{"paper":{"title":"A Necessary and Sufficient Condition for Edge Universality of Wigner matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Ji Oon Lee, Jun Yin","submitted_at":"2012-06-11T15:18:58Z","abstract_excerpt":"In this paper, we prove a necessary and sufficient condition for Tracy-Widom law of Wigner matrices. Consider $N \\times N$ symmetric Wigner matrices $H$ with $H_{ij} = N^{-1/2} x_{ij}$, whose upper right entries $x_{ij}$ $(1\\le i< j\\le N)$ are $i.i.d.$ random variables with distribution $\\mu$ and diagonal entries $x_{ii}$ $(1\\le i\\le N)$ are $i.i.d.$ random variables with distribution $\\wt \\mu$. The means of $\\mu$ and $\\wt \\mu$ are zero, the variance of $\\mu$ is 1, and the variance of $\\wt \\mu $ is finite. We prove that Tracy-Widom law holds if and only if $\\lim_{s\\to \\infty}s^4\\p(|x_{12}| \\ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2251","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"}