A smooth transition from Wishart to GOE

TL;DR

This paper investigates the transition of Wishart matrices to GOE matrices, demonstrating that the phase change is smooth within a critical window where the degrees of freedom scale as n^3.

Contribution

It explicitly computes the total variation distance in the critical window, revealing the smooth nature of the phase transition from Wishart to GOE matrices.

Findings

Transition occurs smoothly when d/n^3 approaches a positive constant c.

Total variation distance between Wishart and GOE matrices is explicitly computed in the critical window.

Phase transition is not abrupt but gradual in the specified scaling regime.

Abstract

It is well known that an $n \times n$ Wishart matrix with $d$ degrees of freedom is close to the appropriately centered and scaled Gaussian Orthogonal Ensemble (GOE) if $d$ is large enough. Recent work of Bubeck, Ding, Eldan, and Racz, and independently Jiang and Li, shows that the transition happens when $d = \Theta ( n^{3} )$. Here we consider this critical window and explicitly compute the total variation distance between the Wishart and GOE matrices when $d / n^{3} \to c \in (0, \infty)$. This shows, in particular, that the phase transition from Wishart to GOE is smooth.