Large Deviations of Cover Time of Tori in Dimensions $d\ge 3$

We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^d$, $d\ge 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $\gamma \in (0,1)$ times its expected value, with exponents matching the upper bound from [Goodman-den Hollander, Probab. Theory Related Fields (2014)] and [Comets-Gallesco-Popov-Vachkovskaia, Electron. J. Probab. (2013)]. Moreover, we derive sharp asymptotics for $\gamma \in (\frac{d+2}{2d},1)$. The strong coupling of the random walk on the torus and random interlacements developed in a recent work [Prévost-Rodriguez-Sousi, arXiv:2309.03192] serves as an important ingredient in the proofs.