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Additionally if $p_n$ satisfies the inequality $np_n > \\exp(c\\sqrt{\\log n})$ for some constant $c$, then the above convergence is shown to"},"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":"1707.03675","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-07-12T12:25:57Z","cross_cats_sorted":[],"title_canon_sha256":"fa16c140060b63075653127baca0156e6304227a8c346b1bb5675b91664abb11","abstract_canon_sha256":"922ccaf23a1d8085818365ac722851c9048141331c741bed64f57a3ec05cc1b8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:37.767042Z","signature_b64":"qSd4qfNEutlB8gaNcWFP6/Xj3VieOTfYwxAz0R3wFDOA+puN/ok34Qtqh78CzfyhX5u07BSRgcz6mmR9DDpABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dd8afd46d3deeb3b9e9e3dbd69fbb5de770d9d6d125277c96b98df404c7244bd","last_reissued_at":"2026-05-18T00:13:37.766415Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:37.766415Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The circular law for sparse non-Hermitian matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anirban Basak, Mark Rudelson","submitted_at":"2017-07-12T12:25:57Z","abstract_excerpt":"For a class of sparse random matrices of the form $A_n =(\\xi_{i,j}\\delta_{i,j})_{i,j=1}^n$, where $\\{\\xi_{i,j}\\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\\{\\delta_{i,j}\\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, we prove that the empirical spectral distribution of $A_n/\\sqrt{np_n}$ converges weakly to the circular law, in probability, for all $p_n$ such that $p_n=\\omega({\\log^2n}/{n})$. Additionally if $p_n$ satisfies the inequality $np_n > \\exp(c\\sqrt{\\log n})$ for some constant $c$, then the above convergence is shown to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03675","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":"1707.03675","created_at":"2026-05-18T00:13:37.766500+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.03675v2","created_at":"2026-05-18T00:13:37.766500+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.03675","created_at":"2026-05-18T00:13:37.766500+00:00"},{"alias_kind":"pith_short_12","alias_value":"3WFP2RWT33VT","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"3WFP2RWT33VTXHU6","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"3WFP2RWT","created_at":"2026-05-18T12:30:58.224056+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/3WFP2RWT33VTXHU6HW6WT65V3Z","json":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z.json","graph_json":"https://pith.science/api/pith-number/3WFP2RWT33VTXHU6HW6WT65V3Z/graph.json","events_json":"https://pith.science/api/pith-number/3WFP2RWT33VTXHU6HW6WT65V3Z/events.json","paper":"https://pith.science/paper/3WFP2RWT"},"agent_actions":{"view_html":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z","download_json":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z.json","view_paper":"https://pith.science/paper/3WFP2RWT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.03675&json=true","fetch_graph":"https://pith.science/api/pith-number/3WFP2RWT33VTXHU6HW6WT65V3Z/graph.json","fetch_events":"https://pith.science/api/pith-number/3WFP2RWT33VTXHU6HW6WT65V3Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z/action/storage_attestation","attest_author":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z/action/author_attestation","sign_citation":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z/action/citation_signature","submit_replication":"https://pith.science/pith/3WFP2RWT33VTXHU6HW6WT65V3Z/action/replication_record"}},"created_at":"2026-05-18T00:13:37.766500+00:00","updated_at":"2026-05-18T00:13:37.766500+00:00"}