\neq o$ then at
level-0 the origin is in a cycle of length at least 3.  Standard
diagrammatic estimates then allow for the replacement in \refeq{Pi1Tbd}
of a factor $T_p'$ by a constant multiple of $T_p$.  This improves the
bound \refeq{Pi1Tbd} from $O(\beta)$ to $O(\beta^2)$, by \refeq{Tbd}.

We are left with the contribution to \refeq{Pi1defa} due to $u=o$, namely
\begin{align}\lbeq{Pi1defu0}
\sum_{x,v}p\,D(v)~\mE_{\sss0}\big[\mP_{\sss1}\big(E'(v,x;\tilde C^{\sss
 (o,v)}_{\sss0}(o))\big)\big].
\end{align}
If $ x\notin\tilde C^{\sss(o,v)}_{\sss0}(o)$, then to obtain a non-zero
contribution to $\mP_{\sss1}(E'(v,x;\tilde C^{\sss(o,v)}_{\sss0}(o)))$,
$x$ must be in an occupied cycle of length at least 3, in level-1
(in the language of \cite[Section~3]{BCHSS04b}, the sausage containing
$x$ must consist of a cycle containing both $x$ and an endpoint of the
last pivotal bond for the connection from $o$ to $x$).  In this case,
in \refeq{Pi1Tbd}, we may again replace a factor $T_p'$ by a constant
multiple of $T_p$, and again this contribution is $O(\beta^2)$.  We are
left to consider
\begin{align}\lbeq{Pi1defux}
\sum_{x,v}p\,D(v)~\mE_{\sss0}\Big[\ind{x\in\tilde C^{\sss(o,v)}_0(o)}
 ~\mP_{\sss1}\big(E'(v,x;\tilde C^{\sss(o,v)}_{\sss0}(o))\big)\Big].
\end{align}
This is as far as the analogy with the argument in \cite[Section~4.2]{HS04a}
goes.  We now need to adapt the proof there to compute the asymptotics of
$\hat\pi^{\sss(1)}_{\pc}$ when $L\to\infty$.

If $x\in\tilde C^{\sss(o,v)}_{\sss0}(o)$, and if $v\cntd x$, then
$v\ctx{\tilde C^{\sss(o,v)}_{\sss0}(o)}x$.  We next claim that the
intersection with the second event in \refeq{E'def} leads to an error
term.  We write
\begin{align*}
&I_{\sss0}[x\in\tilde C^{\sss(o,v)}_{\sss0}(o)]~I_{\sss1}[E'(v,x;\tilde
 C^{\sss(o,v)}_{\sss0}(o))]\\[5pt]
&\qquad=I_{\sss0}[x\in\tilde C^{\sss(o,v)}_{\sss0}(o)]~I_{\sss1}[v\cntd
 x]\nn\\
&\qquad\qquad\times\Big(1-I_{\sss1}\big[\exists(u',v')
 \text{ occupied \& pivotal for }v\cntd x\text{ s.t. }
 v\ctx{\tilde C^{\sss(o,v)}_{\sss0}(o)}u'\big]\Big),
\end{align*}

\noindent
where we write $I_{\sss0}$ and $I_{\sss1}$ for the indicator functions on
levels 0 and 1, respectively.  The latter term can be bounded by
\begin{align}
\sum_{(u',v')}\sum_zI_{\sss0}[z\in\tilde C^{\sss(o,v)}_{\sss0}(o)]~I_{\sss1}
 \big[\{v\cntd z\}\circ\{z\cntd u'\}\circ\{(u',v')\text{ occupied}\}\circ
 \{v'\cntd x\}\big],
\end{align}
which, using the BK inequality, yields a bound of the form
\begin{align}
\sum_{x,v,z}\sum_{(u',v')}p\,D(v)~\mP_{\sss0}(o\cntd x,~o\cntd z)~\tau_p
 (z-v)~\tau_p(u'-z)~p\,D(v'-u')~\tau_p(x-v').
\end{align}
By the tree-graph inequality \cite{an84}
\begin{align}
\mP_{\sss0}(o\cntd x,~o\cntd z)\leq\sum_y\tau_p(y)~\tau_p(x-y)~\tau_p(z-y),
\end{align}
so that we end up with
\begin{align}
\sum_{x,z,y}(p\,D*\tau_p)(y)~\tau_p(x-y)~\tau_p(z-y)~\tau_p(z-v)~(\tau_p*p
 \,D*\tau_p)(x-z)\leq T_p^2=O(\beta^2),
\end{align}
which indeed is an error term.  Thus, using the identity
\begin{align}
\{x\in\tilde C^{\sss(o,v)}_{\sss0}(o)\}=\{o\cntd x
 \text{ without using }(o,v)\},
\end{align}
we end up with
\begin{align}\lbeq{pi1asya}
\hat\pi_p^{\sss(1)}=\sum_{v,x}p\,D(v)~\tau_p^{\sss(o,v)}(x)~\tau_p(x-v)
 +O(\beta^2),
\end{align}
where
\begin{align}\lbeq{tau(o,v)def}
\tau_p^{\sss(o,v)}(x)=\mP(o\cntd x\text{ without using }(o,v)).
\end{align}
Note that we can think of $\tau_p^{\sss(o,v)}(x)$ as the two-point function
on $\Zd$, where the bond $(o,v)$ is removed. We will denote the resulting
graph with vertex set $\Z^d$ and edge set
$\big\{\{x,y\}:x,y\in\Zd,\{x,y\}\ne\{o,v\}\big\}$ by $\zd_{\sss(o,v)}$,
so that $\tau_p^{\sss(o,v)}(x)$ is the two-point function on
$\zd_{\sss(o,v)}$.  We will use this observation to compute
$\tau_p^{\sss(o,v)}(x)$.

