We study the decay rate of large deviation probabilities of occupation times, up to time t, for the voter model η: ℤ2×[0, ∞)→{0, 1} with simple random walk transition kernel, starting from a Bernoulli product distribution with density ρ∈(0, 1). In [Probab. Theory Related Fields 77 (1988) 401–413], Bramson, Cox and Griffeath showed that the decay rate order lies in [log(t), log2(t)].
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In this paper, we establish the true decay rates depending on the level. We show that the decay rates are log2(t) when the deviation from ρ is maximal (i.e., η≡0 or 1), and log(t) in all other situations. This answers some conjectures in [Probab. Theory Related Fields 77 (1988) 401–413] and confirms nonrigorous analysis carried out in [Phys. Rev. E 53 (1996) 3078–3087], [J. Phys. A 31 (1998) 5413–5429] and [J. Phys. A 31 (1998) L209–L215].