For d ≥ 3 and γ > (d+2)/(2d), the probability that the torus cover time is at most γ times its mean is exp(-(1+o(1)) N^{d(1-γ)}).
Cover times in the discrete cylinder
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abstract
This article proves that, in terms of local times, the rescaled and recentered cover times of finite subsets of the discrete cylinder by simple random walk converge in law to the Gumbel distribution, as the cardinality of the set goes to infinity. As applications we obtain several other results related to covering in the discrete cylinder. Our method is new and involves random interlacements, which were introduced by Sznitman in arXiv:0704.2560. To enable the proof we develop a new stronger coupling of simple random walk in the cylinder and random interlacements, which is also of independent interest.
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Large Deviations of Cover Time of Tori in Dimensions $d\geq 3$
For d ≥ 3 and γ > (d+2)/(2d), the probability that the torus cover time is at most γ times its mean is exp(-(1+o(1)) N^{d(1-γ)}).