We investigate the main term in the right-hand side of \refeq{pi1asya}
further.
%
%When $x=v$, then
%by the BK inequality,
%\begin{align}
%\mP_p(o\cntd x\text{ without using }(o,v))
%&= \mP_p(\cup_{y\neq v} \{(o,y)\text{ occupied}\}\circ \{y\cntd x\}
%\text{ without using }(o,v))\nn\\
%&\leq \mP_p(\cup_{y\neq v} \{(o,y)\text{ occupied}\}\circ \{y\cntd x\})
%\leq \sum_{y\neq v} pD(y) \tau_p(x-y).
%\end{align}
%On the other hand,
%\begin{align}
%\mP_p(o\cntd x\text{ without using }(o,v))
%&\geq \mP_p(\cup_{y\neq v} \{(o,y)\text{ occ.\ and piv.\ for }y\cntd x
%\text{ without using }(o,v))\nn\\
%&= \sum_{y\neq v} pD(y) \mP_p(y\cntd x \text{ without using }(o,v)\text{ or }(o,y))\nn\\
%&\geq \sum_{y\neq v} pD(y) \tau_p(x-y)-
%\sum_{y\neq v} pD(y) \mP_p((o,v)\text{ or }(o,y) \text{ piv.\ for }y\cntd x).
%\end{align}
%The second term clearly leads to an error term. {\bf More detail!}
%
%Furthermore, when $x\neq v$, we have that
%\eq
%\tau_p(x)-\mP_p(o\cntd x\text{ without using }(o,v))\leq
%pD(v)\tau_p(x)=O(\beta) \tau_p(x),
%\en
%which leads to an error term, and we can replace
%$\mP_p(o\cntd x\text{ without using }(o,v))$ by $\tau_p(x)$.
%Thus, we obtain that
%\begin{align}
%\hat{\pi}_p^\smallsup{1} &= \sum_{v,x}
% pD(v)\Big(\tau_p(x)[1-\delta_{v,x}]+\delta_{v,x}(pD*\tau_p)(x)\Big)\tau_p(x-v)\nn\\
%&=p(D*\tau_p)(o)+\sum_{v\neq x}
%pD(v)(pD*\tau_p)(x)\tau_p(x-v)+O(\beta^2).
%\end{align}
%{\bf To be completed!}
%
Russo's formula, together with the BK inequality, yields that
\begin{align}
\partial_p\tau_p(x)&=\sum_{(y,z)}D(z-y)~\mP((y,z)\text{ pivotal for }
 o\cntd x)\leq(\tau_p*D*\tau_p)(x),\\
\partial_p\tau_p^{\smallsup{o,v}}(x)&=\sum_{(y,z)}D(z-y)~\mP((y,z)
 \text{ pivotal for }o\cntd x\text{ in }\zd_{\smallsup{o,v}})
 \leq(\tau_p*D*\tau_p)(x).
\end{align}
Therefore, we obtain that for $p=\pc$,
\begin{align}
\hat{\pi}_{\pc}^\smallsup{1} &=\sum_{x,v}\pc D(v) \tau_{\pc}^{\smallsup{o,v}}(x)\tau_{\pc}(x-v)
+O(\beta^2)\nn\\
&=\sum_{x,v}D(v) \tau_{1}^{\smallsup{o,v}}(x)\tau_{1}(x-v)
+O((\pc-1)T_{\pc})+O(\beta^2)\nn\\
&=\sum_{x,v}D(v) \tau_{1}^{\smallsup{o,v}}(x)\tau_{1}(x-v)
+O(\beta^2),
\end{align}
since $\pc=1+O(\beta)$. Furthermore, an argument similar to the one for
$\hat{\pi}_p^{\smallsup{0}}$ shows that
\begin{align}
\tau_1(x)&=G(x)+O\big((G*g*G)(x)\big),\lbeq{taup1}\\
\tau_1^{\sss(o,v)}(x)&=G(x)\,(1-\delta_{v,x})+\delta_{v,x}(D^{*2}*G)(x)
 +O\big((G*g*G)(x)\big)+O\big(D*(G-\delta_o)(x)\big).
\lbeq{taupv1}
\end{align}
where we recall $G(x)=\sum_{n=0}^\infty D^{*n}(x)$ and define
\begin{align}
g(x)=G(x)(D*G)(x).
\end{align}
We will prove \refeq{taup1}--\refeq{taupv1} in full detail below, and first
complete the proof subject to \refeq{taup1}--\refeq{taupv1}.  Using
\refeq{taup1}--\refeq{taupv1}, together with the fact that for $u\neq o$,
we have $G(u)=(D*G)(u)$, we end up with
\begin{align}\lbeq{pi1asyconc}
\hat\pi_{\pc}^\smallsup{1}&=\sum_{x\neq v}G(x)\,D(v)\,G(x-v)+\sum_x(D^{*2}*G)
 (x)\,D(x)+O(\beta^2)+O((D*G^{*3}*g)(o))\nn\\
&=\sum_{n=2}^\infty(n-1)D^{*n}(o)+\sum_{n=3}^\infty D^{*n}(o)+O(\beta^2)
 =D^{*2}(o) +\sum_{n=3}^{\infty} n D^{*n}(o)+O(\beta^2),
\end{align}
where we use
\begin{align}
(D*G^{*3}*g)(o)\leq\|D*G^{*3}\|_\infty\,\|g\|_1\leq O(\beta^2),
\end{align}
for $d>6$, by \refeq{d-gauss}.

This completes the proof subject to \refeq{taup1}--\refeq{taupv1} and
Lemma~\ref{lem-T}.
\qed

\newcommand{\Flow}{F_{\succ}}

\begin{proof}[Proof of \refeq{taup1}--\refeq{taupv1}]
We start by proving \refeq{taup1}, and then adapt the argument to prove
\refeq{taupv1}.  To see \refeq{taup1}, we recall the arbitrary ordering of
the elements in $\Scal_x$ introduced above \refeq{pi-reexprup}.  Then we
have that
\eq
\tau_1(x)=\sum_{\omega\in\Scal_x}\mP_p(\omega\text{ occupied};\Flow(\omega)),
\en
where $\Flow(\omega)$ is the event that $\omega$ is the lowest occupied path
in $\Scal_x$.  Thus, we can write
\eq
\tau_1(x) = \sum_{\omega\in \Scal_x} \mathbb{P}_p(\omega \text{ occupied}) -
\sum_{\omega\in \Scal_x} \mathbb{P}_p(\omega \text{ occupied}; \Flow(\omega)^c).
\lbeq{tau1rew}
\en
The former term equals
\eq
\lbeq{sawpaths}
\delta_{o,x}+(1-\delta_{o,x})\sum_{\omega\in \Scal_x} \prod_{i=0}^{|\omega|-1}D(\omega(i+1)-\omega(i)).
\en
Clearly, by using inclusion-exclusion on the fact that $\omega$ is self-avoiding,
as in \refeq{saw-pi1exp}, \refeq{sawpaths} equals
\eq
G(x)+O((G*G)(x)(G(o)-1)),
\en
which is a contribution to the error in \refeq{taup1} when we note that
$G(o)-1=(D*G)(o)$. Similarly, the second term in \refeq{tau1rew} is bounded
by $O\big((G*g*G)(x)\big)$ using the fact that there must exist a $u\in\Zd$
such that there exist bond disjoint occupied paths from $o$ to $u$, two
occupied paths from $u$ to $v$ (of which at least one is non-vanishing) and
one from $v$ to $x$. Thus, by the BK inequality, this term is bounded by
\[ \sum_{u,v}G(u)\,G(v-u)\,(D*G)(v-u)\,G(x-v)=(G*g*G)(x). \]

