Jonathan Breuer: Sine kernel asymptotics and eigenvalue repulsion for Jacobi matrices with purely singular measures

 
Abstract: The Christoffel-Darboux (CD) kernel associated with a measure has played a central role both in understanding the distribution of random matrix eigenvalues and in understanding fine properties of eigenvalues of truncated Jacobi matrices (a generalization of one dimensional discrete Schroedinger operators). In particular, convergence of the scaled kernel to the sine kernel, which has been shown to occur for many `nice' measures, is connected both to universality in random matrix theory and to very strong repulsion between the Jacobi eigenvalues. Physical intuition associates this type of behavior with absolute continuity of the measure. In this talk we present a family of Jacobi matrices whose spectral measures are purely singular with respect to Lebesgue measure, and for which the CD kernel has sine kernel asymptotics.