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We construct a diffusive matrix model for the $\beta$-Wishart (or Laguerre)ensemble for general continuous $\beta\in [0,2]$, which preserves invarianceunder 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 densitiesparametrized by $c$. As $c$ is varied, this density interpolates continuouslybetween the Mar\vcenko-Pastur ($c\to \infty$ limit) and the Gamma law ($c\to 0$limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtainas a byproduct the correction to the Mar\vcenko-Pastur density in the bulk upto order 1/M for all $\beta$ and up to order $1/M^2$ for the particular cases$\beta=1,2$.

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