{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:JEVL4IPRTOEWGUWZGQBKNSXLQ7","short_pith_number":"pith:JEVL4IPR","schema_version":"1.0","canonical_sha256":"492abe21f19b896352d93402a6caeb87f66256d5d8c04c6b140916253565049a","source":{"kind":"arxiv","id":"1402.2207","version":2},"attestation_state":"computed","paper":{"title":"Bulk behaviour of Schur-Hadamard products of symmetric random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arup Bose, Soumendu Sundar Mukherjee","submitted_at":"2014-02-10T16:45:03Z","abstract_excerpt":"We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSDs of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples."},"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":"1402.2207","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-10T16:45:03Z","cross_cats_sorted":[],"title_canon_sha256":"e145f9e80a7ac92d015ab10500275999e738e0afd3b921e8b788acf6729fd9bc","abstract_canon_sha256":"15f557ae9299574a3f3921c409f1034eac5227f65bb6dbd21b5e71828982336e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:18.830188Z","signature_b64":"sAv5U6CzrBvIJa87LEC03K40v+2pBhVhWFy7etjVl/nEWgphWHOQubzGiJ4HfvQuP4TO7jIaulZ3Z6gM7OvmBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"492abe21f19b896352d93402a6caeb87f66256d5d8c04c6b140916253565049a","last_reissued_at":"2026-05-18T02:56:18.829727Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:18.829727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bulk behaviour of Schur-Hadamard products of symmetric random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Arup Bose, Soumendu Sundar Mukherjee","submitted_at":"2014-02-10T16:45:03Z","abstract_excerpt":"We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSDs of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.2207","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1402.2207","created_at":"2026-05-18T02:56:18.829803+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.2207v2","created_at":"2026-05-18T02:56:18.829803+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.2207","created_at":"2026-05-18T02:56:18.829803+00:00"},{"alias_kind":"pith_short_12","alias_value":"JEVL4IPRTOEW","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"JEVL4IPRTOEWGUWZ","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"JEVL4IPR","created_at":"2026-05-18T12:28:33.132498+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/JEVL4IPRTOEWGUWZGQBKNSXLQ7","json":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7.json","graph_json":"https://pith.science/api/pith-number/JEVL4IPRTOEWGUWZGQBKNSXLQ7/graph.json","events_json":"https://pith.science/api/pith-number/JEVL4IPRTOEWGUWZGQBKNSXLQ7/events.json","paper":"https://pith.science/paper/JEVL4IPR"},"agent_actions":{"view_html":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7","download_json":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7.json","view_paper":"https://pith.science/paper/JEVL4IPR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.2207&json=true","fetch_graph":"https://pith.science/api/pith-number/JEVL4IPRTOEWGUWZGQBKNSXLQ7/graph.json","fetch_events":"https://pith.science/api/pith-number/JEVL4IPRTOEWGUWZGQBKNSXLQ7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7/action/storage_attestation","attest_author":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7/action/author_attestation","sign_citation":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7/action/citation_signature","submit_replication":"https://pith.science/pith/JEVL4IPRTOEWGUWZGQBKNSXLQ7/action/replication_record"}},"created_at":"2026-05-18T02:56:18.829803+00:00","updated_at":"2026-05-18T02:56:18.829803+00:00"}