Wigner random matrices with non-symmetrically distributed 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 above 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. Our bound improves the earlier results by Z.F\"{u}redi and J.Koml\'{o}s (1981), and the recent bound obtained by Van Vu (2005).