The Annals of Applied Probability

Genealogies in expanding populations

Rick Durrett and Wai-Tong (Louis) Fan

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Abstract

The goal of this paper is to prove rigorous results for the behavior of genealogies in a one-dimensional long range biased voter model introduced by Hallatschek and Nelson [Theor. Pop. Biol. 73 (2008) 158–170]. The first step, which is easily accomplished using results of Mueller and Tribe [Probab. Theory Related Fields 102 (1995) 519–545], is to show that when space and time are rescaled correctly, our biased voter model converges to a Wright–Fisher SPDE. A simple extension of a result of Durrett and Restrepo [Ann. Appl. Probab. 18 (2008) 334–358] then shows that the dual branching coalescing random walk converges to a branching Brownian motion in which particles coalesce after an exponentially distributed amount of intersection local time. Brunet et al. [Phys. Rev. E (3) 76 (2007) 041104, 20] have conjectured that genealogies in models of this type are described by the Bolthausen–Sznitman coalescent, see [Proc. Natl. Acad. Sci. USA 110 (2013) 437–442]. However, in the model we study there are no simultaneous coalescences. Our third and most significant result concerns “tracer dynamics” in which some of the initial particles in the biased voter model are labeled. We show that the joint distribution of the labeled and unlabeled particles converges to the solution of a system of stochastic partial differential equations. A new duality equation that generalizes the one Shiga [In Stochastic Processes in Physics and Engineering (Bielefeld, 1986) (1988) 345–355 Reidel] developed for the Wright–Fisher SPDE is the key to the proof of that result.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 6 (2016), 3456-3490.

Dates
Received: July 2015
Revised: January 2016
First available in Project Euclid: 15 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1481792590

Digital Object Identifier
doi:10.1214/16-AAP1181

Mathematical Reviews number (MathSciNet)
MR3582808

Zentralblatt MATH identifier
1358.92065

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 92C50: Medical applications (general)

Keywords
Biased voter model SPDE duality scaling limit genealogies

Citation

Durrett, Rick; Fan, Wai-Tong (Louis). Genealogies in expanding populations. Ann. Appl. Probab. 26 (2016), no. 6, 3456--3490. doi:10.1214/16-AAP1181. https://projecteuclid.org/euclid.aoap/1481792590


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