{"paper":{"title":"Circular law for non-central random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Djalil Chafai (LAMA)","submitted_at":"2007-09-01T06:57:30Z","abstract_excerpt":"Let $(X_{jk})_{j,k\\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\\la_{n,1},...,\\la_{n,n}$ be the eigenvalues of $(\\frac{1}{\\sqrt{n}}X_{jk})_{1\\leq j,k\\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\\frac{1}{n}(\\de_{\\la_{n,1}}+...+\\de_{\\la_{n,n}})$ converges weakly as $n\\to\\infty$ to the uniform law over the unit disc $\\{z\\in\\dC;|z|\\leq1\\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\\leq j,k\\leq n}$ provided"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.0036","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"}