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Local semicircle law and level repulsion for Wigner random matrices

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Consider ensembles of N by N hermitian random matrices with independent and identically distributed entries (up to the symmetry constraints), scaled so that the typical distance between successive eigenvalues is of the order 1/N. In this talk, I am going to discuss some properties of the spectrum of these matrices as N tends to infinity. In particular, I am going to present a proof of the validity of the semicircle law for the eigenvalue density on energy scales of the order K/N, in the limit of large but fixed K (independent of N). This is the smallest scale on which the semicircle law can be expected to hold. Moreover, I am going to discuss some upper bounds on the probability of finding eigenvalues in a given interval, which show the phenomenon of level repulsion. This is a joint work with L. Erdos and H.-T. Yau.

This talk is part of the Probability series.

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