Coalescents with Simultaneous Multiple Collisions
Jason Schweinsberg, University of California, Berkeley
Abstract
We study a family of coalescent processes that undergo ``simultaneous
multiple collisions,'' meaning that many clusters of particles can
merge into a single cluster at one time, and many such mergers can occur
simultaneously. This family of processes, which we obtain from simple
assumptions about the rates of different types of mergers,
essentially coincides with a family of processes that Mohle and
Sagitov obtain as a limit of scaled ancestral
processes in a population model with exchangeable family sizes.
We characterize the possible merger rates in terms of a single
measure, show how these coalescents can be constructed from a Poisson
process, and discuss some basic properties of these processes. This
work generalizes some work of Pitman, who provides similar analysis
for a family of coalescent processes in which many clusters can
coalesce into a single cluster, but almost surely no two such mergers
occur simultaneously.
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