{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:IF7373AD5URJXYF7TWE7LXDLHA","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":"e832f452909ad2589b99a10caf4cf50e2c02d75160f9ecd9a7470b5bd463bbc5","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.PR","submitted_at":"2007-01-14T19:14:02Z","title_canon_sha256":"546fe5bddb3401db17d4048c8f8bd0954d42db6768817ad4754d7dc9f45f6485"},"schema_version":"1.0","source":{"id":"math/0701390","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0701390","created_at":"2026-07-04T14:58:44Z"},{"alias_kind":"arxiv_version","alias_value":"math/0701390v1","created_at":"2026-07-04T14:58:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701390","created_at":"2026-07-04T14:58:44Z"},{"alias_kind":"pith_short_12","alias_value":"IF7373AD5URJ","created_at":"2026-07-04T14:58:44Z"},{"alias_kind":"pith_short_16","alias_value":"IF7373AD5URJXYF7","created_at":"2026-07-04T14:58:44Z"},{"alias_kind":"pith_short_8","alias_value":"IF7373AD","created_at":"2026-07-04T14:58:44Z"}],"graph_snapshots":[{"event_id":"sha256:fa87f7609d928e727ce8521efc45ae0949706137553e2c4be5e78f6393a9dab2","target":"graph","created_at":"2026-07-04T14:58:44Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0701390/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the problem of generating a sample from the stationary distribution of a Markov chain, given a method to simulate the chain. We give an approximation algorithm for the case of a random walk on a regular graph with n vertices that runs in expected time O^*(\\sqrt{n} x L^2-mixing time). This is close to the best possible, since \\sqrt{n} is a lower bound on the worst-case expected running time of any algorithm.","authors_text":"Ben Morris, Itai Benjamini","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2007-01-14T19:14:02Z","title":"The birthday problem and Markov chain Monte Carlo"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701390","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":"sha256:93174bee271a5a9341b4bfc383022bfca644d1bf9702acfb60b076c3fbd1f0db","target":"record","created_at":"2026-07-04T14:58:44Z","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":"e832f452909ad2589b99a10caf4cf50e2c02d75160f9ecd9a7470b5bd463bbc5","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.PR","submitted_at":"2007-01-14T19:14:02Z","title_canon_sha256":"546fe5bddb3401db17d4048c8f8bd0954d42db6768817ad4754d7dc9f45f6485"},"schema_version":"1.0","source":{"id":"math/0701390","kind":"arxiv","version":1}},"canonical_sha256":"417fbfec03ed229be0bf9d89f5dc6b383b896d180066e4a9df5fd1a63d5cea77","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"417fbfec03ed229be0bf9d89f5dc6b383b896d180066e4a9df5fd1a63d5cea77","first_computed_at":"2026-07-04T14:58:44.471951Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:58:44.471951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RvSkwpPqyyvPxY8wNPzN//E6C+ShmKuET683NhQN+RK93zeS0EAKbumW+jYddN5KgZn1c6pMIU6/Yq0NxTLYAw==","signature_status":"signed_v1","signed_at":"2026-07-04T14:58:44.472299Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0701390","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:93174bee271a5a9341b4bfc383022bfca644d1bf9702acfb60b076c3fbd1f0db","sha256:fa87f7609d928e727ce8521efc45ae0949706137553e2c4be5e78f6393a9dab2"],"state_sha256":"6298d47373ff069a87e349ed5227b9a689cb2ba6ad772303e3bccb8818fa631c"}