The proof of \refeq{taupv1} follows the same ideas. In
\refeq{tau1rew} and \refeq{sawpaths}, we only need to sum over
self-avoiding walk paths that do not use the bond
$(o,v)$. When $x=v$, this means that $|\omega|\geq 2$, so that we obtain
\eq
\tau_1^{\smallsup{o,v}}(v)=(D^{*2}*G)(v)+O((G*g*G)(v)) + O(D(v)(G(o)-1)).
\en
When $x\neq v$, we can use inclusion-exclusion on the fact that the bond
$(o,v)$ is not used, and obtain
\eq
\tau_1^{\smallsup{o,v}}(x) = \tau_1(x)+O(D(v) G(x-v)),
\en
and then use \refeq{taup1}.
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem-T}]
We use \cite[(5.20)]{HS90a}, which states that uniformly in $p\in[1,\pc)$
and for $L$ large enough
\begin{align}\lbeq{IRbd}
\hat\tau_p(k)\leq\frac{1+o(1)}{1-\hat{D}(k)},
\end{align}
where $o(1)$ tends to 0 when $L\rightarrow \infty$. %In the sequel,
%we will assume that
%\eq
%\hat\tau_p(k) \leq \frac{2}{1-\hat{D}(k)}.
%
%\en
We also use the standard bound (see e.g.\ \cite{BCHSS04b}) that for $x\neq 0$,
\eq
\tau_p(x) \leq (pD*\tau_p)(x).
\lbeq{tauDtau}
\en
We then follow the proof as in \cite{BCHSS04b}.  For $T_p$, we fix $x$ and
extract the term in \refeq{Tdef} due to the case where every argument of
$\tau_p$ is $o$, which is $pD(x)\leq p\,C\beta$ (see \refeq{Dprop}).
This gives
\begin{align}
T_p(x)\leq p\,C\beta+p\sum_{(u,y,z)\ne(x,o,o)}\tau_p(y)\,\tau_p(z-y)\,D(u)\,
 \tau_p(x+z-u).
\end{align}
Therefore, by \refeq{tauDtau},
\begin{align}
T_p\leq p\,C\beta+3p^2\sup_x(D^{*2}*\tau_p^{*3})(x),
\end{align}
where the factor 3 comes from the 3 factors $\tau_p$ whose argument can
differ from $o$.  In terms of the Fourier transform, this gives
\begin{align}
T_p\leq p\,C\beta +3p^2\,\sup_x\int_{\Box_\pi}\frac{d^d k}{(2\pi)^d}~\hat
 D(k)^2\,\hat\tau_p(k)^3\,e^{-ik\cdot x}=p\,C\beta+3p^2\int_{\Box_\pi}
 \frac{d^dk}{(2\pi)^d}~\hat D(k)^2\,\hat\tau_p(k)^3,
\end{align}
where $\Box_\pi=[-\pi,\pi]^d$ and we use $\hat{\tau}_p(k)\geq 0$
\cite{an84}.  For $L\gg1$, by \refeq{IRbd} we obtain
    \begin{align}\lbeq{Tpintbd}
    T_p\leq p\,C\beta+4p^2\int_{\Box_\pi}\frac{d^d k}{(2\pi)^d}~\frac{\hat
    D(k)^2}{[1-\hat D(k)]^3}.
    \end{align}
Let $\hat B_{1/L}=\{k\in\Box_\pi:cL^2|k|^2\leq\eta\}$.  Using \refeq{Dprop},
we estimate the contribution to the integral in \refeq{Tpintbd} from
$k\in\Box_\pi\setminus\hat B_{1/L}$ by
    \begin{align}
    \int_{\Box_\pi\setminus\hat B_{1/L}}\frac{d^d k}{(2\pi)^d}~\frac{\hat D
    (k)^2}{[1-\hat D(k)]^3}\leq\eta^{-3}\int_{\Box_\pi}\frac{d^d k}{(2\pi)^d}
    ~\hat D(k)^2=O(\beta).
    \end{align}
On the other hand, the contribution from $k\in\hat B_{1/L}$ is, again using
\refeq{Dprop}, bounded by
    \begin{align}
    \int_{\hat B_{1/L}}\frac{d^d k}{(2\pi)^d}~\frac{\hat{D}(k)^2}{[1-\hat{D}(k)]
    ^3}\leq\int_{\hat B_{1/L}}\frac{d^d k}{(2\pi)^d}~(cL^2|k|^2)^{-3}=O(\beta).
    \end{align}
This proves the bound on $T_p$.

The bound on $T_p'$ is a consequence of $T_p' \leq 1+3T_p$.  Here the term
1 is due to the contribution to \refeq{Tdef} where the arguments of the
three factors of $\tau_p$ in $T_p'$ in \refeq{Tdef} are equal to $o$.
If at least one of these arguments is nonzero, then we can use
\refeq{tauDtau} for the corresponding two-point function.
\end{proof}




\appendix
\section{Bounds on $D^{*n}(x)$ and $q^{*n}(x)$}\label{s:lclt}

In this appendix, we prove \refeq{q-gauss} for any $\vep\in (0,1]$
assuming \refeq{Dprop}. The inequality \refeq{d-gauss} follows
by taking $\vep=1$.

