{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RJEFWPNDAJ7KIML3CWMZHXENMC","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":"521b03eaa86c11dc8fe95731efdb928ab33dbd21d7b0c3fd96648e8ecafdd745","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-30T21:11:24Z","title_canon_sha256":"b274ce83c60bacc920665ca9b5865f7a0bafaebff1d66d325d9c2e02f0a542cb"},"schema_version":"1.0","source":{"id":"1201.6373","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.6373","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"arxiv_version","alias_value":"1201.6373v1","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.6373","created_at":"2026-05-18T04:03:31Z"},{"alias_kind":"pith_short_12","alias_value":"RJEFWPNDAJ7K","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RJEFWPNDAJ7KIML3","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RJEFWPND","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:09f827dffe28334b23cb5646db90f4517a2e36d02ad7279fffe562306975dfa0","target":"graph","created_at":"2026-05-18T04:03:31Z","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":"There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^{d-1} joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs.","authors_text":"Alexander E. Holroyd, James Martin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-30T21:11:24Z","title":"Stochastic Domination and Comb Percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.6373","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:ebb1456143629bafad317c0e246a58042522a91c2b443adcf66e6c2da9b7e136","target":"record","created_at":"2026-05-18T04:03:31Z","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":"521b03eaa86c11dc8fe95731efdb928ab33dbd21d7b0c3fd96648e8ecafdd745","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-30T21:11:24Z","title_canon_sha256":"b274ce83c60bacc920665ca9b5865f7a0bafaebff1d66d325d9c2e02f0a542cb"},"schema_version":"1.0","source":{"id":"1201.6373","kind":"arxiv","version":1}},"canonical_sha256":"8a485b3da3027ea4317b159993dc8d60b501ceb81107c9f39a592e7d68fc8df4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8a485b3da3027ea4317b159993dc8d60b501ceb81107c9f39a592e7d68fc8df4","first_computed_at":"2026-05-18T04:03:31.721291Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:31.721291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cDundStA04GiBzDXrSFxi6ghOcNKW5BRXuFUVYyPEDfM607NOgLsOAx6Jvp3lWmeH1CSmmacK7YukBAokr72Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:31.721979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.6373","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ebb1456143629bafad317c0e246a58042522a91c2b443adcf66e6c2da9b7e136","sha256:09f827dffe28334b23cb5646db90f4517a2e36d02ad7279fffe562306975dfa0"],"state_sha256":"bf50eed5dfef7ff3f4ef90fe055cc839f2ebd600cd2a0decf1d0ca65efc153bd"}