Intermediate range migration in the two-dimensional stepping stone model

TL;DR

This paper analyzes the coalescence times of lineages in a two-dimensional stepping stone model with intermediate migration range, bridging the gap between finite and long-range migration cases, and confirms a conjecture about mixing behavior.

Contribution

It provides limit theorems for coalescence times in the intermediate migration range and verifies a conjecture on the conditions for homogeneous mixing.

Findings

Coalescence times follow specific limit distributions in the intermediate range.

Homogeneous mixing occurs if and only if migration range exceeds a threshold of (log L)^{1/2}.

The model bridges finite and long-range migration cases with new asymptotic results.

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