First, we note that
\begin{align}
q^{*n}(x)=(1-\vep)^n\delta_{o,x}+\sum_{j=0}^{n-1}(1-\vep)^{n-1-j}
 \vep(D*q^{*j})(x),\lbeq{q-dec}
\end{align}
where we suppose that the empty sum equals zero.  When
$n\leq N\equiv\vep^{-1}$, we use \refeq{q-dec} to obtain
    \begin{align}\lbeq{qbd-small}
    q^{*n}(x)\leq(1-\vep)^n\delta_{o,x}+\sum_{l=0}^{n-1}(1-\vep)^l
    \vep\,\|D\|_\infty\leq(1-\vep)^n\delta_{o,x}+O(\beta)\leq (1-\vep)^n\delta_{o,x}
    +\frac{O(\beta)}{[1\vee (n\vep)]^{d/2}},
    \end{align}
as required. On the other hand, when $n>N$, we use
    \begin{align}
    q^{*n}(x)=(1-\vep)^n\delta_{o,x}+(1-\vep)^{n-1}n\vep D(x)+\sum_{j=0}^{n-2}
    (n-1-j)(1-\vep)^{n-2-j}\vep^2(D^{*2}*q^{*j})(x),\lbeq{q-dec2}
    \end{align}
which is obtained by substituting \refeq{q-dec} into $q^{*j}$ in the
right-hand side of \refeq{q-dec}.  Since the
second term in the right-hand side is bounded by $O(\beta)n\vep e^{-n\vep}
\leq O(\beta)[1\vee (n\vep)]^{-d/2}$, it suffices to investigate the third term.
Let $S_1$ be the sum over $j<N$, and let $S_2$ be the remaining sum, i.e.,
    \begin{align}
    S_1&=\sum_{0\leq j<N}(n-1-j)(1-\vep)^{n-2-j}\vep^2(D^{*2}*q^{*j})(x),\\
    S_2&=\sum_{N\leq j\leq n-2}(n-1-j)(1-\vep)^{n-2-j}\vep^2(D^{*2}*q^{*j})(x).
    \lbeq{S2}
    \end{align}
For $S_1$, we use
\refeq{qbd-small} to obtain
    \begin{align}\lbeq{S1-bd}
    S_1
    %\leq(1-\vep)^{n-2}\vep^2D^{*2}(o)\sum_{j=0}^{N-1}(n-1-j)
    %+O(\beta)\vep^2\sum_{j=0}^{N-1}(n-1-j)(1-\vep)^{n-2-j}\nn\\
    \leq(1-\vep)^{n-2}\vep^2D^{*2}(o)\sum_{l=n-N}^{n-1}l+O(\beta)\,\vep^2
    \sum_{l=n-N}^{n-1}l\,(1-\vep)^{l-1}\leq O(\beta)\,(n\vep)^2e^{-n\vep}.
    \end{align}
For $S_2$, we recall the definition $\hat B_{1/L}=\{k\in\Box_\pi:cL^2|k|^2\leq\eta\}$
below \refeq{Tpintbd}, and let $\hat B_{1/L}^+=\{k\in\hat B_{1/L}:\hat q(k)\geq0\}$.
Note that
    \begin{align}
    (D^{*2}*q^{*j})(x)\leq\int_{\hat B_{1/L}^+}\frac{d^dk}{(2\pi)^d}~\hat
    q(k)^j+\int_{\Box_\pi\setminus\hat B_{1/L}^+}\frac{d^dk}{(2\pi)^d}~
    \hat D(k)^2|\hat q(k)|^j.\lbeq{D*q-prebd}
    \end{align}
Recall \refeq{Dprop}.  For the first integral, we use
    \begin{align}
    \hat q(k)=1-\vep[1-\hat D(k)]\leq e^{-\vep[1-\hat D(k)]}\leq e^{-c\vep
    L^2|k|^2},\lbeq{q-bd}
    \end{align}
while for the second integral in \refeq{D*q-prebd}, we use, noting that without loss of generality,
we may assume that $\eta\leq 1$,
    \begin{align}\lbeq{|q|-bd}
    |\hat q(k)|\leq|1-\vep(2-\eta)|\vee(1-\vep\eta)\leq1-\vep\eta.
    \end{align}
Therefore, we have
    \begin{align}
    (D^{*2}*q^{*j})(x)&\leq\int_{\hat B_{1/L}^+}\frac{d^dk}{(2\pi)^d}~e^{-cj
    \vep L^2|k|^2}+(1-\vep\eta)^j\int_{\Box_\pi\setminus\hat B_{1/L}^+}\frac
    {d^dk}{(2\pi)^d}~\hat D(k)^2\nn\\
    &=O(\beta)\,(j\vep)^{-d/2}+O(\beta)\,(1-\vep\eta)^j.\lbeq{D*q-bd}
    \end{align}
Substituting \refeq{D*q-bd} into \refeq{S2} and separating the sum of
the first term in \refeq{D*q-bd} into the sum over $N\leq j<\frac{n}2-1$
and the sum over $\frac{n}2-1\leq j\leq n-2$, we obtain
    \begin{align}
    S_2&\leq O(\beta)\,(n\vep-\vep-1)\,(1-\vep)^{\frac{n}2-1}\,\vep\sum_{N\leq
    j<\frac{n}2-1}(j\vep)^{-d/2}\nn\\
    &\qquad+O(\beta)\,(\tfrac{n\vep}2-\vep)^{-d/2}\,\vep^2\sum_{\frac{n}2-1\leq
    j\leq n-2}(n-1-j)\,(1-\vep)^{n-2-j}\nn\\
    &\qquad+O(\beta)\,(1-\vep\eta)^{n-2}\,\vep^2\sum_{N\leq j\leq n-2}(n-1-j)\nn\\
    &\leq O(\beta)\,(n\vep)e^{-n\vep/2}+O(\beta)\,(n\vep)^{-d/2}+O(\beta)\,(n
    \vep)^2\,e^{-\eta n\vep}.\lbeq{S2-bd}
    \end{align}
The proof of \refeq{q-gauss} is completed by
combining \refeq{qbd-small}--\refeq{q-dec2}, \refeq{S1-bd} and \refeq{S2-bd},
and $(n\vep)^2 e^{-n\vep/2}\leq C [1\vee (n\vep)]^{-d/2}$ for all $n\geq 1/\vep$.
\qed



\section{Computation for the spread-out uniform model}
\label{s:uniform}
In this appendix, we compute the model-dependent terms of $\pc-1$ in
\refeq{sawcp-critpt}--\refeq{perc-critpt} when the probability distribution
$D$ is defined as in \refeq{dex}.  Recalling \refeq{U-def}, we have
\begin{align}\lbeq{D*D}
D^{*2}(o)=\frac1{(2L+1)^d-1}=\frac\beta{2^d}+O(\beta L^{-1})=\beta\,
 U^{\star2}(o)+O(\beta L^{-1}).
\end{align}
This relation can be extended as follows:

\begin{prop}\label{prop:DtoU}
Let $D$ be the function defined in \refeq{dex}.  For $\alpha=0,1$,
as $L\to\infty$,
\begin{align}
\sum_{n=3}^\infty(n+1)^\alpha\,D^{*n}(o)&=\beta\sum_{n=3}^\infty
 (n+1)^\alpha\,U^{\star n}(o)+O(\beta L^{-1}),\lbeq{nD*n}\\
\sum_{n=2}^\infty D^{*2n}(o)&=\beta\sum_{n=2}^\infty U^{\star 2n}
 (o)+O(\beta L^{-1}),\lbeq{D*2n}
\end{align}
where $d>4+2\alpha$ in \refeq{nD*n} and $d>4$ in \refeq{D*2n}.
\end{prop}

Theorem \ref{thm:uniform} is an immediate consequence of \refeq{D*D} and
Proposition~\ref{prop:DtoU}.  Note further that the coefficients of
$\beta$ in \refeq{nD*n}--\refeq{D*2n} are bounded if $d>2+2\alpha$ and
$d>2$, respectively, which suggests that also for $d=3+2\alpha$ and
$d=4+2\alpha$ the leading order contributions should be given by the
first terms in \refeq{nD*n}--\refeq{D*2n}.

