## Concentration of the spectral measure for large matrices.(English)Zbl 0969.15010

Deviations of the sum $$T(C)=f(\lambda_1)+\dots+f(\lambda_N)$$ from its expectation values are studied for large selfadjoint random matrices $$C$$, where $$\lambda_i$$ are the eigenvalues of $$C$$, $$i=1,\dots,N$$ and $$f(\cdot)$$ is a convex Lipschitz function. It is proved that if $$C$$ has independent entries in the upper triangle with a compact support, then for $$T(C)$$ the probability of deviations $$\delta$$ larger $$\delta_0=\delta(c_1/N)$$ is less than $$\exp (-c_2N^2(\delta-\delta_0)^2)$$ ($$c_i$$, $$i=1,2,3,$$ are some constants determined by the Lipschitz constant and the compactness measure). If the distribution of entries satisfies a logarithmic Sobolev inequality, then for $$T(C)$$ the probability of deviations larger than $$\delta$$ is not larger than $$\exp(-c_3N^2\delta^2)$$. These assertions are refined for non convex functions $$f(\cdot)$$, non-Gaussian Wigner and Wishart matrices. Also, upper bounds for the deviations of $$T(C)$$ are provided for some non-commutative functionals of random matrices.

### MSC:

 15B52 Random matrices (algebraic aspects) 15A18 Eigenvalues, singular values, and eigenvectors 60F10 Large deviations
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