On the lower bound of the spectral norm of symmetric random matrices with independent entries

Abstract

We show that the spectral radius of an N × N random symmetric matrix with i.i.d. Bounded centered but non-symmetrically distributed entries is bounded from below by 2σ−o(N−6/11+ε), where σ2 is the variance of the matrix entries and ε is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any ε > 0, one has ||AN|| = 2σ + o(N−6/11+ε) with probability going to 1 as N → ∞. © 2008 Applied Probability Trust.