The Complete Convergence Theorem for Coexistent Threshold Voter Models

Abstract

We consider the $d$-dimensional threshold voter model. It is known that, except in the one-dimensional nearest-neighbor case, coexistence occurs (nontrivial invariant measures exist). In fact, there is a nontrivial limit $\eta_\infty^{1/2}$ obtained by starting from the product measure with density 1/2. We show that in these coexistent cases, $$\eta_t \Rightarrow \alpha\delta_0 + \beta\delta_1 + (1 - \alpha - \beta)\eta^{1/2}_\infty \quad \text{as } t \to \infty,$$ where $\alpha=P(\tau_0<\infty), \beta=P(\tau_1<\infty), \tau_0$ and $\tau_1$ are the first hitting times of the all-zero and all-one configurations, respectively, and $\Rightarrow$ denotes weak convergence.