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

Abstract

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 $\beta =1,2$.

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References in corpus (7)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Exact distribution of the maximal height of p vicious walkers
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Power-law deformation of Wishart-Laguerre ensembles of random matrices
• On invariant 2x2 β-ensembles of random matrices

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