Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$

We obtain sharp asymptotic probabilities for upward deviations of the maximum local time of discrete- and continuous-time simple random walks on $\mathbb{Z}^d$, $d\ge 3$.

For downward deviations, we prove the sharp continuous-time asymptotics and the discrete-time upper bound.

Together with the loop-pruning paper~\cite{li2026loopprune_inprep}, which proves the matching discrete-time lower bound via a loop-pruning construction, this yields the sharp downward-deviation asymptotics in discrete time as well.

We also derive Gumbel-type consequences at the logarithmic scale.