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

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