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

We construct a diffusive matrix model for the $β$-Wishart (or Laguerre) ensemble for general continuous $β\in [0,2]$, 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\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 $β$ and up to order $1/M^2$ for the particular cases $β=1,2$.

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