Intermediate range migration in the two-dimensional stepping stone model

J. Theodore Cox

doi:10.1214/09-AAP639

Intermediate range migration in the two-dimensional stepping stone model

J. Theodore Cox

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We consider the stepping stone model on the torus of side L in ℤ² 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.

Original language	English (US)
Pages (from-to)	785-805
Number of pages	21
Journal	Annals of Applied Probability
Volume	20
Issue number	3
DOIs	https://doi.org/10.1214/09-AAP639
State	Published - Jun 2010

Keywords

Coalescence times
Hitting times
Stepping stone model
Torus random walk

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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AB - We consider the stepping stone model on the torus of side L in ℤ2 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.

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KW - Torus random walk

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Intermediate range migration in the two-dimensional stepping stone model

Abstract

Keywords

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