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\documentclass[imslayout,noinfoline]{imsart}
\RequirePackage[OT1]{fontenc}
\RequirePackage{amsfonts,amsthm,amsmath,natbib}
\RequirePackage[colorlinks]{hyperref}
\usepackage[usenames]{color}

\startlocaldefs
\newcommand{\nc}{\newcommand}
\nc{\be}{\begin{equation}}
\nc{\ee}{\end{equation}}
\nc{\ds}{\displaystyle}
\nc{\vep}{\varepsilon}
\nc{\Z}{\mathbb{Z}}
\nc{\R}{\mathbb{R}}
\nc{\T}{\mathbb{T}}
\nc\To{\Rightarrow}
\nc\E{\mathcal{E}}
\nc\A{\mathcal{A}}
\nc{\cD}{\mathcal{D}}
\nc{\fl}[1]{\lfloor #1 \rfloor}
\endlocaldefs

\newcommand{\MP}[1]{\marginpar{\small\bf #1}}

\begin{document}
\begin{frontmatter}
\title{Intermediate Range Migration in the two-dimensional
  Stepping Stone Model}
\runtitle{Stepping Stone Model}
\thankstext{T1}{Research supported in part by NSF Grant 0803517}
\begin{aug}
  \author{J. Theodore
    Cox\protect\thanksref{T1}\ead[label=e1]{jtcox@syr.edu}}, 
  \runauthor{J. Theodore Cox}

\affiliation{Syracuse University}

\address{Mathematics Department, Syracuse University,
    Syracuse NY 13244\\ \printead{e1}} 
\end{aug}

\begin{abstract} We consider the stepping stone model on the torus of side $L$ in $\Z^2$ in the limit $L\to\infty$, and study the time it takes two lineages tracing backwards in time to coalesce. Our work fills a gap between the finite range migration case of \cite{ZCD} and the long range case of \cite{MW06}, where the migration range is a positive fraction of $L$. We obtain limit theorems for the intermediate case, and verify a conjecture in \cite{Du08} that the model is homogeneously mixing if and only if the migration range is of larger order than $(\log L)^{1/2}$.
\end{abstract}

\begin{keyword}[class=AMS]
\kwd[Primary ]{60K35}
\kwd{60G50,92D10}
\kwd[; secondary ]{82C41}
\end{keyword}

\begin{keyword}
\kwd{Stepping stone model}
\kwd{torus random walk}
\kwd{hitting times}
\kwd{coalescence times}
\end{keyword}
\end{frontmatter}

\section{Introduction}

The subject of this paper is the stepping stone model of population genetics, and in particular the contrast between recent results of \cite{MW06} and \cite{ZCD} in the two-dimensional setting. There is a vast literature on the many variants of the stepping stone model dating back to the seminal work of Mal\`ecot \cite{Ma49} and Kimura \cite{Ki53}.  (A few sources for background and references are \cite{Du08}, \cite {Na89} and \cite{Wi04}).)  We will begin by describing the version of the model we consider here, generally following the setup in \cite{ZCD}.

...

Following two lineages backward in time amounts to following two random walks until they meet. The difference between the lineage locations is also a random walk, and $T_0$ is just the time it takes this difference walk to hit 0. On account of this, we will now focus on the following random walk setting. 
For $k>0$ let 
\[
\Lambda_k= [-k/2,k/2]^2\cap\Z^2 
\]
and for any $A\subset\R^2$ let 
\[
A'=A\setminus\{0\} \ .
\] For $r>0$ let $B(r) = \{x\in\R^2:\|x\|_\infty\le
r\}$. Let $M_1,M_2,\dots$ be a sequence of positive integers
and assume that $q_{M_L}:\Z^2\to[0,1]$ satisfies

\medskip\noindent
\textbf{(P0)} $\ds
\quad \begin{aligned}
&q_{M_L}(x)=0 \text{ for }x\notin \Lambda'_{M_L}\,,
\  \sum\nolimits_x q_{M_L}(x)=1\;, 
q_{M_L} \text{ is symmetric, }\\
&\text{and }\sigma_{M_L}^2 \equiv \sum\nolimits_x x^2_1 q_{M_L}(x) 
= \sum\nolimits_x x^2_2q_{M_L}(x) >0 \ .
\end{aligned}$

With the above notation, the Matsen and Wakely result is as
follows. Fix $0<c<1$ and let $M_L=cL$ and
$q_{M_L}=u_{M_L}$. Then 
as $L\to\infty$, 
\be\label{mfexplaw}
H_L/L^2 \To \E(1) \text{ uniformly in }
X^L_0\in\T_L' \;,
\ee
where $\To$ indicates the law of the left-hand side converges
weakly to the distribution on right-hand side, and $\E(\beta)$
is the exponential distribution with mean $\beta$.  On the
other hand, if $M_L\equiv M$ is fixed, so there is a single
jump distribution $q_M$, then by Theorem~1 of of \cite{ZCD},
if $0<\alpha< 1$ and $|X^L_0|\approx L^{\alpha}$ as
$L\to\infty$, then
\be\label{myexplaw}
\dfrac{H_L}{L^2\log L} \To (1-\alpha)\delta_0 + \alpha
\E(1/\pi \sigma^2_M) \,.
\ee
Here $x_L\approx L^\alpha$ means $x_L\in \T_{L^\alpha\log
  L}\setminus \T_{L^\alpha/\log L}$.

...

