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

TL;DR

This paper investigates how voter models on large finite sets converge to the Wright-Fisher diffusion, identifying key mixing conditions that ensure this convergence through a martingale perspective.

Contribution

It introduces a new approach emphasizing the martingale property to establish convergence, and identifies mixing conditions based on eigenvalues for voter models on growing graphs.

Findings

Convergence of voter densities to Wright-Fisher diffusion under certain conditions

Mixing conditions related to eigenvalues ensure convergence

Applicable to a broad class of voter models on growing graphs

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.