Voter model in a random environment in ℤ^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 ℤ^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.

Keywords

Voter model / random walk / random environment / duality

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