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 $2\text{*}\sigma -o(N^{-6\mathrm{/}11+ϵ}),$ where ${\sigma }^2$ 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 $$\mathrm{\parallel }{A}_N\mathrm{\parallel }=2\text{*}\sigma +o(N^{-6\mathrm{/}11+ϵ})$$ with probability going to 1 as $N\to \mathrm{\infty }\mathrm{.}$