The Spectral Radius of Large Random Matrices

Let ᵢⱼ\, i = 1,2,…, j = 1,2,…, be iid random variables with Em₁₁ = 0 and Em²₁₁ = σ². For each n define Mₙ = ᵢⱼ\1 ≤ i, j ≤ n, the n × n matrix whose $(i, j)$ component is mᵢⱼ. We show that lim n → ∞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.