{"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"}