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

We study voter models defined on large finite 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. We show by examples that these conditions are satisfied by a large class of voter models on growing finite graphs.