Special Seminar

We discuss the distribution of the extreme eigenvalues for several classes of large (N X N) Hermitian random matrices. For a class of sparse matrices, the distribution is approximately Tracy--Widom. For band matrices with band width W(N) , the distribution is approximately Tracy--Widom if and only if W(N) >> N^{5/6} . These results are coherent with the Thouless criterion for localization, adapted to band matrices by Fyodorov and Mirlin (which we shall hopefully also discuss).