{"paper":{"title":"Index Distribution of Cauchy Random Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","stat.OT"],"primary_cat":"cond-mat.stat-mech","authors_text":"Gr\\'egory Schehr, Pierpaolo Vivo, Ricardo Marino, Satya N. Majumdar","submitted_at":"2013-12-08T13:41:23Z","abstract_excerpt":"Using a Coulomb gas technique, we compute analytically the probability $\\mathcal{P}_\\beta^{(C)}(N_+,N)$ that a large $N\\times N$ Cauchy random matrix has $N_+$ positive eigenvalues, where $N_+$ is called the index of the ensemble. We show that this probability scales for large $N$ as $\\mathcal{P}_\\beta^{(C)}(N_+,N)\\approx \\exp\\left[-\\beta N^2 \\psi_C(N_+/N)\\right]$, where $\\beta$ is the Dyson index of the ensemble. The rate function $\\psi_C(\\kappa)$ is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis. We find that the rate functi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2211","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"}