\begin{proof}
For $x\in\Zd$, define
\begin{align}
D_o(x)=\frac{\indic{\|x\|_\infty\leq L}}{(2L+1)^d},
\end{align}
to be a regularized version of $D$ in \refeq{dex}.  It is obvious that
$D_o^{*2}(o)$ satisfies the same estimate as in \refeq{D*D}.  We note that
\begin{align}
D^{*m}(o)-D_o^{*m}(o)&=\sum_{j=1}^m\big((D-D_o)*D^{*(j-1)}*D_o^{*(m-j)}
 \big)(o)\nn\\
&=\sum_{j=1}^m\Bigg[\sum_{x:0<\|x\|_\infty\leq L}\frac{\big(D^{*(j-1)}
 *D_o^{*(m-j)}\big)(x)} {(2L+1)^d[(2L+1)^d-1]}-\frac{\big(D^{*(j-1)}*
 D_o^{*(m-j)}\big)(o)}{(2L+1)^d}\Bigg]\nn\\
&=\frac1{(2L+1)^d}\sum_{j=1}^m\Big[\big(D^{*j}*D_o^{*(m-j)}\big)(o)-
 \big(D^{*(j-1)}*D_o^{*(m-j)}\big)(o)\Big].\lbeq{D-Do}
\end{align}
By this identity and the fact that $D_o^{*n}(x)$ also satisfies
\refeq{d-gauss}, we can approximate the expressions in the left-hand side
of \refeq{nD*n}--\refeq{D*2n} by the corresponding expressions defined in
terms of $D_o$ instead of $D$, up to $O(\beta^2)$ when $d>4+2\alpha$ and
$d>4$, respectively.  For example, for \refeq{nD*n} with $\alpha=0$, we
use \refeq{d-gauss} to obtain
\begin{align}
&\bigg|\sum_{n=3}^\infty D^{*n}(o)-\sum_{n=3}^\infty D_o^{*n}(o)\bigg|\nn\\
&\qquad\leq\frac1{(2L+1)^d}\sum_{j=1}^\infty\;\sum_{n=j\vee3}^\infty\Big[
 \big(D^{*j}*D_o^{*(n-j)}\big)(o)+\big(D^{*(j-1)}*D_o^{*(n-j)}\big)(o)
 \Big]\nn\\
&\qquad=\frac1{(2L+1)^d}\Big((D+\delta_o)*\big(D_o^{*2}+D*D_o+D^{*2}*G\big)
 *G_o\Big)(o)=O(\beta^2),\lbeq{sumD-sumDo}
\end{align}
where $\delta_o(x)=\delta_{o,x}$ and $G_o(x)=\sum_{n=0}^\infty D_o^{*n}(x)$.
Therefore, to prove Proposition~\ref{prop:DtoU}, it suffices to show that,
for $\alpha=0,1$,
\begin{align}
\sum_{n=3}^\infty(n+1)^\alpha\,D_o^{*n}(o)&=\beta\sum_{n=3}^\infty
 (n+1)^\alpha\,U^{\star n}(o)+O(\beta L^{-1}),\lbeq{nDo*n}\\
\sum_{n=2}^\infty D_o^{*2n}(o)&=\beta\sum_{n=2}^\infty U^{\star 2n}
 (o)+O(\beta L^{-1}).\lbeq{Do*2n}
\end{align}

We prove \refeq{nDo*n} for $\alpha=0$ by comparing the Fourier transform
of the first term in the right-hand side of \refeq{nDo*n}, i.e.,
\begin{align}\lbeq{Usum-fourier}
\beta\sum_{n=3}^\infty U^{\star n}(o)=\beta\int_{\mR^d}\frac{d^dk}
 {(2\pi)^d}~\frac{\hat U(k)^3}{1-\hat U(k)},
\end{align}
with the Fourier transform of the left-hand side of \refeq{nDo*n}, i.e.,
\begin{align}
\sum_{n=3}^\infty D_o^{*n}(o)=\int_{\Box_\pi}\frac{d^dk}{(2\pi)^d}~\frac
 {\hat D_o(k)^3}{1-\hat D_o(k)}&=\beta_o\int_{\Box_{
 (L+\frac12)\pi}}\frac{d^dk}{(2\pi)^d}~\frac{\hat D_o\big(\frac{k}{L+\frac
 12}\big)^3}{1-\hat D_o\big(\frac{k}{L+\frac12}\big)},\lbeq{Dosum-fourier}
%&=\beta\int_{\Box_{(L+\frac12)\pi}}\frac{d^dk}{(2\pi)^d}~\frac{\hat D_o
% \big(\frac{k}{L+\frac12}\big)^3}{1-\hat D_o\big(\frac{k}{L+\frac12}\big)}
% +O(\beta L^{-1})
\end{align}
where $\Box_\ell=[-\ell,\ell]^d$ and
\begin{align}
\beta_o=(L+\tfrac12)^{-d}=\beta+O(\beta L^{-1}),\lbeq{betao}
\end{align}
and also
\begin{align}
\hat U(k)&=\int_{\mR^d}d^dx~U(x)~e^{ik\cdot x}=\prod_{j=1}^d\frac{\sin
 k_j}{k_j},\lbeq{U-fourier}\\
\hat D_o(k)&=\sum_{x\in\Zd}D_o(x)~e^{ik\cdot x}=\prod_{j=1}^d\frac{\sin
 [(2L+1)\frac{k_j}2]}{(2L+1)\sin\frac{k_j}2}=\frac{\hat U((L+\tfrac12)k)}
 {\hat U(\frac{k}2)}.\lbeq{Do-fourier}
\end{align}
The simple product formula \refeq{Do-fourier} is the main advantage of
using $D_o$ instead of $D$.  It follows from \refeq{Do-fourier} that
$\hat D_o\big(\frac{k}{L+\frac12}\big)=\hat U(k)/\hat U(\frac{k}{2L+1})$,
which approximates $\hat U(k)$ for large $L$.  We write
\begin{align}\lbeq{I12}
\beta\sum_{n=3}^\infty U^{\star n}(o)-\sum_{n=3}^\infty D_o^{*n}(o)=
 (\beta-\beta_o)\sum_{n=3}^\infty U^{\star n}(o)+\beta_o(I_1+I_2),
\end{align}
where, by \refeq{betao}, the first term is $O(\beta L^{-1})$, and
\begin{align}\lbeq{I1I2}
I_1=\int_{\mR^d\setminus\Box_{(L+\frac12)\pi}}\frac{d^dk}{(2\pi)^d}~
 \frac{\hat U(k)^3}{1-\hat U(k)},&&
I_2=\int_{\Box_{(L+\frac12)\pi}}\frac{d^dk}{(2\pi)^d}~\Bigg[\frac{
 \hat U(k)^3}{1-\hat U(k)}-\frac{\hat D_o\big(\frac{k}{L+\frac12}
 \big)^3}{1-\hat D_o\big(\frac{k}{L+\frac12}\big)}\Bigg].
\end{align}
We prove below that each $I_j$ is $O(L^{-1})$ if $d>2$.  This suffices
to prove \refeq{nDo*n} for $\alpha=0$.  If fact, we will prove that
each $I_j$ is $O(L^{-2}\log L)$ if $d>2$, which also identifies the
coefficient of $\beta L^{-1}$.

