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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1309.3265","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-12T19:57:06Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"e49fb3e33e890fc4852368753c2a585e35eba8920f71ff56f258fe9f8a13c185","abstract_canon_sha256":"6a38788c73d2ee0269e0a873ee3b0df7e111fe59e8b0ad8673bc995b74f6e6dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:32.411435Z","signature_b64":"UZbI4JEZ6bIdlihrUsUbCKisXQubqBJEaZ3hJ3mwrcnBrDxp9cfdCGS4bsBDlLwOtz0ElbWhN/BKOnMALvBSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e33a0d828be0ca9bbf94d750b2b59e49742338003cb60321d63d4f43c6881fbf","last_reissued_at":"2026-05-18T03:13:32.410638Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:32.410638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1309.3265","created_at":"2026-05-18T03:13:32.410758+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.3265v1","created_at":"2026-05-18T03:13:32.410758+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3265","created_at":"2026-05-18T03:13:32.410758+00:00"},{"alias_kind":"pith_short_12","alias_value":"4M5A3AUL4DFJ","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4M5A3AUL4DFJXP4U","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4M5A3AUL","created_at":"2026-05-18T12:27:34.582898+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF","json":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF.json","graph_json":"https://pith.science/api/pith-number/4M5A3AUL4DFJXP4U25ILFNM6JF/graph.json","events_json":"https://pith.science/api/pith-number/4M5A3AUL4DFJXP4U25ILFNM6JF/events.json","paper":"https://pith.science/paper/4M5A3AUL"},"agent_actions":{"view_html":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF","download_json":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF.json","view_paper":"https://pith.science/paper/4M5A3AUL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.3265&json=true","fetch_graph":"https://pith.science/api/pith-number/4M5A3AUL4DFJXP4U25ILFNM6JF/graph.json","fetch_events":"https://pith.science/api/pith-number/4M5A3AUL4DFJXP4U25ILFNM6JF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF/action/storage_attestation","attest_author":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF/action/author_attestation","sign_citation":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF/action/citation_signature","submit_replication":"https://pith.science/pith/4M5A3AUL4DFJXP4U25ILFNM6JF/action/replication_record"}},"created_at":"2026-05-18T03:13:32.410758+00:00","updated_at":"2026-05-18T03:13:32.410758+00:00"}