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Random matrices: Universality of local eigenvalue statistics

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and $k$-point correlation and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.