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On the norm of random matrices. (English. Russian original) Zbl 0847.15010
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 $$\lambda^{(N)}_{\max}$$.

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