{"paper":{"title":"Loop pruning and downward deviations for maximum local time of discrete-time simple random walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Loop pruning transfers the continuous-time lower bound to prove sharp asymptotics for downward deviations of discrete random walk local times.","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xinyi Li, Yushu Zheng","submitted_at":"2026-05-15T15:46:45Z","abstract_excerpt":"We study downward deviations of the maximum local time of the discrete-time simple random walk on $\\mathbb{Z}^d$, $d\\ge 3$. In our previous paper \\cite{li2026ldmaxlocal}, the corresponding upper bound was established, while the matching lower bound was left open. In the present paper, we prove this lower bound and hence obtain the sharp asymptotic formula for the downward-deviation probability. To provide a discrete-time analogue of the jump-chain/holding-time structure used in the continuous-time argument, we introduce a new random structure which we name as {\\it loop-pruned random walk} and "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove this lower bound and hence obtain the sharp asymptotic formula for the downward-deviation probability.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The loop-pruning decomposition and associated random structure accurately isolate the necessary path properties to transfer the continuous-time lower-bound argument to discrete time without introducing uncontrolled errors (introduced in the present paper to handle the discrete-time case).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The paper establishes the lower bound 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