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

We construct a diffusive matrix model for the beta-Wishart (or Laguerre) ensemble for general continuous beta is an element of [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 Marcenko-Pastur (c -> infinity limit) and the Gamma law (c -> 0 limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtain as a byproduct the correction to the Marcenko-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.