To estimate each $I_j$, we use the following properties
of $\hat U(k)$ and $\hat D_o\big(\frac{k}{L+\frac12}\big)$ that follow
from the standard estimates for the trigonometric functions: for any $k$,
\begin{align}\lbeq{U-summary}
|\hat U(k)|\leq\prod_{j=1}^d(1\vee|k_j|)^{-1},&&
1-\hat U(k)&\geq c_1(1\wedge|k|^2),
\end{align}
and for $k\in\Box_{(L+\frac12)\pi}$,
\begin{gather}
\big|\hat D_o\big(\tfrac{k}{L+\frac12}\big)\big|\leq c_2\prod_{j=1}^d
 (1\vee|k_j|)^{-1},\qquad\qquad
1-\hat D_o\big(\tfrac{k}{L+\frac12}\big)\geq c_3(1\wedge|k|^2),
 \lbeq{Do-summary}\\[5pt]
\big|\hat U(k)-\hat D_o\big(\tfrac{k}{L+\frac12}\big)\big|\leq c_4
 L^{-2}|\hat U(k)|\,|k|^2,\lbeq{U-Do-summary}
\end{gather}
where each $c_i\in(0,\infty)$ is independent of $L$ and $k$.  To see
the first inequality in \refeq{U-summary}, we only need to use
$|\frac{\sin r}r|\leq(1\vee r)^{-1}$ for any $r$.  For the first
inequality in \refeq{Do-summary}, we recall $\hat D_o\big(\frac{k}
{L+\frac12}\big)=\hat U(k)/\hat U(\frac{k}{2L+1})$ and use
$\frac{\sin r}r\geq\frac2\pi$ for any $r\in[-\frac\pi2,\frac\pi2]$,
so that $|\hat U(\frac{k}{2L+1})^{-1}|\leq(\frac\pi2)^d$, and then
use the bound on $|\hat U(k)|$ in \refeq{U-summary}.  The second
inequalities in \refeq{U-summary}--\refeq{Do-summary} follow from
\refeq{U-fourier} and \refeq{Dprop}, respectively.  Finally, for
\refeq{U-Do-summary}, we again use $\frac2\pi\leq\frac{\sin r}r\leq1$
for any $r\in[-\frac\pi2,\frac\pi2]$ to obtain
\begin{align}
\big|\hat U(k)-\hat D_o\big(\tfrac{k}{L+\frac12}\big)\big|&\leq\Big(\frac
 \pi2\Big)^{\!d}\,|\hat U(k)|\,\bigg|\prod_{j=1}^d\frac{\sin\frac{k_j}{2L
 +1}}{\frac{k_j}{2L+1}}-1\bigg|\leq\Big(\frac\pi2\Big)^{\!d}\,|\hat U(k)|
 \sum_{i=1}^d\bigg|\frac{\sin\frac{k_i}{2L+1}}{\frac{k_i}{2L+1}}-1\bigg|,
\end{align}
which is bounded by $|\hat U(k)|\,O(|k|^2L^{-2})$, using
$0\leq\frac{\sin r}r-1+\frac{r^2}{3!}\leq\frac{r^2}{5!}(\frac\pi2)^2$
for any $r\in[-\frac\pi2,\frac\pi2]$.  This completes the proof of
\refeq{U-summary}--\refeq{U-Do-summary}.

First, we consider $I_1$.  Using \refeq{U-summary}, we obtain
\begin{align}
|I_1|\leq c\int_{\mR^d\setminus\Box_{(L+\frac12)\pi}}d^dk~|\hat U(k)|^3
 \leq c'\sum_{i=1}^d\bigg[\int_{(L+\frac12)\pi}^\infty\frac{dk_i}{k_i^3}
 \bigg]\bigg[\prod_{j\ne i}\int_{-\infty}^\infty\frac{dk_j}{(1\vee|k_j|)^3}
 \bigg]=O(L^{-2}).\lbeq{I1-bd}
\end{align}
For $I_2$, we use
\begin{align}
\bigg|\frac{u^3}{1-u}-\frac{d^3}{1-d}\bigg|=\frac{|u-d|\,|u^2+ud+d^2
 -ud(u+d)|}{(1-u)(1-d)}\leq\frac{|u-d|\,(u^2+3|ud|+d^2)}{(1-u)(1-d)},
\end{align}
with $u=\hat U(k)$ and $d=\hat D_o\big(\frac{k}{L+\frac12}\big)$.
Using \refeq{U-summary}--\refeq{U-Do-summary}, we obtain
\begin{align}
|I_2|\leq cL^{-2}\int_{\Box_{(L+\frac12)\pi}}d^dk~\frac{|k|^2\prod_{j=
 1}^d(1\vee|k_j|)^{-3}}{(1\wedge|k|^2)^2}=O(L^{-2}\log L),\lbeq{I2-bd}
\end{align}
for $d>2$.
%\begin{align}
%|I_2|&\leq cL^{-2}\int_{\Box_1}\frac{d^dk}{1\wedge|k|^2}+ c'L^{-2}
% \int_{\Box_{L\pi+\frac\pi2}\setminus\Box_1}d^dk~\frac{|k|^2}{\prod_{j
% =1}^d(1\vee|k_j|)^3}=O(L^{-2}\log L).\lbeq{I2-bd}
%%&\leq O(L^{-2})+\frac{c'}{L^2}\sum_{i=1}^d\bigg[2\int_1^{L\pi+
%% \frac\pi2}\frac{dk_i}{|k_i|}\bigg]\bigg[\prod_{j\ne i}\int_{-L\pi-\frac
%% \pi2}^{L\pi+\frac\pi2}\frac{dk_j}{(1\vee|k_j|)^3}\bigg]\nn\\
%%&\qquad+\frac{c'}{L^2}\sum_{\substack{i,j=1\\ i\ne j}}^d\bigg[2
%% \int_1^{L\pi+\frac\pi2}\frac{dk_i}{|k_i|^3}\bigg]\bigg[\int_{-L\pi-
%% \frac\pi2}^{L\pi+\frac\pi2}dk_j~\frac{k_j^2}{(1\vee|k_j|)^3}\bigg]
%% \bigg[\prod_{l\ne i,j}\int_{-L\pi-\frac\pi2}^{L\pi+\frac\pi2}\frac
%% {dk_l}{(1\vee|k_l|)^3}\bigg].
%\end{align}
The proof of \refeq{nDo*n} for $\alpha=0$ is completed by \refeq{I12},
\refeq{I1-bd} and \refeq{I2-bd}.

