{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:VGJCXTJ3M6ZN3GZ5BKAFFDVFLA","short_pith_number":"pith:VGJCXTJ3","schema_version":"1.0","canonical_sha256":"a9922bcd3b67b2dd9b3d0a80528ea558108740ec87c952de8042e59023b98d9b","source":{"kind":"arxiv","id":"0912.0966","version":5},"attestation_state":"computed","paper":{"title":"Random covariance matrices: Universality of local statistics of eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.SP","authors_text":"Terence Tao, Van Vu","submitted_at":"2009-12-07T20:51:24Z","abstract_excerpt":"We study the eigenvalues of the covariance matrix $\\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\\zeta_{ij})_{1\\leq i\\leq p;1\\leq j\\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite $C_0$th moment for some sufficiently large constant $C_0$. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549--"},"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":"0912.0966","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2009-12-07T20:51:24Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"660e46b87cf49ec5f5333440b942eba72517ede2ff1c2360e853dc01c69da1cc","abstract_canon_sha256":"3e72c5e180cf279b0dee3a5b4a8527c55355f0868a57a7c97f531d1d6603fc17"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:55:00.659308Z","signature_b64":"dsiiRedAbBqvfRhs4bo23ggyfG5UuEySmkNmhKcPwOFu2vfIciiIm1ENVMHbNWAsBbFUVHza/smzcvhqHKy6CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9922bcd3b67b2dd9b3d0a80528ea558108740ec87c952de8042e59023b98d9b","last_reissued_at":"2026-05-18T03:55:00.658604Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:55:00.658604Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random covariance matrices: Universality of local statistics of eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.SP","authors_text":"Terence Tao, Van Vu","submitted_at":"2009-12-07T20:51:24Z","abstract_excerpt":"We study the eigenvalues of the covariance matrix $\\frac{1}{n}M^*M$ of a large rectangular matrix $M=M_{n,p}=(\\zeta_{ij})_{1\\leq i\\leq p;1\\leq j\\leq n}$ whose entries are i.i.d. random variables of mean zero, variance one, and having finite $C_0$th moment for some sufficiently large constant $C_0$. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Universality of local eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549--"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.0966","kind":"arxiv","version":5},"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":"0912.0966","created_at":"2026-05-18T03:55:00.658730+00:00"},{"alias_kind":"arxiv_version","alias_value":"0912.0966v5","created_at":"2026-05-18T03:55:00.658730+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.0966","created_at":"2026-05-18T03:55:00.658730+00:00"},{"alias_kind":"pith_short_12","alias_value":"VGJCXTJ3M6ZN","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"VGJCXTJ3M6ZN3GZ5","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"VGJCXTJ3","created_at":"2026-05-18T12:26:02.257875+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/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA","json":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA.json","graph_json":"https://pith.science/api/pith-number/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/graph.json","events_json":"https://pith.science/api/pith-number/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/events.json","paper":"https://pith.science/paper/VGJCXTJ3"},"agent_actions":{"view_html":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA","download_json":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA.json","view_paper":"https://pith.science/paper/VGJCXTJ3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0912.0966&json=true","fetch_graph":"https://pith.science/api/pith-number/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/graph.json","fetch_events":"https://pith.science/api/pith-number/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/action/storage_attestation","attest_author":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/action/author_attestation","sign_citation":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/action/citation_signature","submit_replication":"https://pith.science/pith/VGJCXTJ3M6ZN3GZ5BKAFFDVFLA/action/replication_record"}},"created_at":"2026-05-18T03:55:00.658730+00:00","updated_at":"2026-05-18T03:55:00.658730+00:00"}