{"paper":{"title":"Asymptotics of the real eigenvalue distribution for the real spherical ensemble","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Peter J. Forrester","submitted_at":"2025-08-06T07:08:42Z","abstract_excerpt":"The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \\times N$ matrices. The expected number of real eigenvalues is known to be of order $\\sqrt{N}$. We consider the probability $p_{N.M}^{\\rm r}$ that there are $M$ real eigenvalues in various regimes. These are when $M$ is proportional to $N$ (large deviations), when $N$ is proportional to $\\sqrt{N}$ (intermediate deviations), and when $M$ is in the neighbourhood of the mean (local central limit theorem). This is done using a Coulomb gas formalism in the la"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.04139","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.04139/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}