Thumbnail

Invariant β-Wishart ensembles, crossover densities and asymptotic corrections to the Marcenko–Pastur law

Allez, Romain; Bouchaud, Jean-Philippe; Najumdar, Satya N.; Vivo, Pierpaolo (2013), Invariant β-Wishart ensembles, crossover densities and asymptotic corrections to the Marcenko–Pastur law, Journal of Physics. A, Mathematical and Theoretical, 46, 1. http://dx.doi.org/10.1088/1751-8113/46/1/015001

View/Open
Type
Article accepté pour publication ou publié
Date
2013
Journal name
Journal of Physics. A, Mathematical and Theoretical
Volume
46
Number
1
Publisher
IOP Publishing
Publication identifier
Metadata
Show full item record
Author(s)
Allez, Romain
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Bouchaud, Jean-Philippe
Science & Finance
Najumdar, Satya N.
Laboratoire de Physique Théorique et Modèles Statistiques [LPTMS]
Vivo, Pierpaolo
Laboratoire de Physique Théorique et Modèles Statistiques [LPTMS]
Abstract (EN)
We construct a diffusive matrix model for the β-Wishart (or Laguerre) ensemble for general continuous β ∈ [0, 2], which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index β with the largest size M of the data matrix as β = 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 Marčenko–Pastur (c → ∞ limit) and the Gamma law (c → 0 limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtain as a byproduct the correction to the Marčenko–Pastur density in the bulk up to order 1/M for all β and up to order 1/M2 for the particular cases β = 1, 2.
Subjects / Keywords
Statistical Mechanics; Probability; marchenko-Pastur law; β-Wishart ensemble