For a jump distribution $q_M$ define the characteristic function
\[
\phi_M(\theta) = \sum_{x\in\Lambda_{M}}e^{i\theta x}q_M(x)
\,,
\quad \theta\in\R^2\,,
\]
where $\theta x=\theta\cdot x$.  We will
assume that the jump distributions $q_{M_L}$ have
characteristic functions $\phi_{M_L}$ which satisfy the 
conditions (P1)--(P3) listed below. These conditions are satisfied
for the uniform distributions $u_{M_L}$ (see the Appendix of
\cite{CDP00}, where $M^2$ in (P2) there should be $M$).  
Proposition~\ref{prop:suff} below shows that they are satisfied
in some generality. Note that the symmetry condition in (P0) implies
each $\phi_M$ is real-valued. The conditions we need are:

\medskip\noindent \textbf{(P1)} There is a $\sigma^2>0$ such
that for all $\vep>0$ there exists $\delta>0$ such that for all
large $L$,
\begin{equation*}
\dfrac{1-\phi_{M_L}(\theta)}{\sigma^2M_L^2|\theta|^2/2} \in
(1-\vep,1+\vep) \text{ for all }\theta\in B'(\delta/M_L)
\,.
\end{equation*}

\section{Proof of Theorem~\ref{thm:uniform}}
We continue to write $M$ for $M_L$. It
suffices to prove that
\be\label{g1}
\lim_{L\to\infty}\sup_{x\in \T_L} L^2\bigl|
P_0(X^L_{s_L}=x) -L^{-2}\bigr| = 0 \;.
\ee
By  pulling out the $y=0$ term from \eqref{ptL} we see that 
\[
L^2 \bigl| P_0(X^L_{s_L}=x) - L^{-2}\bigr|
= \Bigl|\sum_{y\in \T'_L}
\phi^{s_L}_M(2\pi y/L) e^{2\pi i x y/L} \Bigr|
\le \sum_{y\in \T'_L}
\phi^{s_L}_M(2\pi y/L) \,.
\]
The limit \eqref{g1} will follow from showing the
last sum tends to zero as
$L\to\infty$.

...

By the continuity of $\hat q_0$ there exists $\delta''>0$ 
such that if $\theta,\theta'\in
B(\pi)$ and $|\theta-\theta'|<\delta''$ then
\[
\Bigl|
\dfrac1{1-(1-c)\widehat q_0(\theta)} -
\dfrac1{1-(1-c)\widehat q_0(\theta')}\Big| 
<\vep \,.
\]
Assuming $K<\delta'' L$, this implies 
\begin{multline}\label{GLx-mix6}\Bigl|
\dfrac{1}{L^2} \sum_{z\in \Gamma_{L,K}}
\sum_{y\in{z+\T_K}}\dfrac{e^{2\pi i x y/L}}{1-(1-c)\widehat q_0(2\pi y/L)}
\\ - \dfrac{1}{L^2} \sum_{z\in \Gamma_{L,K}}
\dfrac{e^{2\pi ixz/L}}{1-(1-c)\widehat q_0(2\pi z/L)}
\sum_{y\in{\T_K}}e^{2\pi i x y/L} 
\Bigr| < \vep \,.
\end{multline}
Now \eqref{phiest} can be applied, giving
\begin{multline}\label{GLx-mix7}
\Bigl| \dfrac{1}{L^2} \sum_{z\in \Gamma_{L,K}}
\dfrac{e^{2\pi ixz/L}}{1-(1-c)\widehat q_0(2\pi
  z/L)}\sum_{y\in{\T_K}}e^{2\pi i x y/L}  
\Bigr| \\\le
\dfrac{|\Gamma_{L,K}|}{cL^2}
\; \Bigl| \sum_{y\in{\T_K}}e^{2\pi i x y/L} 
\Bigr|
\le \dfrac{(4(K+1)(1+ L/2\pi\ell_L))}{cK^2}\,.
\end{multline}
for all $x\in \T_L\setminus \T_{\ell_L}$. 
Taken together \eqref{GLx-mix3} and \eqref{GLx-mix5}--\eqref{GLx-mix7} 
yield 
\begin{align*}
\limsup_{L\to\infty}&\sup_{x\in \T_L\setminus
  \T_{\ell_L}}\Bigl|\dfrac{1}{L^2}\sum_{{y\in \T'_L}}
   \dfrac{ e^{2\pi i x y/L}} 
{1-\phi_{M_L}(2\pi y/L)} \Bigr| \\
&\le 
\dfrac{12\vep}{c\pi}+
\dfrac{a^2}{c\zeta} + 2\vep 
+ \limsup_{L\to\infty} \Bigl(
\dfrac{8K}{cL} +
\dfrac{(4(K+1)(1+ L/2\pi\ell_L))}{K^2}
\Bigr)\;.
\end{align*}
If set $K=L/\sqrt{\ell}$ then the limsup above is 0.
Let $a\downarrow 0$ and the $\vep\downarrow0$ to finish the
proof.

\begin{thebibliography}{9}
\bibitem{A89} \textsc{Aldous, D.} (1989) Hitting times for
  random walks on vertex-transitive graphs.
  \textit{Math. Proc. Camb. Philos. Soc.} \textbf{106},
  179191.

...

\end{thebibliography}
\end{document}

\section{Garbage}

On the other hand, if $\rho=\infty$  then
\be\label{lim-2}
\dfrac{1}{M^2}\sum_{y\in \T'_{\delta L/M}}
\dfrac{1}{|y|^2} \le \dfrac1{M^2}
\sum_{y\in \T'_{\delta L}}
\dfrac{1}{|y|^2} \to 0 \,.
\ee
Since $t_L=1$ eventually, this and \eqref{1-b} imply
\[
\lim_{L\to\infty} \dfrac1{L^2t_L}\sum_{y\in
      \T'_{\delta L/M}}  
\dfrac{1}{1 - \phi_M(2\pi y/L)}
=0 \,.
\]

\end{document}