Non-Gaussian Limiting Laws for the Entries of Regular Functions of the Wigner Matrices

This paper is a continuation of our paper “Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices”, J. Stat. Phys. (134), 147–159 (2009), in which we proved the Central Limit Theorem for the matrix elements of differential functions of the real symmetric random Gaussian matrices (GOE). Here we consider the real symmetric random Wigner matrices having independent (modulo symmetry conditions) but not necessarily Gaussian entries. We show that in this case the matrix elements of sufficiently smooth functions of these random matrices have in general another limiting law which coincides essentially with the probability law of matrix entries.