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Mean Field Asymptotics for the Planar Stepping Stone Model
Author(s) -
Cox J. T.,
Griffeath David
Publication year - 1990
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-61.1.189
Subject(s) - mathematics , coalescent theory , lattice (music) , partition (number theory) , random walk , stepping stone , planar , combinatorics , discrete mathematics , pure mathematics , computer graphics (images) , computer science , economics , unemployment , economic growth , biochemistry , chemistry , physics , statistics , acoustics , gene , phylogenetic tree
Let ξ t denote coalescing random walks, ζ t the stepping stone model, on the two‐dimensional integer lattice Z 2 . 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 ζ t . Theorem 3 , our main result, says that appropriately scaled block density processes derived from ζ t 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 Z 2 has 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].