The Spectral Radius of Large Random Matrices

Abstract

Let {mij},i=1,2,…,j=1,2,…, be iid random variables with Em11=0 and Em211=σ2. For each n define Mn={mij}1≤i,j≤n, the n×n matrix whose (i,j) component is mij. We show that limsupn→∞ρn≤σ a.s., where ρn is the spectral radius of Mn/n√. Evidence from computer experiments indicates that in fact ρn→σ a.s.