{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:P45W7CKYEK6IPEE63DNBJRJ6B3","short_pith_number":"pith:P45W7CKY","schema_version":"1.0","canonical_sha256":"7f3b6f895822bc87909ed8da14c53e0ed91778edd3b1f71aded5b8b3a7c93d25","source":{"kind":"arxiv","id":"1209.6171","version":1},"attestation_state":"computed","paper":{"title":"Invariant $\\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Jean-Philippe Bouchaud, Pierpaolo Vivo, Romain Allez, Satya N. Majumdar","submitted_at":"2012-09-27T09:26:53Z","abstract_excerpt":"We construct a diffusive matrix model for the $\\beta$-Wishart (or Laguerre) ensemble for general continuous $\\beta\\in [0,2]$, which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index $\\beta$ with the largest size $M$ of the data matrix as $\\beta=2c/M$ (with $c$ a fixed positive constant), we obtain a family of spectral densities parametrized by $c$. As $c$ is varied, this density interpolates continuously between the Mar\\vcenko-Pastur ($c\\to \\infty$ limit) and the Gamma law ($c\\to 0$ limit). Analyzing the full Stieltjes transform (resolvent) equatio"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.6171","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2012-09-27T09:26:53Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"02aebe667426979511aa2f4f6dd4c22c8de46a63c2a2f82c8c5ccfc6bb6852f7","abstract_canon_sha256":"2b98cfbf20f4f892a727dad67d6b97f172964f22a95efb86eceefd29494a0868"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:25.431582Z","signature_b64":"cZmaUKxqPLY/czB7SGmD1qlkLrkmPIVXx7jrXePUj04BfbBjHNmrOs9Q/p6rZhsZyslrLH2dIk3c5rXpQFvFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f3b6f895822bc87909ed8da14c53e0ed91778edd3b1f71aded5b8b3a7c93d25","last_reissued_at":"2026-05-18T03:35:25.431049Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:25.431049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariant $\\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Jean-Philippe Bouchaud, Pierpaolo Vivo, Romain Allez, Satya N. Majumdar","submitted_at":"2012-09-27T09:26:53Z","abstract_excerpt":"We construct a diffusive matrix model for the $\\beta$-Wishart (or Laguerre) ensemble for general continuous $\\beta\\in [0,2]$, which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index $\\beta$ with the largest size $M$ of the data matrix as $\\beta=2c/M$ (with $c$ a fixed positive constant), we obtain a family of spectral densities parametrized by $c$. As $c$ is varied, this density interpolates continuously between the Mar\\vcenko-Pastur ($c\\to \\infty$ limit) and the Gamma law ($c\\to 0$ limit). Analyzing the full Stieltjes transform (resolvent) equatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6171","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.6171","created_at":"2026-05-18T03:35:25.431126+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.6171v1","created_at":"2026-05-18T03:35:25.431126+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.6171","created_at":"2026-05-18T03:35:25.431126+00:00"},{"alias_kind":"pith_short_12","alias_value":"P45W7CKYEK6I","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"P45W7CKYEK6IPEE6","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"P45W7CKY","created_at":"2026-05-18T12:27:18.751474+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3","json":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3.json","graph_json":"https://pith.science/api/pith-number/P45W7CKYEK6IPEE63DNBJRJ6B3/graph.json","events_json":"https://pith.science/api/pith-number/P45W7CKYEK6IPEE63DNBJRJ6B3/events.json","paper":"https://pith.science/paper/P45W7CKY"},"agent_actions":{"view_html":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3","download_json":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3.json","view_paper":"https://pith.science/paper/P45W7CKY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.6171&json=true","fetch_graph":"https://pith.science/api/pith-number/P45W7CKYEK6IPEE63DNBJRJ6B3/graph.json","fetch_events":"https://pith.science/api/pith-number/P45W7CKYEK6IPEE63DNBJRJ6B3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3/action/storage_attestation","attest_author":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3/action/author_attestation","sign_citation":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3/action/citation_signature","submit_replication":"https://pith.science/pith/P45W7CKYEK6IPEE63DNBJRJ6B3/action/replication_record"}},"created_at":"2026-05-18T03:35:25.431126+00:00","updated_at":"2026-05-18T03:35:25.431126+00:00"}