{"paper":{"title":"Random matrices: Localization of the eigenvalues and the necessity of four moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Terence Tao, Van Vu","submitted_at":"2010-05-17T12:28:23Z","abstract_excerpt":"Consider the eigenvalues $\\lambda_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law, one expects that $\\lambda_i(M_n)$ concentrates around $\\gamma_i \\sqrt n$, where $\\int_{-\\infty}^{\\gamma_i} \\rho_{sc} (x) dx = \\frac{i}{n}$ and $\\rho_{sc}$ is the semicircular function.\n  In this paper, we show that if the entries have vanishing third moment, then for all $1\\le i \\le n$\n  $$\\E |\\lambda_i(M_n)-\\sqrt{n} \\gamma_i|^2 = O(\\min(n^{-c} \\min(i,n+1-i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.2901","kind":"arxiv","version":4},"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"}