{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:TJGWI2D5MWKOHB7L2TDJAD4DLE","short_pith_number":"pith:TJGWI2D5","canonical_record":{"source":{"id":"1401.2572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-01-11T22:21:27Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"615ce4fc513e80f948b1c45d91a070b1f88d164b395ae90f722a2acc7e58959f","abstract_canon_sha256":"dab3d2db216a831492ce2585b2c219185f90032adad6f98a68a33bd8e4a0c286"},"schema_version":"1.0"},"canonical_sha256":"9a4d64687d6594e387ebd4c6900f83591d3fec46332a42aa22320c24bd825331","source":{"kind":"arxiv","id":"1401.2572","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.2572","created_at":"2026-05-18T01:45:26Z"},{"alias_kind":"arxiv_version","alias_value":"1401.2572v2","created_at":"2026-05-18T01:45:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.2572","created_at":"2026-05-18T01:45:26Z"},{"alias_kind":"pith_short_12","alias_value":"TJGWI2D5MWKO","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TJGWI2D5MWKOHB7L","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TJGWI2D5","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:TJGWI2D5MWKOHB7L2TDJAD4DLE","target":"record","payload":{"canonical_record":{"source":{"id":"1401.2572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-01-11T22:21:27Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"615ce4fc513e80f948b1c45d91a070b1f88d164b395ae90f722a2acc7e58959f","abstract_canon_sha256":"dab3d2db216a831492ce2585b2c219185f90032adad6f98a68a33bd8e4a0c286"},"schema_version":"1.0"},"canonical_sha256":"9a4d64687d6594e387ebd4c6900f83591d3fec46332a42aa22320c24bd825331","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:45:26.438706Z","signature_b64":"RPjVJyk1ll015KA7OLJHmNLFkAWk1KJRj/Dpp4IB0RXwnPpyS794LnjKwVHh0LHdPuKsm05VcZiCDnFkk9BxDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a4d64687d6594e387ebd4c6900f83591d3fec46332a42aa22320c24bd825331","last_reissued_at":"2026-05-18T01:45:26.438136Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:45:26.438136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.2572","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":"2026-05-18T01:45:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T/AtbDff/vwrjiEIyGO7dx0X3umkSGgU5xmH6olGCodhAD8rHHrnyaRroTzxdYnrH6aiVmPNJvRTfhOoYAuTDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T20:26:12.460670Z"},"content_sha256":"0b71bc76c8f8ece7917bb876c7722933cfe8bcb1dde9813088e239bb67938db9","schema_version":"1.0","event_id":"sha256:0b71bc76c8f8ece7917bb876c7722933cfe8bcb1dde9813088e239bb67938db9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:TJGWI2D5MWKOHB7L2TDJAD4DLE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Eigenvalue statistics for product complex Wishart matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Peter J. Forrester","submitted_at":"2014-01-11T22:21:27Z","abstract_excerpt":"The eigenvalue statistics for complex $N \\times N$ Wishart matrices $X_{r,s}^\\dagger X_{r,s}$, where $ X_{r,s}$ is equal to the product of $r$ complex Gaussian matrices, and the inverse of $s$ complex Gaussian matrices, are considered. In the case $r=s$ the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function ${}_{s+1} F_r$. For finite $N$ the eigenvalue probability density function is computed, and is shown to be an example of a biorth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2572","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":"2026-05-18T01:45:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mDUjT51Ljtay57ucehli5RrKlM/uEQbCVL8n7X1MeR6/YvyXKIvngYl80dtjGiWvaETev7i4kr3r0QRGRCvtDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T20:26:12.460967Z"},"content_sha256":"d3f1388ab67abeccbf3701f3b1d1ae10d1c2400569459ed7e689794f9fb52495","schema_version":"1.0","event_id":"sha256:d3f1388ab67abeccbf3701f3b1d1ae10d1c2400569459ed7e689794f9fb52495"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE/bundle.json","state_url":"https://pith.science/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE/