TY - JOUR

T1 - Mean field asymptotics for the planar stepping stone model

AU - Cox, J. T.

N1 - Funding Information:
The research of the first author was supported in part by NSF Grant DMS-86-01713, and that of the second author by NSF Grant DMS-87-0083. A.M.S. (1980) subject classification: 60K35.

PY - 1990/7

Y1 - 1990/7

N2 - Let ξ, denote coalescing random walks, ς, the stepping stone model, on the two-dimensional integer lattice ℤ2. We prove four theorems, announced previously in [4] and [5], concerning these evolutions. Theorem 1 states that if we spread out the initial locations of m coalescing walks, then, after appropriate rescaling, the partition of (1, 2,., m) that records which sets of particles have coalesced by time s converges to Kingman's coalescent π(s) [10]. Various duality equations then lead to results for the stepping stone model. Theorem 2 identifies an exchangeable limit distribution for power-law thinnings of ς,. Theorem 3, our main result, says that appropriately scaled block density processes derived from ς, converge to a time change of the Wright-Fisher diffusion. We treat stepping stone models with finitely many possible types per site, and also the case in which every site of ℤ2has a different type initially. Finally, Theorem 4 describes the asymptotic number of types represented in a large box centred at the origin, generalizing some results from [1].

AB - Let ξ, denote coalescing random walks, ς, the stepping stone model, on the two-dimensional integer lattice ℤ2. We prove four theorems, announced previously in [4] and [5], concerning these evolutions. Theorem 1 states that if we spread out the initial locations of m coalescing walks, then, after appropriate rescaling, the partition of (1, 2,., m) that records which sets of particles have coalesced by time s converges to Kingman's coalescent π(s) [10]. Various duality equations then lead to results for the stepping stone model. Theorem 2 identifies an exchangeable limit distribution for power-law thinnings of ς,. Theorem 3, our main result, says that appropriately scaled block density processes derived from ς, converge to a time change of the Wright-Fisher diffusion. We treat stepping stone models with finitely many possible types per site, and also the case in which every site of ℤ2has a different type initially. Finally, Theorem 4 describes the asymptotic number of types represented in a large box centred at the origin, generalizing some results from [1].

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U2 - 10.1112/plms/s3-61.1.189

DO - 10.1112/plms/s3-61.1.189

M3 - Article

AN - SCOPUS:0040594934

SN - 0024-6115

VL - s3-61

SP - 189

EP - 208

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 1

ER -