{"paper":{"title":"Invariant $\\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Jean-Philippe Bouchaud, Pierpaolo Vivo, Romain Allez, Satya N. Majumdar","submitted_at":"2012-09-27T09:26:53Z","abstract_excerpt":"We construct a diffusive matrix model for the $\\beta$-Wishart (or Laguerre) ensemble for general continuous $\\beta\\in [0,2]$, which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index $\\beta$ with the largest size $M$ of the data matrix as $\\beta=2c/M$ (with $c$ a fixed positive constant), we obtain a family of spectral densities parametrized by $c$. As $c$ is varied, this density interpolates continuously between the Mar\\vcenko-Pastur ($c\\to \\infty$ limit) and the Gamma law ($c\\to 0$ limit). Analyzing the full Stieltjes transform (resolvent) equatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6171","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"}