Large deviations for voter model occupation times in two dimensions

Abstract

We study the decay rate of large deviation probabilities of occupation times, up to time $t$, for the voter model $η\colon\Z^2\times[0,\infty)\ra\{0,1\}$ with simple random walk transition kernel, starting from a Bernoulli product distribution with density $\rho \in (0,1)$. Bramson, Cox and Griffeath (1988) showed that the decay rate order lies in $[\log (t),{\log }^2(t)]$. In this paper, we establish the true decay rates depending on the level. We show that the decay rates are ${\log }^2(t)$ when the deviation from $\rho$ is maximal (i.e., $\eta \equiv 0$ or 1), and $\log (t)$ in all other situations.

14 pages