On the norm of random matrices (Q1909784)

It is known that the largest eigenvalue \(\lambda^{(N)}_{\max}\) of the Wigner \(N\times N\) random matrices \(W_N\) converges with probability 1 as \(N\to \infty\) to \(2v\), where \(v^2\) is the variance of the matrix entries [see e.g. \textit{Z. D. Bai} and \textit{Y. Q. Yin}, Ann. Probab. 16, No. 4, 1729-1741 (1988; Zbl 0677.60038)]. The proof concerns analysis of moments \(N^{- 1} \text{Tr } W^k_N\) in asymptotics \(k\), \(N\to \infty\). In the present paper, the moment technique gets its further elegant development. The authors consider random matrix ensembles more general than the Wigner one and obtain large deviations-type statements for respective \(\lambda^{(N)}_{\max}\).