{"paper":{"title":"Uniformity of the late points of random walk on Z_n^d for d >= 3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Jason Miller, Perla Sousi","submitted_at":"2013-09-12T19:57:06Z","abstract_excerpt":"Suppose that $X$ is a simple random walk on $\\Z_n^d$ for $d \\geq 3$ and, for each $t$, we let $\\U(t)$ consist of those $x \\in \\Z_n^d$ which have not been visited by $X$ by time $t$. Let $\\tcov$ be the expected amount of time that it takes for $X$ to visit every site of $\\Z_n^d$. We show that there exists $0 < \\alpha_0(d) \\leq \\alpha_1(d) < 1$ and a time $t_* = \\tcov(1+o(1))$ as $n \\to \\infty$ such that the following is true. For $\\alpha > \\alpha_1(d)$ (resp.\\ $\\alpha < \\alpha_0(d)$), the total variation distance between the law of $\\U(\\alpha t_*)$ and the law of i.i.d.\\ Bernoulli random variab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3265","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}