The Annals of Applied Probability

Intermediate range migration in the two-dimensional stepping stone model

J. Theodore Cox

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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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Ann. Appl. Probab., Volume 20, Number 3 (2010), 785-805.

First available in Project Euclid: 18 June 2010

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G50: Sums of independent random variables; random walks 92D10: Genetics {For genetic algebras, see 17D92}
Secondary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Stepping stone model torus random walk hitting times coalescence times


Cox, J. Theodore. Intermediate range migration in the two-dimensional stepping stone model. Ann. Appl. Probab. 20 (2010), no. 3, 785--805. doi:10.1214/09-AAP639.

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