  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
$\beta=1,2$.