The same strategy explained above also applies to the proof of \refeq{nD*n}
for $\alpha=1$ and \refeq{D*2n}, using the following expressions:
\begin{align}
\sum_{n=3}^\infty(n+1)\,D_o^{*n}(o)&=\beta_o\int_{\Box_{(L+\frac12)\pi}}
 \frac{d^dk}{(2\pi)^d}~\frac{\hat D_o\big(\frac{k}{L+\frac12}\big)^3
 \big[4-3\hat D_o\big(\frac{k}{L+\frac12}\big)\big]}{\big[1-\hat D_o
 \big(\frac{k}{L+\frac12}\big)\big]^2},\\
\sum_{n=2}^\infty D_o^{*2n}(o)&=\beta_o\int_{\Box_{(L+\frac12)\pi}}
 \frac{d^dk}{(2\pi)^d}~\frac{\hat D_o\big(\frac{k}{L+\frac12}\big)^4}
 {1-\hat D_o\big(\frac{k}{L+\frac12}\big)^2}.
\end{align}
This completes the proof of Proposition~\ref{prop:DtoU}.
\end{proof}


\section*{Acknowledgements}
The work of AS was supported in part by NSERC of Canada.  The work of RvdH
and AS was supported in part by Netherlands Organisation for Scientific
Research (NWO). %, and was carried out in part at Delft University of
%Technology.
This project was initiated during an extensive visit of RvdH to the
University of British Columbia, Vancouver, Canada. We thank Gordon~Slade
for comments on a preliminary version of the paper, and for pointing us
to the appropriate bound in \cite{HS90a} that implies Lemma \ref{lem-T}.




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\end{thebibliography}



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math0402050.extracted_bib
\begin{thebibliography}{99}

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\newblock {\it J. Statist. Phys.} {\bf 36} (1984): 107--143.

\bibitem{ba91}D. J. Barsky and M. Aizenman.
\newblock Percolation critical exponents under the triangle condition.
\newblock {\it Ann. Probab.} {\bf 19} (1991): 1520--1536.

%\bibitem{bw98}D. J. Barsky and C. C. Wu.
%\newblock Critical exponents for the contact process under the triangle condition.
%\newblock {\it J. Statist. Phys.} {\bf 91} (1998): 95--124.

%\bibitem{bg90}C. Bezuidenhout and G. Grimmett.
%\newblock The critical contact process dies out.
%\newblock {\it Ann. Probab.} {\bf 18} (1990): 1462--1482.

\bibitem{bg91}C. Bezuidenhout and G. Grimmett.
\newblock Exponential decay for subcritical contact and percolation processes.
\newblock {\it Ann. Probab.} {\bf 19} (1991): 984--1009.

\bibitem{bds89}M. Bramson, R. Durrett and G. Swindle.
\newblock Statistical mechanics of crabgrass.
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%\bibitem{BS85}
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%\newblock Self-avoiding walk in 5 or more dimensions.
%\newblock {\em Commun. Math. Phys.} {\bf 97} (1985):125--148.

\bibitem{BCHSS04b}
C.~Borgs, J.T. Chayes, R.~van~der Hofstad, G.~Slade, and J.~Spencer.
\newblock Random subgraphs of finite graphs: {II}. {The} lace expansion
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\newblock Preprint, (2003).

\bibitem{cd83}J. T. Cox and R. Durrett.
\newblock Oriented percolation in dimensions $d\geq4$: bounds and asymptotic
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\newblock {\it Math. Proc. Cambridge Philos. Soc.} {\bf 93} (1983): 151-162.

%\bibitem{Daws93}
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%\newblock Measure-valued {Markov} processes.
%\newblock In {\em Ecole d'Et\'e de Probabilit\'es de Saint--Flour 1991. Lecture
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%\bibitem{durr95}R. Durrett.
%\newblock Ten lectures on particle systems.
%\newblock {\em Lectures on Probability Theory and
%  Statistics. Ecole d'Et\'e de Probabilit\'es de Saint--Flour XXIII-1995}, pages
%  97--201, Berlin, (1995). Springer.
%  \newblock Lecture Notes in Mathematics \#1608.

\bibitem{dp99}R. Durrett and E. Perkins.
\newblock Rescaled contact processes converge to super-Brownian motion in two
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\newblock {\it Probab. Th. Rel. Fields} {\bf 114} (1999): 309--399.


%\bibitem{Dynk88}
%E.B. Dynkin.
%\newblock Representation for functionals of superprocesses by multiple
%  stochastic integrals, with applications to self-intersection local times.
%\newblock {\em Ast\'erisque}, {\bf 157--158}:147--171, (1988).


%\bibitem{Dynk94}
%E.B. Dynkin.
%\newblock {\em An Introduction to Branching Measure-Valued Processes}.
%\newblock American Mathematical Society, Providence, (1994).

%\bibitem{Ethe00}
%A.M. Etheridge.
%\newblock {\em An Introduction to Superprocesses}.
%\newblock American Mathematical Society, Providence, (2000).

%\bibitem{LeGa99c}
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%\newblock {\em Spatial {B}ranching {P}rocesses, {R}andom {S}nakes, and
%  {P}artial {D}ifferential {E}quations}.
%\newblock Birkh\"auser, Basel (1999).

\bibitem{g99}G. Grimmett.
\newblock {\it Percolation}.  Springer, Berlin (1999).

\bibitem{gh01}G. Grimmett and P. Hiemer.
\newblock Directed percolation and random walk.
\newblock {\it In and Out of Equilibrium} (ed. V. Sidoravicius).
Birkh\"auser (2002): 273-297.

\bibitem{HHS01a}
T.~Hara, R.~van~der Hofstad, and G.~Slade.
\newblock Critical two-point functions and the lace expansion for spread-out
  high-dimensional percolation and related models.
\newblock {\em Ann.\ Probab.}, {\bf 31} (2003): 349-408.

\bibitem{HS90a}
T.~Hara and G.~Slade.
\newblock Mean-field critical behaviour for percolation in high dimensions.
\newblock {\em Commun. Math. Phys.}, {\bf 128} (1990): 333--391.

