Decay rates large deviations for the planar voter model occupation time

We study the decay rate of large deviation probabilities of occupation times, up to time t, for the voter model η : Z × [0,∞) → {0, 1} with simple random walk transition kernel, starting from a Bernoulli product distribution with density ρ ∈ (0, 1). In [2], Bramson, Cox and Griffeath showed that the decay rate order lies in [log(t), log(t)]. In this paper, we establish the true decay rates depending on the level. We show that the decay rates are log(t) when the deviation from ρ is maximal (i.e., η ≡ 0 or 1), and log(t) in all other situations. This answers some conjecture in [2] and confirms analysis carried out in [1], [7] and [12]. MSC 2000. Primary 60F10, 60K35; Secondary 60J25.