The Spectral Radius of Large Random Matrices

Abstract

Let $\{m_{ij}\}, i = 1,2,\ldots, j = 1,2,\ldots,$ be iid random variables with $Em_{11} = 0$ and $Em^2_{11} = \sigma^2$. For each $n$ define $M_n = \{m_{ij}\}_{1 \leq i, j \leq n}$, the $n \times n$ matrix whose $(i, j)$ component is $m_{ij}$. We show that $\lim \sup_{n \rightarrow \infty}\rho_n \leq \sigma$ a.s., where $\rho_n$ is the spectral radius of $M_n/\sqrt n$. Evidence from computer experiments indicates that in fact $\rho_n \rightarrow \sigma$ a.s.