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":"2026-08-22T20:26:12Z","links":{"resolver":"https://pith.science/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE","bundle":"https://pith.science/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE/bundle.json","state":"https://pith.science/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TJGWI2D5MWKOHB7L2TDJAD4DLE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:TJGWI2D5MWKOHB7L2TDJAD4DLE","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":"dab3d2db216a831492ce2585b2c219185f90032adad6f98a68a33bd8e4a0c286","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-01-11T22:21:27Z","title_canon_sha256":"615ce4fc513e80f948b1c45d91a070b1f88d164b395ae90f722a2acc7e58959f"},"schema_version":"1.0","source":{"id":"1401.2572","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.2572","created_at":"2026-05-18T01:45:26Z"},{"alias_kind":"arxiv_version","alias_value":"1401.2572v2","created_at":"2026-05-18T01:45:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.2572","created_at":"2026-05-18T01:45:26Z"},{"alias_kind":"pith_short_12","alias_value":"TJGWI2D5MWKO","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TJGWI2D5MWKOHB7L","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TJGWI2D5","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:d3f1388ab67abeccbf3701f3b1d1ae10d1c2400569459ed7e689794f9fb52495","target":"graph","created_at":"2026-05-18T01:45:26Z","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 eigenvalue statistics for complex $N \\times N$ Wishart matrices $X_{r,s}^\\dagger X_{r,s}$, where $ X_{r,s}$ is equal to the product of $r$ complex Gaussian matrices, and the inverse of $s$ complex Gaussian matrices, are considered. In the case $r=s$ the exact form of the global density is computed. The averaged characteristic polynomial for the corresponding generalized eigenvalue problem is calculated in terms of a particular generalized hypergeometric function ${}_{s+1} F_r$. For finite $N$ the eigenvalue probability density function is computed, and is shown to be an example of a biorth","authors_text":"Peter J. Forrester","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-01-11T22:21:27Z","title":"Eigenvalue statistics for product complex Wishart matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2572","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":"sha256:0b71bc76c8f8ece7917bb876c7722933cfe8bcb1dde9813088e239bb67938db9","target":"record","created_at":"2026-05-18T01:45:26Z","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":"dab3d2db216a831492ce2585b2c219185f90032adad6f98a68a33bd8e4a0c286","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2014-01-11T22:21:27Z","title_canon_sha256":"615ce4fc513e80f948b1c45d91a070b1f88d164b395ae90f722a2acc7e58959f"},"schema_version":"1.0","source":{"id":"1401.2572","kind":"arxiv","version":2}},"canonical_sha256":"9a4d64687d6594e387ebd4c6900f83591d3fec46332a42aa22320c24bd825331","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a4d64687d6594e387ebd4c6900f83591d3fec46332a42aa22320c24bd825331","first_computed_at":"2026-05-18T01:45:26.438136Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:45:26.438136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RPjVJyk1ll015KA7OLJHmNLFkAWk1KJRj/Dpp4IB0RXwnPpyS794LnjKwVHh0LHdPuKsm05VcZiCDnFkk9BxDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:45:26.438706Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.2572","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b71bc76c8f8ece7917bb876c7722933cfe8bcb1dde9813088e239bb67938db9","sha256:d3f1388ab67abeccbf3701f3b1d1ae10d1c2400569459ed7e689794f9fb52495"],"state_sha256":"2a4bfc2c82e0dbcef33cfd324f1c38a991f6aec8d068a85271aaf0a80540dca7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CqSnOOZDOZWraH3GtoN8YWKb8q+/JwlPHi2HZx/dNjS7pGOgDEwGI/ykzHQUVRllSsOLqt2i2upeHOf05RL/DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T20:26:12.464791Z","bundle_sha256":"af54bcbf9b64e35f01d05267299cc5e7d3064b9c58bfcba1758ff53ee6c1b3fb"}}