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We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues fxjg N=1 are close to their classical location fjg N=1 determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1=N, the necessary apriori estimate on the location of eigenvalues requires only to know that Ejxj − jj 2  N 1 " on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for sample covariance matrices.

The local relaxation flow approach to universality of the local statistics for random matrices