The Annals of Applied Probability

One-dimensional stepping stone models, sardine genetics and Brownian local time

Richard Durrett and Mateo Restrepo

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Consider a one-dimensional stepping stone model with colonies of size M and per-generation migration probability ν, or a voter model on ℤ in which interactions occur over a distance of order K. Sample one individual at the origin and one at L. We show that if /L and L/K2 converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0.

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Ann. Appl. Probab., Volume 18, Number 1 (2008), 334-358.

First available in Project Euclid: 9 January 2008

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D10: Genetics {For genetic algebras, see 17D92}

Stepping stone model voter model Brownian local time coalescent


Durrett, Richard; Restrepo, Mateo. One-dimensional stepping stone models, sardine genetics and Brownian local time. Ann. Appl. Probab. 18 (2008), no. 1, 334--358. doi:10.1214/07-AAP451.

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