{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:G6UE4JIEBITQ3DSE2ERWOZOIJB","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":"a08ef8b4314ab284565d671f354f92f0228d6b9c2eeb64be2b04811797aa037a","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-26T17:42:26Z","title_canon_sha256":"f1722d239b986bdf1eb3c45e0ead36f7131842f8e621250291c8cdbef3101943"},"schema_version":"1.0","source":{"id":"1804.10191","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10191","created_at":"2026-05-17T23:50:28Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10191v3","created_at":"2026-05-17T23:50:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10191","created_at":"2026-05-17T23:50:28Z"},{"alias_kind":"pith_short_12","alias_value":"G6UE4JIEBITQ","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_16","alias_value":"G6UE4JIEBITQ3DSE","created_at":"2026-05-18T12:32:25Z"},{"alias_kind":"pith_short_8","alias_value":"G6UE4JIE","created_at":"2026-05-18T12:32:25Z"}],"graph_snapshots":[{"event_id":"sha256:61fe210529fb036b9ffd8cbbbdf1f1505c1f42a194a4eea069f017bb63970a3c","target":"graph","created_at":"2026-05-17T23:50:28Z","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":"We prove that Bernoulli bond percolation on any nonamenable, Gromov hyperbolic, quasi-transitive graph has a phase in which there are infinitely many infinite clusters, verifying a well-known conjecture of Benjamini and Schramm (1996) under the additional assumption of hyperbolicity. In other words, we show that $p_c<p_u$ for any such graph. Our proof also yields that the triangle condition $\\nabla_{p_c}<\\infty$ holds at criticality on any such graph, which is known to imply that several critical exponents exist and take their mean-field values. This gives the first family of examples of one-e","authors_text":"Tom Hutchcroft","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-26T17:42:26Z","title":"Percolation on hyperbolic graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10191","kind":"arxiv","version":3},"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:caad4a608b518cc80fc4c59037e9f9a5a4a02a64e1fbcfe531d5a5981f97c624","target":"record","created_at":"2026-05-17T23:50:28Z","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":"a08ef8b4314ab284565d671f354f92f0228d6b9c2eeb64be2b04811797aa037a","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-04-26T17:42:26Z","title_canon_sha256":"f1722d239b986bdf1eb3c45e0ead36f7131842f8e621250291c8cdbef3101943"},"schema_version":"1.0","source":{"id":"1804.10191","kind":"arxiv","version":3}},"canonical_sha256":"37a84e25040a270d8e44d1236765c84852487da5b5bcce5fa4285b94b9acad03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37a84e25040a270d8e44d1236765c84852487da5b5bcce5fa4285b94b9acad03","first_computed_at":"2026-05-17T23:50:28.090708Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:28.090708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GFOmWBDQEfOUAwIeZq1N5U8bG9HbC3LgFoj+hvHgOr3CjVEQ4asjTnXDPyC7SUasyFP7VHn/ONGNb9+IwtdXBg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:28.091111Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.10191","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:caad4a608b518cc80fc4c59037e9f9a5a4a02a64e1fbcfe531d5a5981f97c624","sha256:61fe210529fb036b9ffd8cbbbdf1f1505c1f42a194a4eea069f017bb63970a3c"],"state_sha256":"82a83a37c89c5cecb5c695bb7c6b90ceb952bc6d41a8c7cbcfc8ab0db3482394"}