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On the convergence of densities of finite voter models to the Wright-Fisher diffusion. (English. French summary) Zbl 1336.60184

Summary: 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 the density processes. We show by examples that these conditions are satisfied by a large class of voter models on growing finite graphs.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J60 Diffusion processes
60F05 Central limit and other weak theorems
82C22 Interacting particle systems in time-dependent statistical mechanics
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