Intermediate range migration in the two-dimensional stepping stone model

Details Intermediate range migration in the two-dimensional stepping stone model Intermediate range migration in the two-dimensional stepping stone model

2010-10-12

arxiv.org/abs/1010.2356v1

Annals of Applied Probability 2010, Vol. 20, No. 3, 785-805

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP639 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP639

We consider the stepping stone model on the torus of side $L$ in $\mathbb{Z}^2$ in the limit $L\to\infty$, and study the time it takes two lineages tracing backward in time to coalesce. Our work fills a gap between the finite range migration case of [Ann. Appl. Probab. 15 (2005) 671--699] and the long range case of [Genetics 172 (2006) 701--708], where the migration range is a positive fraction of $L$. We obtain limit theorems for the intermediate case, and verify a conjecture in [Probability Models for DNA Sequence Evolution (2008) Springer] that the model is homogeneously mixing if and only if the migration range is of larger order than $(\log L)^{1/2}$.

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