  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. $

Probability;Combinatorics