Some Limit Theorems for Voter Model Occupation Times

Abstract

Let $\eta_t$ be the (basic) voter model on $\mathbb{Z}^d$. We consider the occupation time functionals $\int^t_0 f(\eta_s)ds$ for certain functions $f$ and initial distributions. The first result is a pointwise ergodic theorem in the case $d = 2$, extending the work of Andjel and Kipnis. The second result is a central limit type theorem for $f(\eta) = \eta(0)$ and initial distributions: (i) $\delta_\eta$, for a class of states $\eta, d \geq 2$, and (ii) $\nu_\theta$, the extremal invariant measures, $d \geq 3$.