{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4M5A3AUL4DFJXP4U25ILFNM6JF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6a38788c73d2ee0269e0a873ee3b0df7e111fe59e8b0ad8673bc995b74f6e6dd","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-12T19:57:06Z","title_canon_sha256":"e49fb3e33e890fc4852368753c2a585e35eba8920f71ff56f258fe9f8a13c185"},"schema_version":"1.0","source":{"id":"1309.3265","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.3265","created_at":"2026-05-18T03:13:32Z"},{"alias_kind":"arxiv_version","alias_value":"1309.3265v1","created_at":"2026-05-18T03:13:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.3265","created_at":"2026-05-18T03:13:32Z"},{"alias_kind":"pith_short_12","alias_value":"4M5A3AUL4DFJ","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4M5A3AUL4DFJXP4U","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4M5A3AUL","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:222a0ce359da707781372a92bdd7a12deef4d329b43318c6a2c98a2c354d984c","target":"graph","created_at":"2026-05-18T03:13:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Jason Miller, Perla Sousi","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-12T19:57:06Z","title":"Uniformity of the late points of random walk on Z_n^d for d >= 3"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3265","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3c3e01d2b506ca852177be582c8d1079edecd658b808826defb8cfe9610bd23a","target":"record","created_at":"2026-05-18T03:13:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6a38788c73d2ee0269e0a873ee3b0df7e111fe59e8b0ad8673bc995b74f6e6dd","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-12T19:57:06Z","title_canon_sha256":"e49fb3e33e890fc4852368753c2a585e35eba8920f71ff56f258fe9f8a13c185"},"schema_version":"1.0","source":{"id":"1309.3265","kind":"arxiv","version":1}},"canonical_sha256":"e33a0d828be0ca9bbf94d750b2b59e49742338003cb60321d63d4f43c6881fbf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e33a0d828be0ca9bbf94d750b2b59e49742338003cb60321d63d4f43c6881fbf","first_computed_at":"2026-05-18T03:13:32.410638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:32.410638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UZbI4JEZ6bIdlihrUsUbCKisXQubqBJEaZ3hJ3mwrcnBrDxp9cfdCGS4bsBDlLwOtz0ElbWhN/BKOnMALvBSBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:32.411435Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.3265","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3c3e01d2b506ca852177be582c8d1079edecd658b808826defb8cfe9610bd23a","sha256:222a0ce359da707781372a92bdd7a12deef4d329b43318c6a2c98a2c354d984c"],"state_sha256":"a7d538b773807a3a6a71f97f95ff07e5c428aae6f6d23d3171e92362ce9c4e5b"}