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

Summary · qwen2.5:32b

The study provides precise asymptotic probabilities for upward and downward deviations of the maximum local time of simple random walks on $\mathbb{Z}^d$ for dimensions $d\ge 3$, both in discrete- and continuous-time settings, with Gumbel-type distributions identified at the logarithmic scale. The research matters as it completes the understanding of local time deviations through a loop-pruning construction that proves matching discrete-time lower bounds. A concrete detail is the derivation of sharp continuous-time asymptotics for downward deviations alongside establishing discrete-time upper bounds.

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arXiv:2604.10214v3 Announce Type: replace Abstract: 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.