{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:YWCEG22TV26QWYQJ73NKZHP3AU","short_pith_number":"pith:YWCEG22T","schema_version":"1.0","canonical_sha256":"c584436b53aebd0b6209fedaac9dfb05035026a608ffd5b4c84e82eedc367ab6","source":{"kind":"arxiv","id":"1506.04436","version":2},"attestation_state":"computed","paper":{"title":"Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. Tikhomirov, H. K\\\"osters","submitted_at":"2015-06-14T20:45:18Z","abstract_excerpt":"For fixed $l,m \\ge 1$, let $\\mathbf{X}_n^{(0)},\\mathbf{X}_n^{(1)},\\dots,\\mathbf{X}_n^{(l)}$ be independent random $n \\times n$ matrices with independent entries, let $\\mathbf{F}_n^{(0)} := \\mathbf{X}_n^{(0)} (\\mathbf{X}_n^{(1)})^{-1} \\cdots (\\mathbf{X}_n^{(l)})^{-1}$, and let $\\mathbf{F}_n^{(1)},\\dots,\\mathbf{F}_n^{(m)}$ be independent random matrices of the same form as $\\mathbf{F}_n^{(0)}$. We investigate the limiting spectral distributions of the matrices $\\mathbf{F}_n^{(0)}$ and $\\mathbf{F}_n^{(1)} + \\dots + \\mathbf{F}_n^{(m)}$ as $n \\to \\infty$. Our main result shows that the sum $\\mathbf"},"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":"1506.04436","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-14T20:45:18Z","cross_cats_sorted":[],"title_canon_sha256":"677643c47861355026fec07712bd5b9d5618d015ee555e1400406771e0225522","abstract_canon_sha256":"b790bc69c58ea436360e952cfcf4da0d05c7dd5a68e85025d9dd6fa3bdf4e40a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:37.895143Z","signature_b64":"IdW3ez27XDCf7tLOxcmXxnPZHAqIEI5l0cWFVAJVuxmjlcZDNw7vf2AwxnrJdkPRKVp3+D1+fY5DGd48T/PiBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c584436b53aebd0b6209fedaac9dfb05035026a608ffd5b4c84e82eedc367ab6","last_reissued_at":"2026-05-18T01:36:37.894512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:37.894512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"A. Tikhomirov, H. K\\\"osters","submitted_at":"2015-06-14T20:45:18Z","abstract_excerpt":"For fixed $l,m \\ge 1$, let $\\mathbf{X}_n^{(0)},\\mathbf{X}_n^{(1)},\\dots,\\mathbf{X}_n^{(l)}$ be independent random $n \\times n$ matrices with independent entries, let $\\mathbf{F}_n^{(0)} := \\mathbf{X}_n^{(0)} (\\mathbf{X}_n^{(1)})^{-1} \\cdots (\\mathbf{X}_n^{(l)})^{-1}$, and let $\\mathbf{F}_n^{(1)},\\dots,\\mathbf{F}_n^{(m)}$ be independent random matrices of the same form as $\\mathbf{F}_n^{(0)}$. We investigate the limiting spectral distributions of the matrices $\\mathbf{F}_n^{(0)}$ and $\\mathbf{F}_n^{(1)} + \\dots + \\mathbf{F}_n^{(m)}$ as $n \\to \\infty$. Our main result shows that the sum $\\mathbf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.04436","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":"1506.04436","created_at":"2026-05-18T01:36:37.894617+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.04436v2","created_at":"2026-05-18T01:36:37.894617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.04436","created_at":"2026-05-18T01:36:37.894617+00:00"},{"alias_kind":"pith_short_12","alias_value":"YWCEG22TV26Q","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"YWCEG22TV26QWYQJ","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"YWCEG22T","created_at":"2026-05-18T12:29:52.810259+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2312.14883","citing_title":"Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators","ref_index":34,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU","json":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU.json","graph_json":"https://pith.science/api/pith-number/YWCEG22TV26QWYQJ73NKZHP3AU/graph.json","events_json":"https://pith.science/api/pith-number/YWCEG22TV26QWYQJ73NKZHP3AU/events.json","paper":"https://pith.science/paper/YWCEG22T"},"agent_actions":{"view_html":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU","download_json":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU.json","view_paper":"https://pith.science/paper/YWCEG22T","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.04436&json=true","fetch_graph":"https://pith.science/api/pith-number/YWCEG22TV26QWYQJ73NKZHP3AU/graph.json","fetch_events":"https://pith.science/api/pith-number/YWCEG22TV26QWYQJ73NKZHP3AU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU/action/storage_attestation","attest_author":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU/action/author_attestation","sign_citation":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU/action/citation_signature","submit_replication":"https://pith.science/pith/YWCEG22TV26QWYQJ73NKZHP3AU/action/replication_record"}},"created_at":"2026-05-18T01:36:37.894617+00:00","updated_at":"2026-05-18T01:36:37.894617+00:00"}