{"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"}