Title: Local semicircle law and level repulsion for Wigner random
matrices
Abstract:
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 project with L. Erdos and H.-T. Yau.