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

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.