{"paper":{"title":"The sparse circular law under minimal assumptions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Konstantin Tikhomirov, Mark Rudelson","submitted_at":"2018-07-21T05:04:06Z","abstract_excerpt":"The circular law asserts that the empirical distribution of eigenvalues of appropriately normalized $n\\times n$ matrix with i.i.d. entries converges to the uniform measure on the unit disc as the dimension $n$ grows to infinity. Consider an $n\\times n$ matrix $A_n=(\\delta_{ij}^{(n)}\\xi_{ij}^{(n)})$, where $\\xi_{ij}^{(n)}$ are copies of a real random variable of unit variance, variables $\\delta_{ij}^{(n)}$ are Bernoulli ($0/1$) with ${\\mathbb P}\\{\\delta_{ij}^{(n)}=1\\}=p_n$, and $\\delta_{ij}^{(n)}$ and $\\xi_{ij}^{(n)}$, $i,j\\in[n]$, are jointly independent. In order for the circular law to hold "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.08085","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":""},"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"}