{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:NIJW5N4P245ASYMV74XPHERDI2","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":"47e29040f4d7477625fe7d553aca6e17d38ddeb3ae0d5b6adfbd36d8dab0bc76","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-05-15T18:57:16Z","title_canon_sha256":"8be4ff029133e7d19455a53600f8c28e5ec3c335af841f7049b3c61c3ca4289d"},"schema_version":"1.0","source":{"id":"1505.04159","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.04159","created_at":"2026-05-18T01:00:40Z"},{"alias_kind":"arxiv_version","alias_value":"1505.04159v1","created_at":"2026-05-18T01:00:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.04159","created_at":"2026-05-18T01:00:40Z"},{"alias_kind":"pith_short_12","alias_value":"NIJW5N4P245A","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_16","alias_value":"NIJW5N4P245ASYMV","created_at":"2026-05-18T12:29:32Z"},{"alias_kind":"pith_short_8","alias_value":"NIJW5N4P","created_at":"2026-05-18T12:29:32Z"}],"graph_snapshots":[{"event_id":"sha256:eec69b28e7b50da756d8f3a2d0f07118abf826d9cabfd9f53fdda492762c642b","target":"graph","created_at":"2026-05-18T01:00:40Z","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":"This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\\mathbb Z^2$ is continuous for $q\\in\\{2,3,4\\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients:\n  1. The fact that the two-point function for the free stat","authors_text":"Hugo Duminil-Copin, Vincent Tassion, Vladas Sidoravicius","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-05-15T18:57:16Z","title":"Continuity of the phase transition for planar random-cluster and Potts models with $1\\le q\\le4$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.04159","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":"sha256:337b5ea05f3edc4042b94336e6edda348ee57228eae8adf9988e5a91f457518c","target":"record","created_at":"2026-05-18T01:00:40Z","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":"47e29040f4d7477625fe7d553aca6e17d38ddeb3ae0d5b6adfbd36d8dab0bc76","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-05-15T18:57:16Z","title_canon_sha256":"8be4ff029133e7d19455a53600f8c28e5ec3c335af841f7049b3c61c3ca4289d"},"schema_version":"1.0","source":{"id":"1505.04159","kind":"arxiv","version":1}},"canonical_sha256":"6a136eb78fd73a096195ff2ef39223468ea8b525d3c0b147a70839c23bdeb9bf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a136eb78fd73a096195ff2ef39223468ea8b525d3c0b147a70839c23bdeb9bf","first_computed_at":"2026-05-18T01:00:40.642099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:00:40.642099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nsjpRYrtSa/M2f220ZRyqjrt67RdhPLi3QBp9ZUyfdD2ETgoxRAgJ8E7AY2rJ6Q8Uhdrk9569dZkziNmQCtvCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:00:40.642526Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.04159","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:337b5ea05f3edc4042b94336e6edda348ee57228eae8adf9988e5a91f457518c","sha256:eec69b28e7b50da756d8f3a2d0f07118abf826d9cabfd9f53fdda492762c642b"],"state_sha256":"76427a763c60686ddb9e519dab2172410e7ed95ed129c5b886ddde645251057a"}