On the convergence of densities of finite voter models to the Wright–Fisher diffusion

Abstract

We study voter models defined on large sets. Through a perspective emphasizing the martingale property of voter density processes, we prove that in general, their convergence to the Wright-Fisher diffusion only involves certain averages of the voter models over a small number of spatial locations. This enables us to identify suitable mixing conditions on the underlying voting kernels, one of which may just depend on their eigenvalues in some contexts, to obtain the convergence of density processes. Our examples show that these conditions are satisfied by a large class of voter models on growing finite graphs.2000 Mathematics Subject Classification. Primary: 60K35, 82C22, Secondary: 60F05, 60J60.

From Coalescing Random Walks on a Torus to Kingman’s Coalescent

“…It would certainly be interesting to extend the results presented here to the random graphs covered by Oliveira [14] or to non-reversible dynamics, but also to consider the dynamics which keeps track of the total number of particles which coalesced with each particle present at a given time. This later dynamics is related to a Wright-Fisher diffusion, already examined by Cox [6] and Chen, Choi and Cox [4].…”

“…The difference with respect to P N 0 is that the random walk is not speeded-up by 2 under P N 0 . An elementary random walk estimation yields that the right hand side multiplied by log N vanishes as N → ∞ if we choose S N = N 2 /(log N ) 4 . Hence, wit this definition for S N , for all 1 ≤ i ≤ n,…”

Wright–Fisher diffusions in stochastic spatial evolutionary games with death–birth updating

“…For these two approximations, the reader may recall the fact that the weak convergence of absorbing processes does not guarantee the weak convergence of their times to absorption in general. By proving a stronger tightness property of the Radon-Nikodym derivative processes at selection strengths of order O(1/N ), we show that Oliveira's result on the convergence in the Wasserstein distance of order 1 of times to absorption in [27,28] and the convergence of absorbing probabilities in [6] under voter models carry to the corresponding convergences under the evolutionary games. These are included in the second main result, Theorem 5.2, where the major theme is around convergences of occupation measures of the game density processes.…”

“…Since voting kernels are symmetric, the proof of 2 • ) can be obtained by the same argument as that of [6,Corollary 5.2]. It can be detailed as follows.…”

“…In this way, we can view the game density processes in terms of the voter models, even with the limit of large population size, if the corresponding Radon-Nikodym derivative processes satisfy appropriate tightness properties. Then the proof of Theorem 4.6 turns to and relies heavily on the main result in [6] that diffusion approximations of the voter density processes hold on large spatial structures where the underlying voting kernels are subject to appropriate, but mild, mixing conditions; an extension in [5] is used when mutation is present. In these cases, the limiting voter density processes are given by the Wright-Fisher diffusions, which originally arise from the Moran processes, namely, the voter models on complete graphs (cf.…”