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

TL;DR

This paper establishes a lower bound on the spectral radius of large symmetric random matrices with independent, bounded, centered entries, showing it converges to twice the standard deviation with high probability.

Contribution

It provides a new lower bound on the spectral norm of symmetric random matrices with independent entries, complementing previous upper bounds and confirming the spectral radius's asymptotic behavior.

Findings

Spectral radius bounded below by 2σ minus a small order term

Spectral norm converges to 2σ with high probability

Results hold for matrices with non-symmetrically distributed entries

Abstract

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by $ 2 \*\sigma - o(N^{-6/11+\epsilon}), $ where $\sigma^2 $ is the variance of the matrix entries and $\epsilon $ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\epsilon >0, $ one has $$ \|A_N\| =2 \*\sigma + o(N^{-6/11+\epsilon}) $$ with probability going to 1 as $N \to \infty. $