{"paper":{"title":"Number statistics for $\\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","cond-mat.quant-gas","math-ph","math.MP","math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Gregory Schehr, Pierpaolo Vivo, Ricardo Marino, Satya N. Majumdar","submitted_at":"2016-01-13T09:24:01Z","abstract_excerpt":"Let $\\mathcal{P}_{\\beta}^{(V)} (N_{\\cal I})$ be the probability that a $N\\times N$ $\\beta$-ensemble of random matrices with confining potential $V(x)$ has $N_{\\cal I}$ eigenvalues inside an interval ${\\cal I}=[a,b]$ of the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically $\\mathcal{P}_{\\beta}^{(V)} (N_{\\cal I})$ for large $N$. We show that this probability scales for large $N$ as $\\mathcal{P}_{\\beta}^{(V)} (N_{\\cal I})\\approx \\exp\\left(-\\beta N^2 \\psi^{(V)}(N_{\\cal I} /N)\\right)$, where $\\beta$ is the Dyson index o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03178","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"}