Intermediate range migration in the two-dimensional stepping stone model

Abstract

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 $(\log L{)}^{1/2}$.

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

References in corpus (2)

• The stepping stone model. II: Genealogies and the infinite sites model
• One-dimensional stepping stone models, sardine genetics and Brownian local time