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)}).
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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)}).