Intermediate range migration in the two-dimensional stepping stone model

We consider the stepping stone model on the torus of side L in .2 in the limit L.., 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.