Voter model in a random environment in {\mathrm{\mathbb{Z}}}^{\mathit{d}}

Abstract

We consider the voter model with flip rates determined by {μ_e, e ∈ E_d}, where E_d is the set of all non-oriented nearest-neighbour edges in the Euclidean lattice {\mathrm{\mathbb{Z}}}^{\mathit{d}}. Suppose that {μ_e, e ∈ E_d} are independent and identically distributed (i.i.d.) random variables satisfying μ_e≥1. We prove that when d = 2, almost surely for all random environments, the voter model has only two extremal invariant measures: δ₀ and δ₁.

Key words： Voter model; random walk; random environment; duality

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