{"paper":{"title":"Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP","math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Alexandre Krajenbrink, Pierre Le Doussal","submitted_at":"2018-11-01T17:26:55Z","abstract_excerpt":"The Airy$_\\beta$ point process, $a_i \\equiv N^{2/3} (\\lambda_i-2)$, describes the eigenvalues $\\lambda_i$ at the edge of the Gaussian $\\beta$ ensembles of random matrices for large matrix size $N \\to \\infty$. We study the probability distribution function (PDF) of linear statistics ${\\sf L}= \\sum_i t \\varphi(t^{-2/3} a_i)$ for large parameter $t$. We show the large deviation forms $\\mathbb{E}_{{\\rm Airy},\\beta}[\\exp(-{\\sf L})] \\sim \\exp(- t^2 \\Sigma[\\varphi])$ and $P({\\sf L}) \\sim \\exp(- t^2 G(L/t^2))$ for the cumulant generating function and the PDF. We obtain the exact rate function $\\Sigma["},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.00509","kind":"arxiv","version":1},"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"}