{"paper":{"title":"Random matrices: Law of the determinant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hoi H. Nguyen, Van Vu","submitted_at":"2011-12-04T13:55:51Z","abstract_excerpt":"Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\\det A_n|$ satisfies a central limit theorem. More precisely, \\begin{eqnarray*}\\sup_{x\\in {\\mathbf {R}}}\\biggl|{\\mathbf {P}}\\biggl(\\frac{\\log(|\\det A_n|)-({1}/{2})\\log (n-1)!}{\\sqrt{({1}/{2})\\log n}}\\le x\\biggr)-{\\mathbf {P}}\\bigl(\\mathbf {N}(0,1)\\le x\\bigr)\\biggr|\\\\\\qquad\\le\\log^{-{1}/{3}+o(1)}n.\\end{eqnarray*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0752","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"}