{"paper":{"title":"Loop Equation Analysis of the Circular $ \\beta $ Ensembles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.MP"],"primary_cat":"math-ph","authors_text":"N.S. Witte, P.J. Forrester","submitted_at":"2014-09-21T20:20:36Z","abstract_excerpt":"We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\\rm Re}\\,\\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\\zeta)=(\\zeta+z)/(\\zeta-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6038","kind":"arxiv","version":2},"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"}