# Invariant $\beta$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law

### Romain Allez 1, Jean-Philippe Bouchaud 2, Satya N. Majumdar 3, Pierpaolo Vivo 3

#### Journal of Physics A: Mathematical and Theoretical46 (2013) 015001

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) equation, we obtain as a byproduct the correction to the Mar\vcenko-Pastur density in the bulk up to order 1/M for all $\beta$ and up to order $1/M^2$ for the particular cases $\beta=1,2$.

• 1 : CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
CNRS : UMR7534 – Université Paris IX - Paris Dauphine
• 2 : Science et Finance
Science et Finance
• 3 : Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
CNRS : UMR8626 – Université Paris XI - Paris Sud