{"paper":{"title":"The logarithmic law of random determinant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Guangming Pan, Wang Zhou, Zhigang Bao","submitted_at":"2012-08-29T02:14:35Z","abstract_excerpt":"Consider the square random matrix $A_n=(a_{ij})_{n,n}$, where $\\{a_{ij}:=a_{ij}^{(n)},i,j=1,\\ldots,n\\}$ is a collection of independent real random variables with means zero and variances one. Under the additional moment condition \\[\\sup_n\\max_{1\\leq i,j\\leq n}\\mathbb{E}a_{ij}^4<\\infty,\\] we prove Girko's logarithmic law of $\\det A_n$ in the sense that as $n\\rightarrow\\infty$ \\begin{eqnarray*}\\frac{\\log|\\det A_n|-(1/2)\\log(n-1)!}{\\sqrt{(1/2)\\log n}}\\stackrel{d}{ \\longrightarrow}N(0,1).\\end{eqnarray*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.5823","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"}