{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:P7V6M4BRSREZ4LWWIZFOHMLEE7","short_pith_number":"pith:P7V6M4BR","schema_version":"1.0","canonical_sha256":"7febe6703194499e2ed6464ae3b16427f5ba4d061cc36a04f5ada6541275331e","source":{"kind":"arxiv","id":"math/0703872","version":2},"attestation_state":"computed","paper":{"title":"Long Range Percolation Mixing Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.PR","authors_text":"Ariel Yadin, Itai Benjamini, Noam Berger","submitted_at":"2007-03-29T08:26:38Z","abstract_excerpt":"We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic factors)."},"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":"math/0703872","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2007-03-29T08:26:38Z","cross_cats_sorted":["math-ph","math.CO","math.MP"],"title_canon_sha256":"cc3216ea0d67535d208b2a4220014f58ddebe33b091fe07215fc3e96eac00440","abstract_canon_sha256":"c9b5288242992ff4a051a19ba348b49c21bbf0ba92770754b52f92539e022cd3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:41:21.742142Z","signature_b64":"LmnKianTuZKeLh5PNnXapi9ccdTOeZ2rkqK9fFsQMttsuc3ofYSdO2DhasVWgpHcxRGiyKGwXzTP5ZV+Ai5RAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7febe6703194499e2ed6464ae3b16427f5ba4d061cc36a04f5ada6541275331e","last_reissued_at":"2026-07-04T15:41:21.740819Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:41:21.740819Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Long Range Percolation Mixing Time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CO","math.MP"],"primary_cat":"math.PR","authors_text":"Ariel Yadin, Itai Benjamini, Noam Berger","submitted_at":"2007-03-29T08:26:38Z","abstract_excerpt":"We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic factors)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703872","kind":"arxiv","version":2},"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/math/0703872/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0703872","created_at":"2026-07-04T15:41:21.740877+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0703872v2","created_at":"2026-07-04T15:41:21.740877+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0703872","created_at":"2026-07-04T15:41:21.740877+00:00"},{"alias_kind":"pith_short_12","alias_value":"P7V6M4BRSREZ","created_at":"2026-07-04T15:41:21.740877+00:00"},{"alias_kind":"pith_short_16","alias_value":"P7V6M4BRSREZ4LWW","created_at":"2026-07-04T15:41:21.740877+00:00"},{"alias_kind":"pith_short_8","alias_value":"P7V6M4BR","created_at":"2026-07-04T15:41:21.740877+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/P7V6M4BRSREZ4LWWIZFOHMLEE7","json":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7.json","graph_json":"https://pith.science/api/pith-number/P7V6M4BRSREZ4LWWIZFOHMLEE7/graph.json","events_json":"https://pith.science/api/pith-number/P7V6M4BRSREZ4LWWIZFOHMLEE7/events.json","paper":"https://pith.science/paper/P7V6M4BR"},"agent_actions":{"view_html":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7","download_json":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7.json","view_paper":"https://pith.science/paper/P7V6M4BR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0703872&json=true","fetch_graph":"https://pith.science/api/pith-number/P7V6M4BRSREZ4LWWIZFOHMLEE7/graph.json","fetch_events":"https://pith.science/api/pith-number/P7V6M4BRSREZ4LWWIZFOHMLEE7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7/action/storage_attestation","attest_author":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7/action/author_attestation","sign_citation":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7/action/citation_signature","submit_replication":"https://pith.science/pith/P7V6M4BRSREZ4LWWIZFOHMLEE7/action/replication_record"}},"created_at":"2026-07-04T15:41:21.740877+00:00","updated_at":"2026-07-04T15:41:21.740877+00:00"}