{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Z5G26ZLEOMQJDBJZBTWKGLL3WD","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":"31013de7549a1ee6c5b81936b4c08f3594af090dab77949fd8cbc02c9c79da44","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-06T08:44:26Z","title_canon_sha256":"200d5364b2a25534a03636aa06688de84eac278bc9db81ee2201786ff4cca0d8"},"schema_version":"1.0","source":{"id":"1507.01356","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.01356","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"arxiv_version","alias_value":"1507.01356v1","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.01356","created_at":"2026-05-18T01:37:17Z"},{"alias_kind":"pith_short_12","alias_value":"Z5G26ZLEOMQJ","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"Z5G26ZLEOMQJDBJZ","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"Z5G26ZLE","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:fcc061fdb4ef6843332401ed1247b0233b33a1013375c2bdec0a684b9a50b611","target":"graph","created_at":"2026-05-18T01:37:17Z","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":"The critical surface for random-cluster model with cluster-weight $q\\ge 4$ on isoradial graphs is identified using parafermionic observables. Correlations are also shown to decay exponentially fast in the subcritical regime. While this result is restricted to random-cluster models with $q\\ge 4$, it extends the recent theorem of the two first authors to a large class of planar graphs. In particular, the anisotropic random-cluster model on the square lattice is shown to be critical if $\\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and $p_h$ denote the horizontal and vertical edge-weights respect","authors_text":"Hugo Duminil-Copin, Stanislav Smirnov, Vincent Beffara","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-06T08:44:26Z","title":"On the critical parameters of the $q\\ge4$ random-cluster model on isoradial graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01356","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:ab4a469525221e7b28f572b547976c09e9461d23e0fb7f74c9a6e8388f4ceeca","target":"record","created_at":"2026-05-18T01:37:17Z","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":"31013de7549a1ee6c5b81936b4c08f3594af090dab77949fd8cbc02c9c79da44","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-07-06T08:44:26Z","title_canon_sha256":"200d5364b2a25534a03636aa06688de84eac278bc9db81ee2201786ff4cca0d8"},"schema_version":"1.0","source":{"id":"1507.01356","kind":"arxiv","version":1}},"canonical_sha256":"cf4daf656473209185390ceca32d7bb0f97e5b607d8eb598fda635e589388069","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf4daf656473209185390ceca32d7bb0f97e5b607d8eb598fda635e589388069","first_computed_at":"2026-05-18T01:37:17.207858Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:17.207858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cLtiKoV7nvtfUIPK5e4Pcf2vq1lz5ZUYKdsibYMJ3qNOyA4m9RuqVzjRym8nvofOrQ8YTVqIYAfnZqB4XbrkAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:17.208620Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.01356","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ab4a469525221e7b28f572b547976c09e9461d23e0fb7f74c9a6e8388f4ceeca","sha256:fcc061fdb4ef6843332401ed1247b0233b33a1013375c2bdec0a684b9a50b611"],"state_sha256":"949fa7b2cdbfe530a7efb43e2a72b94f8ca655d5a001112e962f6e68b59dde60"}