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

TL;DR

This paper introduces a diffusive matrix model for the $eta$-Wishart ensemble that interpolates between the Marčenko-Pastur law and the Gamma distribution, providing detailed asymptotic corrections to the spectral density.

Contribution

It constructs a $eta$-Wishart ensemble model for all $eta o [0,2]$ and derives asymptotic spectral density corrections, bridging classical laws.

Findings

Interpolates between Marčenko-Pastur and Gamma distributions as parameter varies.

Provides correction terms to the spectral density up to order 1/M and 1/M^2.

Offers a unified framework for spectral density analysis across different $eta$ values.

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 $\beta=1,2$.