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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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