{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:TGJWQS4FMTTOZ2GCREXA2YVUKA","short_pith_number":"pith:TGJWQS4F","canonical_record":{"source":{"id":"1708.06113","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-08-21T08:21:54Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"9eeff09420b74a8a12cb1d5c988f59a01678b3bd472ed8af66568a1e8d176b8e","abstract_canon_sha256":"901eba6d957b406d784472f0011337563891cfaf9e028d450a302cdadf90cb0d"},"schema_version":"1.0"},"canonical_sha256":"9993684b8564e6ece8c2892e0d62b4502203625ab6f6ceb904182977262aaaa7","source":{"kind":"arxiv","id":"1708.06113","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.06113","created_at":"2026-05-18T00:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"1708.06113v2","created_at":"2026-05-18T00:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.06113","created_at":"2026-05-18T00:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"TGJWQS4FMTTO","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"TGJWQS4FMTTOZ2GC","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"TGJWQS4F","created_at":"2026-05-18T12:31:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:TGJWQS4FMTTOZ2GCREXA2YVUKA","target":"record","payload":{"canonical_record":{"source":{"id":"1708.06113","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-08-21T08:21:54Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"9eeff09420b74a8a12cb1d5c988f59a01678b3bd472ed8af66568a1e8d176b8e","abstract_canon_sha256":"901eba6d957b406d784472f0011337563891cfaf9e028d450a302cdadf90cb0d"},"schema_version":"1.0"},"canonical_sha256":"9993684b8564e6ece8c2892e0d62b4502203625ab6f6ceb904182977262aaaa7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:00.436240Z","signature_b64":"7756/s6AWE0ujRuezGax7d7a2Me4THt2QrTwXZB+RtV+QdtdDmrp4Q0WsiYaKowvXO6pge7MMT17mLGMb1bcAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9993684b8564e6ece8c2892e0d62b4502203625ab6f6ceb904182977262aaaa7","last_reissued_at":"2026-05-18T00:05:00.435791Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:00.435791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.06113","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:05:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cpa6uhhwGBW78zQlXCAfXDpm0wFxQSwg10hJG/Ra9Je/8ePTleO3Nc//s/T7NR/YS1nRIWrVQXYZ7+psQc7vCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T11:29:58.278265Z"},"content_sha256":"d69bc25da73f7f492bbc4f2e6ddb19121bb813e1ee7d281af1165425f04cc58a","schema_version":"1.0","event_id":"sha256:d69bc25da73f7f492bbc4f2e6ddb19121bb813e1ee7d281af1165425f04cc58a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:TGJWQS4FMTTOZ2GCREXA2YVUKA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlev\\'e II system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Dan Dai, Shuai-Xia Xu","submitted_at":"2017-08-21T08:21:54Z","abstract_excerpt":"We study Fredholm determinants of the Painlev\\'e II and Painlev\\'e XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlev\\'e II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06113","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:05:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0+u403kQSyr1j4d5noLOH/rjEw+dMm+yJfcT+wSL8IUWEDX2/MZiypkd9U+E2h812/XLAepZrumL6eZbaGr5DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T11:29:58.278634Z"},"content_sha256":"46137345825839791b9ed8d05ea4ed4f1b7448ac124380f73ec1913c1111179b","schema_version":"1.0","event_id":"sha256:46137345825839791b9ed8d05ea4ed4f1b7448ac124380f73ec1913c1111179b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA/bundle.json","state_url":"https://pith.science/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T11:29:58Z","links":{"resolver":"https://pith.science/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA","bundle":"https://pith.science/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA/bundle.json","state":"https://pith.science/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TGJWQS4FMTTOZ2GCREXA2YVUKA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:TGJWQS4FMTTOZ2GCREXA2YVUKA","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":"901eba6d957b406d784472f0011337563891cfaf9e028d450a302cdadf90cb0d","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-08-21T08:21:54Z","title_canon_sha256":"9eeff09420b74a8a12cb1d5c988f59a01678b3bd472ed8af66568a1e8d176b8e"},"schema_version":"1.0","source":{"id":"1708.06113","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.06113","created_at":"2026-05-18T00:05:00Z"},{"alias_kind":"arxiv_version","alias_value":"1708.06113v2","created_at":"2026-05-18T00:05:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.06113","created_at":"2026-05-18T00:05:00Z"},{"alias_kind":"pith_short_12","alias_value":"TGJWQS4FMTTO","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"TGJWQS4FMTTOZ2GC","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"TGJWQS4F","created_at":"2026-05-18T12:31:46Z"}],"graph_snapshots":[{"event_id":"sha256:46137345825839791b9ed8d05ea4ed4f1b7448ac124380f73ec1913c1111179b","target":"graph","created_at":"2026-05-18T00:05:00Z","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 study Fredholm determinants of the Painlev\\'e II and Painlev\\'e XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlev\\'e II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.","authors_text":"Dan Dai, Shuai-Xia Xu","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-08-21T08:21:54Z","title":"Tracy-Widom distributions in critical unitary random matrix ensembles and the coupled Painlev\\'e II system"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.06113","kind":"arxiv","version":2},"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:d69bc25da73f7f492bbc4f2e6ddb19121bb813e1ee7d281af1165425f04cc58a","target":"record","created_at":"2026-05-18T00:05:00Z","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":"901eba6d957b406d784472f0011337563891cfaf9e028d450a302cdadf90cb0d","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-08-21T08:21:54Z","title_canon_sha256":"9eeff09420b74a8a12cb1d5c988f59a01678b3bd472ed8af66568a1e8d176b8e"},"schema_version":"1.0","source":{"id":"1708.06113","kind":"arxiv","version":2}},"canonical_sha256":"9993684b8564e6ece8c2892e0d62b4502203625ab6f6ceb904182977262aaaa7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9993684b8564e6ece8c2892e0d62b4502203625ab6f6ceb904182977262aaaa7","first_computed_at":"2026-05-18T00:05:00.435791Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:00.435791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7756/s6AWE0ujRuezGax7d7a2Me4THt2QrTwXZB+RtV+QdtdDmrp4Q0WsiYaKowvXO6pge7MMT17mLGMb1bcAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:00.436240Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.06113","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d69bc25da73f7f492bbc4f2e6ddb19121bb813e1ee7d281af1165425f04cc58a","sha256:46137345825839791b9ed8d05ea4ed4f1b7448ac124380f73ec1913c1111179b"],"state_sha256":"66c91a66d3cb20ac95bf8c984f1efc4cc5826f3420aaafd9497180a4ea6583b0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8CBSEwy/qeZjSby3lgshDCv3ZKCuPewUaFs+1X/5DTIhgcos2XwYWdQrEOGo+kuL+G+dcYIIkHNL/NZeEpQVDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T11:29:58.280711Z","bundle_sha256":"c19f780f183013cf6eee551c240188a2dfb0560877160aa97b24039ff21dad1e"}}