{"paper":{"title":"Chen--Stein Method for the Uncovered Set of Random Walk on $\\mathbb Z_n^d$ for $d \\ge 3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Perla Sousi, Sam Olesker-Taylor","submitted_at":"2019-11-13T16:20:21Z","abstract_excerpt":"Let $X$ be a simple random walk on $\\mathbb{Z}_n^d$ with $d\\geq 3$ and let $t_{\\rm{cov}}$ be the expected cover time. We consider the set of points $\\mathcal{U}_\\alpha$ of $\\mathbb{Z}_n^d$ that have not been visited by the walk by time $\\alpha t_{\\rm{cov}}$ for $\\alpha\\in (0,1)$. It was shown in [MS17] that there exists $\\alpha_1(d)\\in (0,1)$ such that for all $\\alpha>\\alpha_1(d)$ the total variation distance between the law of the set $\\mathcal{U}_\\alpha$ and an i.i.d. sequence of Bernoulli random variables indexed by $\\mathbb{Z}_n^d$ with success probability $n^{-\\alpha d}$ tends to $0$ as $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.05581","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1911.05581/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}