On the norm of random matrices. (English. Russian original) Zbl 0847.15010

Boutet de Monvel, A.; Shcherbina, M. V.

Math. Notes 57, No. 5, 475-484 (1995); translation from Mat. Zametki 57, No. 5, 688-698 (1995).

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. Z. D. Bai and 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 \(\lambda^{(N)}_{\max}\).

Reviewer: A.Khorunzhy (Khar’kov)

MSC:

15B52	Random matrices (algebraic aspects)
60H25	Random operators and equations (aspects of stochastic analysis)
60F10	Large deviations

Keywords:

norm; limiting theorem; moment method; Wigner random matrices; largest eigenvalue; large deviations