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

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 Mar-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 Mar-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.