%\bibitem{hs94}
%T.~Hara and G.~Slade.
%\newblock Mean-field behaviour and the lace expansion.
%\newblock {\it Probability and Phase Transition} (ed. G. R. Grimmett).
%Kluwer, Dordrecht (1994): 87--122.

\bibitem{hs95}
T.~Hara and G.~Slade.
\newblock The self-avoiding-walk and percolation critical points in high dimensions.
\newblock {\it Combin. Probab. Comput.}, {\bf 4} (1995): 197--215.

%\bibitem{HS00a}T. Hara and G. Slade.
%\newblock The scaling limit of the incipient infinite cluster in high-dimensional
%percolation. I. Critical exponents.
%\newblock {\it J. Statist. Phys.} {\bf 99} (2000): 1075--1168.
%
%\bibitem{HS00b}T. Hara and G. Slade.
%\newblock The scaling limit of the incipient infinite cluster in high-dimensional
%percolation. II. Integrated super-Brownian excursion.
%\newblock {\it J. Math. Phys.} {\bf 41} (2000): 1244--1293.

\bibitem{h03}R. van der Hofstad.
\newblock The derivative of the lace expansion coefficients for
unoriented percolation.
\newblock Unpublished document (2003).

%\bibitem{hhs98}R. van der Hofstad, F. den Hollander and G. Slade.
%\newblock A new inductive approach to the lace expansion for self-avoiding walks.
%\newblock {\it Probab. Th. Rel. Fields} {\bf 111} (1998): 253--286.
%
%\bibitem{hhs01}R. van der Hofstad, F. den Hollander and G. Slade.
%\newblock Construction of the incipient infinite cluster for the spread-out
%oriented percolation above 4+1 dimensions.
%\newblock {\it Commun. Math. Phys.} {\bf 231} (2002): 435--461.

\bibitem{hsa04}R. van der Hofstad and A. Sakai.
\newblock Gaussian scaling for the critical spread-out
contact process above the upper critical dimension.
\newblock {\it Preprint} (2003). To appear in {\it Electr.\ Journ.\ Probab.}

\bibitem{hs02}R. van der Hofstad and G. Slade.
\newblock A generalised inductive approach to the lace expansion.
\newblock {\it Probab. Th. Rel. Fields} {\bf 122} (2002): 389--430.

\bibitem{hs01}R. van der Hofstad and G. Slade.
\newblock Convergence of critical oriented percolation to super-Brownian motion
above 4+1 dimensions.
\newblock {\it Ann. Inst. H. Poincar\'e Probab. Statist.} {\bf 39} (2003): 413--485.

\bibitem{hs03}R. van der Hofstad and G. Slade.
\newblock The lace expansion on a tree with application to networks of
self-avoiding walks.
\newblock {\it Adv. Appl. Math.} {\bf 30} (2003): 471--528.

\bibitem{HS04a}R. van der Hofstad and G. Slade.
\newblock Expansion in $n^{-1}$ for percolation critical values on the
$n$-cube and ${\mathbb Z}^n$: the first three terms.
\newblock {\it Preprint} (2004). To appear in {\it Combin. Probab. Comput.}

\bibitem{HS04b}R. van der Hofstad and G. Slade.
\newblock Asymptotic expansions in $n^{-1}$ for percolation critical values
on the $n$-cube and ${\mathbb Z}^n$.
\newblock {\it Preprint} (2004).

\bibitem{k80}H. Kesten.
\newblock The critical probability of bond percolation on the square
lattice equals $\frac12$.
\newblock {\it Commun. Math. Phys.} {\bf 74} (1980): 41--59.

\bibitem{Ligg99}T. Liggett.
\newblock {\em Stochastic Interacting Systems: Contact, Voter and Exclusion Processes}.
\newblock Springer, Berlin (1999).

\bibitem{l97}T. Liggett.
\newblock Stochastic models of interacting systems.
\newblock {\it Ann. Probab.} {\bf 25} (1997): 1--29.


\bibitem{ms93}N. Madras and G. Slade.
\newblock {\it The Self-Avoiding Walk}.
\newblock Birkh\"auser, Boston (1993).

\bibitem{ny93}B. G. Nguyen and W.-S. Yang.
\newblock Triangle condition for oriented percolation in high dimensions.
\newblock {\it Ann. Probab.} {\bf 21} (1993): 1809--1844.

%\bibitem{ny95}B. G. Nguyen and W.-S. Yang.
%\newblock Gaussian limit for critical oriented percolation in high dimensions.
%\newblock {\it J. Statist. Phys.} {\bf 78} (1995): 841--876.

\bibitem{p94}M. D. Penrose.
\newblock Self-avoiding walks and trees in spread-out lattices.
\newblock {\it J. Statist. Phys.} {\bf 77} (1994): 3--15.

%\bibitem{Perk02}
%E.~Perkins.
%\newblock Dawson--{Watanabe} superprocesses and measure-valued diffusions.
%\newblock In P.L. Bernard, editor, {\em Lectures on Probability Theory and
%  Statistics. Ecole d'Et\'e de Probabilit\'es de Saint--Flour XXIX-1999}, pages
%  125--329, Berlin, (2002). Springer.
%\newblock Lecture Notes in Mathematics \#1781.

%\bibitem{s00}A. Sakai.
%\newblock Analyses of the critical behavior for the contact process based on
%a percolation structure.
%\newblock {\it Ph.D. thesis} (2000).

\bibitem{s01}A. Sakai.
\newblock Mean-field critical behavior for the contact process.
\newblock {\it J. Statist. Phys.} {\bf 104} (2001): 111--143.

%\bibitem{s02}A. Sakai.
%\newblock Hyperscaling inequalities for the contact process and oriented percolation.
%\newblock {\it J. Statist. Phys.} {\bf 106} (2002): 201--211.

%\bibitem{s03}A. Sakai.
%\newblock Gaussian Scaling for the Critical Contact Process with Rapid Stirring
%above Four Dimensions.
%\newblock In preparation.

%\bibitem{Slad02}
%G.~Slade.
%\newblock Scaling limits and super-{Brownian} motion.
%\newblock {\em Notices A.M.S.}, {\bf 49}(9):1056--1067, (2002).

%\bibitem{Scho98}R. Schonmann.
%\newblock The triangle condition for contact processes on homogeneous trees.
%\newblock {\it J. Statist. Phys.} {\bf 90} (1998): 1429--1440.

%\bibitem{wu95}C. C. Wu.
%\newblock The contact process on a tree: behavior near the first phase transition.
%\newblock {\it Stochastic Process. Appl.} {\bf 57} (1995): 99--112.

\end{thebibliography}