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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Abstract

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 Mν/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.

Article information

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

Dates
First available in Project Euclid: 9 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1199890025

Digital Object Identifier
doi:10.1214/07-AAP451

Mathematical Reviews number (MathSciNet)
MR2380901

Zentralblatt MATH identifier
1145.92024

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

Keywords
Stepping stone model voter model Brownian local time coalescent

Citation

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. https://projecteuclid.org/euclid.aoap/1199890025


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