{"paper":{"title":"Circular law for random discrete matrices of given row sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Hoi H. Nguyen, Van Vu","submitted_at":"2012-03-27T11:47:32Z","abstract_excerpt":"Let $M_n$ be a random matrix of size $n\\times n$ and let $\\lambda_1,...,\\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\\mu_{M_n}$ of $M_n$ is defined as $$\\mu_{M_n}(s,t)=\\frac{1}{n}# \\{k\\le n, \\Re(\\lambda_k)\\le s; \\Im(\\lambda_k)\\le t\\}.$$\n  The circular law theorem in random matrix theory asserts that if the entries of $M_n$ are i.i.d. copies of a random variable with mean zero and variance $\\sigma^2$, then the empirical spectral distribution of the normalized matrix $\\frac{1}{\\sigma\\sqrt{n}}M_n$ of $M_n$ converges almost surely to the uniform distribution $\\mu_\\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5941","kind":"arxiv","version":1},"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"}