{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:H74TKA22NEP5LG5T4XI32PWFYI","short_pith_number":"pith:H74TKA22","canonical_record":{"source":{"id":"0912.3422","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-12-17T18:38:27Z","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"title_canon_sha256":"5d5d076759bf1542700a3c89c25f9ea067df6200819002f2cf2ce86c6c742e9f","abstract_canon_sha256":"321ef5158b8900fd1a8108a188579e6223d8daad7b4ed4a7af4d159e01de7c50"},"schema_version":"1.0"},"canonical_sha256":"3ff935035a691fd59bb3e5d1bd3ec5c20115b0398a8c9c100187cb3a157c416b","source":{"kind":"arxiv","id":"0912.3422","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.3422","created_at":"2026-05-18T03:23:56Z"},{"alias_kind":"arxiv_version","alias_value":"0912.3422v2","created_at":"2026-05-18T03:23:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.3422","created_at":"2026-05-18T03:23:56Z"},{"alias_kind":"pith_short_12","alias_value":"H74TKA22NEP5","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"H74TKA22NEP5LG5T","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"H74TKA22","created_at":"2026-05-18T12:25:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:H74TKA22NEP5LG5T4XI32PWFYI","target":"record","payload":{"canonical_record":{"source":{"id":"0912.3422","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-12-17T18:38:27Z","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"title_canon_sha256":"5d5d076759bf1542700a3c89c25f9ea067df6200819002f2cf2ce86c6c742e9f","abstract_canon_sha256":"321ef5158b8900fd1a8108a188579e6223d8daad7b4ed4a7af4d159e01de7c50"},"schema_version":"1.0"},"canonical_sha256":"3ff935035a691fd59bb3e5d1bd3ec5c20115b0398a8c9c100187cb3a157c416b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:23:56.404972Z","signature_b64":"WdPbnio/WyvojHWbR5EjyJv82dROdgxHem/cCumFzAMNabw3aGcR+l5DgL3Cw0gOYenIfgImEFrWQ8Mh0AU9Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ff935035a691fd59bb3e5d1bd3ec5c20115b0398a8c9c100187cb3a157c416b","last_reissued_at":"2026-05-18T03:23:56.404424Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:23:56.404424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.3422","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":"2026-05-18T03:23:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hEvhuCJRbJ6F5wGLNUOsvaXQE4RQaRF6MvioMwoL8dgtyOADQlhMMGFzUk9KsPBN0Bph7ZdfyCwlI5E8Wa3mAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T04:42:23.977093Z"},"content_sha256":"be8a1a9dc5607e11dff4a0553efe90295f26730e3da70c103bf035e368843256","schema_version":"1.0","event_id":"sha256:be8a1a9dc5607e11dff4a0553efe90295f26730e3da70c103bf035e368843256"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:H74TKA22NEP5LG5T4XI32PWFYI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectrum of the Product of Independent Random Gaussian Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.other","math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"B. Waclaw, R. A. Janik, Z. Burda","submitted_at":"2009-12-17T18:38:27Z","abstract_excerpt":"We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\\bar{z}) = 1/(M\\pi\\sigma^2} |z|^{-2+2/M} for |z|<\\sigma, and is zero for |z|> \\sigma. The parameter \\sigma corresponds to the radius of the circular support and is related to the amplitude of the Gaussian fluctuations. This form of the eigenvalue density is highly universal. It is identical for products of Gaussian Hermitian, non-Hermitian, real or complex random matrices. I"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.3422","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":"2026-05-18T03:23:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"USxrzh/GTJKlYPoZR9ajETnti7H4xRaNgyufKcjhu9FqyGLNY/1wcoQvMW0U6lY61BMIPtt0aYDUzQksfWx0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T04:42:23.977459Z"},"content_sha256":"1732266b7b97ebd510ed0e8697a39e8cf9f3caef1f53c1e07c463874825c4ecb","schema_version":"1.0","event_id":"sha256:1732266b7b97ebd510ed0e8697a39e8cf9f3caef1f53c1e07c463874825c4ecb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H74TKA22NEP5LG5T4XI32PWFYI/bundle.json