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Spectral statistics of large random matrices

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Large random matrices exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. I review recent results on the spectral universality of random matrices. I also describe two types of phase transition in random matrix models: one associated with heavy-tailed entries, and the other associated with finite-rank deformations. (Joint work with L. Erdos, H.T. Yau, and J. Yin.)

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