LetA=(aij) be ann ×n matrix whose entries fori≧j are independent random variables andaji=aij. Suppose that everyaij is bounded and for everyi>j we haveEaij=μ,D2aij=σ2 andEaii=v.
E. P. Wigner determined the asymptotic behavior of the eigenvalues ofA (semi-circle law). In particular, for anyc>2σ with probability 1-o(1) all eigenvalues except for at mosto(n) lie in the intervalI=(−c√n,c√n).
We show that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalue λ1 is outsideI, and