","state_url":"https://pith.science/pith/H74TKA22NEP5LG5T4XI32PWFYI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H74TKA22NEP5LG5T4XI32PWFYI/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":"2026-06-02T04:42:23Z","links":{"resolver":"https://pith.science/pith/H74TKA22NEP5LG5T4XI32PWFYI","bundle":"https://pith.science/pith/H74TKA22NEP5LG5T4XI32PWFYI/bundle.json","state":"https://pith.science/pith/H74TKA22NEP5LG5T4XI32PWFYI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H74TKA22NEP5LG5T4XI32PWFYI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:H74TKA22NEP5LG5T4XI32PWFYI","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":"321ef5158b8900fd1a8108a188579e6223d8daad7b4ed4a7af4d159e01de7c50","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-12-17T18:38:27Z","title_canon_sha256":"5d5d076759bf1542700a3c89c25f9ea067df6200819002f2cf2ce86c6c742e9f"},"schema_version":"1.0","source":{"id":"0912.3422","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.3422","created_at":"2026-05-18T03:23:56Z"},{"alias_kind":"arxiv_version","alias_value":"0912.3422v2","created_at":"2026-05-18T03:23:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.3422","created_at":"2026-05-18T03:23:56Z"},{"alias_kind":"pith_short_12","alias_value":"H74TKA22NEP5","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"H74TKA22NEP5LG5T","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"H74TKA22","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:1732266b7b97ebd510ed0e8697a39e8cf9f3caef1f53c1e07c463874825c4ecb","target":"graph","created_at":"2026-05-18T03:23:56Z","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 show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\\bar{z}) = 1/(M\\pi\\sigma^2} |z|^{-2+2/M} for |z|<\\sigma, and is zero for |z|> \\sigma. The parameter \\sigma corresponds to the radius of the circular support and is related to the amplitude of the Gaussian fluctuations. This form of the eigenvalue density is highly universal. It is identical for products of Gaussian Hermitian, non-Hermitian, real or complex random matrices. I","authors_text":"B. Waclaw, R. A. Janik, Z. Burda","cross_cats":["cond-mat.other","math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-12-17T18:38:27Z","title":"Spectrum of the Product of Independent Random Gaussian Matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.3422","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:be8a1a9dc5607e11dff4a0553efe90295f26730e3da70c103bf035e368843256","target":"record","created_at":"2026-05-18T03:23:56Z","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":"321ef5158b8900fd1a8108a188579e6223d8daad7b4ed4a7af4d159e01de7c50","cross_cats_sorted":["cond-mat.other","math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2009-12-17T18:38:27Z","title_canon_sha256":"5d5d076759bf1542700a3c89c25f9ea067df6200819002f2cf2ce86c6c742e9f"},"schema_version":"1.0","source":{"id":"0912.3422","kind":"arxiv","version":2}},"canonical_sha256":"3ff935035a691fd59bb3e5d1bd3ec5c20115b0398a8c9c100187cb3a157c416b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3ff935035a691fd59bb3e5d1bd3ec5c20115b0398a8c9c100187cb3a157c416b","first_computed_at":"2026-05-18T03:23:56.404424Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:23:56.404424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WdPbnio/WyvojHWbR5EjyJv82dROdgxHem/cCumFzAMNabw3aGcR+l5DgL3Cw0gOYenIfgImEFrWQ8Mh0AU9Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:23:56.404972Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.3422","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:be8a1a9dc5607e11dff4a0553efe90295f26730e3da70c103bf035e368843256","sha256:1732266b7b97ebd510ed0e8697a39e8cf9f3caef1f53c1e07c463874825c4ecb"],"state_sha256":"1f31d0450467cb35a619866d68b2d2d6e14984da9f4f7199d56567df47c1b1cc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A9FUd3cyAhFyi2LGh7As9ddDCncZFsdZJZz0SzY7usvgGHEZzhcZQ5aq6MT5fQz+gCMyOpsHreSRZcspnDIIAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T04:42:23.979427Z","bundle_sha256":"3f7ba38f2b15f047b2ab6c5e6d33f96196adbe6c384d5d90cedb9ac8105003ae"}}