Distribution of subdominant eigenvalues of random matrices (Q1594519)

Random matrices \(B\) of increasing size \(n\times n\) with independent entries having expectations \(1/n\) and variance not greater \(c/n^2\) are considered, where \(c\geq 0\) is a constant. It is shown that for any \(\varepsilon>0\) and positive \(p<1\) there exists a number \(n_0=n_0 (\varepsilon, p)\) such that for all \(n>n_0\) the spectral radius of \(B\) is \(\varepsilon\)-close to 1, while \(n-1\) eigenvalues of \(B\) lie in an open disc of radius \(\varepsilon\) around 0 in the complex plane with the probability \(p'>p\).