Large deviations for voter model occupation times in two dimensions
Abstract
Description
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 $ρ\in(0,1)$. 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 $ρ$ is maximal (i.e., $η\equiv 0$ or 1), and $\log(t)$ in all other situations.
14 pages
14 pages