{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:2TH5EFBQKGPBDAIU52COO2P3UH","short_pith_number":"pith:2TH5EFBQ","schema_version":"1.0","canonical_sha256":"d4cfd21430519e118114ee84e769fba1cfa2f7e76b7fbbafeae3dfc70301328a","source":{"kind":"arxiv","id":"2009.13726","version":3},"attestation_state":"computed","paper":{"title":"Rank of Sparse Bernoulli Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Han Huang","submitted_at":"2020-09-29T02:05:06Z","abstract_excerpt":"Let $ A_n $ be an $n \\times n$ random matrix with i.i.d Bernoulli($p$) entries. For a fixed positive integer $\\beta$, suppose $p$ satisfies $$ \\frac{ \\log(n) }{ n } \\le p \\le c_\\beta $$ where $c_\\beta \\in ( 0, 1/2 )$ is a $\\beta$-dependentvalue. For $t \\ge 0$, $$ \\mathbb{P} \\left\\{ s_{ n - \\beta + 1}(A) \\le t n^{-2\\beta + \\mathfrak{n}(1) }(pn)^{-7} \\right\\} = t + ( 1 + o_\\mathfrak{n}(1) ) \\mathbb{P} \\bigg\\{ \\mbox{either $\\beta$ rows or $\\beta$ columns of $A_n$ equal $\\vec{0}$} \\bigg\\}. $$"},"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":"2009.13726","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2020-09-29T02:05:06Z","cross_cats_sorted":[],"title_canon_sha256":"1bf75ba15041ee4fd2667e69d80b9ce508447521f0599b33916136749daa80eb","abstract_canon_sha256":"6f67e4af3a0db31844f3e6693f71f5e8d3cb4fc88c4ebab0ef6027902bb5d5c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:04:32.398917Z","signature_b64":"SsXerLyih2O5Y5ua6qQP23nSGsq5mBP34gcbe5F9pvTJY6excBCJTaGIAkA6FqhFgkKY/Z0ZnCPPKJBilxFNDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4cfd21430519e118114ee84e769fba1cfa2f7e76b7fbbafeae3dfc70301328a","last_reissued_at":"2026-07-05T11:04:32.398422Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:04:32.398422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rank of Sparse Bernoulli Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Han Huang","submitted_at":"2020-09-29T02:05:06Z","abstract_excerpt":"Let $ A_n $ be an $n \\times n$ random matrix with i.i.d Bernoulli($p$) entries. For a fixed positive integer $\\beta$, suppose $p$ satisfies $$ \\frac{ \\log(n) }{ n } \\le p \\le c_\\beta $$ where $c_\\beta \\in ( 0, 1/2 )$ is a $\\beta$-dependentvalue. For $t \\ge 0$, $$ \\mathbb{P} \\left\\{ s_{ n - \\beta + 1}(A) \\le t n^{-2\\beta + \\mathfrak{n}(1) }(pn)^{-7} \\right\\} = t + ( 1 + o_\\mathfrak{n}(1) ) \\mathbb{P} \\bigg\\{ \\mbox{either $\\beta$ rows or $\\beta$ columns of $A_n$ equal $\\vec{0}$} \\bigg\\}. $$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.13726","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.13726/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2009.13726","created_at":"2026-07-05T11:04:32.398479+00:00"},{"alias_kind":"arxiv_version","alias_value":"2009.13726v3","created_at":"2026-07-05T11:04:32.398479+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.13726","created_at":"2026-07-05T11:04:32.398479+00:00"},{"alias_kind":"pith_short_12","alias_value":"2TH5EFBQKGPB","created_at":"2026-07-05T11:04:32.398479+00:00"},{"alias_kind":"pith_short_16","alias_value":"2TH5EFBQKGPBDAIU","created_at":"2026-07-05T11:04:32.398479+00:00"},{"alias_kind":"pith_short_8","alias_value":"2TH5EFBQ","created_at":"2026-07-05T11:04:32.398479+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2503.08139","citing_title":"The eigenvalue gap of inhomogeneous symmetric discrete random matrix","ref_index":17,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH","json":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH.json","graph_json":"https://pith.science/api/pith-number/2TH5EFBQKGPBDAIU52COO2P3UH/graph.json","events_json":"https://pith.science/api/pith-number/2TH5EFBQKGPBDAIU52COO2P3UH/events.json","paper":"https://pith.science/paper/2TH5EFBQ"},"agent_actions":{"view_html":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH","download_json":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH.json","view_paper":"https://pith.science/paper/2TH5EFBQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2009.13726&json=true","fetch_graph":"https://pith.science/api/pith-number/2TH5EFBQKGPBDAIU52COO2P3UH/graph.json","fetch_events":"https://pith.science/api/pith-number/2TH5EFBQKGPBDAIU52COO2P3UH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH/action/storage_attestation","attest_author":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH/action/author_attestation","sign_citation":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH/action/citation_signature","submit_replication":"https://pith.science/pith/2TH5EFBQKGPBDAIU52COO2P3UH/action/replication_record"}},"created_at":"2026-07-05T11:04:32.398479+00:00","updated_at":"2026-07-05T11:04:32.398479+00:00"}