COALESCENT PROCESSES ARISING IN THE STUDY OF DIFFUSIVE CLUSTERING
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Andreas Greven, Vlada Limic and Anita Winter
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This paper studies spatial coalescents on $\Z^2$.
In our setting, the partition elements are located at the sites of
$\Z^2$ and undergo local delayed coalescence and migration. That is,  
pairs of partition elements located at the same site coalesce 
into one partition element
after exponential waiting times. In addition, the partition 
elements perform independent random walks.  
The system starts in either locally finite configurations or in
configurations containing countably
many partition elements  per site.
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Our goal is to determine the longtime behavior with an initial
population of countably many individuals per site restricted to a box 
$[-t^{\alpha/2}, t^{\alpha/2}]^2 \cap \Z^2$
and observed at time $t^\beta$ with $1 \geq \beta \geq \alpha\ge 0$. 
We study both asymptotics, as $t\to\infty$,
for a fixed value of $\alpha$
as the parameter $\beta\in[\alpha,1]$ varies,
and for a fixed $\beta=1$, 
as the parameter $\alpha\in [0,1]$ varies.
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A new random object, the so-called {\em coalescent with rebirths}, 
is constructed and shown to arise in the limit. For sake of completeness, and in view of 
future applications we introduce this spatial coalescent with
rebirths and study its longtime asymptotics as well.
The present paper is the basis for forthcoming work \cite{glw2}, where the
genealogies in interacting Moran models and
Fisher-Wright diffusions on $\Z^2$ are studied. In \cite{glw2} the
coalescent with rebirth is needed to describe 
the ``complete'' genealogical forests, i.e., the genealogical
structures which include also the fossils.