The Tracy-Widom law for the Largest Eigenvalue of F Type Matrix

TL;DR

This paper proves that the distribution of the largest eigenvalue of a certain F-type matrix converges to the Tracy-Widom law under specific conditions, extending universality results in random matrix theory.

Contribution

It establishes the Tracy-Widom universality for the largest eigenvalue of F-type matrices with independent entries under moment conditions.

Findings

Largest eigenvalue follows Tracy-Widom distribution asymptotically

Universality holds for matrices with independent entries under moment conditions

Results extend to complex and real matrix cases

Abstract

Let $\mathbb{A}_p=\frac{\mathbb{Y}\mathbb{Y}^*}{m}$ and $\mathbb{B}_p=\frac{\mathbb{X}\mathbb{X}^*}{n}$ be two independent random matrices where $\mathbb{X}=(X_{ij})_{p \times n}$ and $\mathbb{Y}=(Y_{ij})_{p \times m}$ respectively consist of real (or complex) independent random variables with $\mathbb{E}X_{ij}=\mathbb{E}Y_{ij}=0$, $\mathbb{E}|X_{ij}|^2=\mathbb{E}|Y_{ij}|^2=1$. Denote by $\lambda_{1}$ the largest root of the determinantal equation $\det(\lambda \mathbb{A}_p-\mathbb{B}_p)=0$. We establish the Tracy-Widom type universality for $\lambda_{1}$ under some moment conditions on $X_{ij}$ and $Y_{ij}$ when $p/m$ and $p/n$ approach positive constants as $p\rightarrow\infty$.