A smooth transition from Wishart to GOE

TL;DR

The paper analyzes the transition from Wishart to GOE in the critical window where $d/n^3 \to c$, proving that the total variation distance between $W(n,d)$ and the centered/scaled GOE $M(n,d)$ converges to $Erf\left( \frac{1}{4*sqrt{3}*sqrt{c}} \right)$. The approach combines a density-based comparison of $W(n,d)$ and $M(n,d)$, a fourth-order Taylor expansion of the log-density ratio, and a central limit theorem for moments of the GOE eigenvalues to derive a Gaussian limit that reduces to the $Erf$ formula. The result confirms a smooth Wishart-GOE phase transition in the critical regime and links the limiting behavior across the regimes $c \to 0$ and $c \to \infty$. This advances understanding in random matrix theory by providing an explicit, finite-parameter description of the transition and its robustness properties near criticality.

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 $d = Θ( n^{3} )$. 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.

Figures (1)

• Figure 1: The limiting total variation distance as a function of $c$, when $c$ is close to $0$.

Theorems & Definitions (2)

• Theorem 1.1
• Theorem 1.2