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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.2251","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-06-11T15:18:58Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"1e4b2a30eb21756e151372bc1e7af3dc0556f7399a0546b8c2ba0b2458105cdb","abstract_canon_sha256":"772bb4fc4e4013a1e5405c60e06d0bb974ce865ce8f6a0cb47f3d475e3d53d90"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:32.607058Z","signature_b64":"T8c9O3yxXYNjeFeVFKHeDA2WYUSJsBi+y1fByGMgsxlav8dHV+tKOhML2XQMSq4Gdnvf1+eGXIOSo+ae6sJmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65e66e03bbe72f4a5bac58728213ae9dffc9365970c93c8169e0bab0b37fd5a3","last_reissued_at":"2026-05-18T02:29:32.606632Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:32.606632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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