{"paper":{"title":"How many eigenvalues of a Gaussian random matrix are positive?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Antonello Scardicchio, C\\'eline Nadal, Pierpaolo Vivo, Satya N. Majumdar","submitted_at":"2010-12-06T09:53:30Z","abstract_excerpt":"We study the probability distribution of the index ${\\mathcal N}_+$, i.e., the number of positive eigenvalues of an $N\\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large $\\mathcal{N}_+$ with the fraction $0\\le c=\\mathcal{N}_+/N\\le 1$ of positive eigenvalues fixed, the index distribution $\\mathcal{P}({\\mathcal N}_+=cN,N)\\sim\\exp[-\\beta N^2 \\Phi(c)]$ where $\\beta$ is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function $\\Phi(c)$ is computed explicitly for all $0\\leq c \\leq 1$. It is independent of $\\beta$ and displays"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1107","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"}