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

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 $p\rightarrow\infty$.