Operator limits of random matrices

Tuesday, June 19, 2012 - 3:30pm - 4:45pm
Lind 305
Benedek Valko (University of Wisconsin, Madison)
By the Hilbert-Polya conjecture the critical zeros of the Riemann zeta function correspond to the eigenvalues of a self adjoint operator. By a conjecture of Dyson and Montgomery the critical zeros (after a certain rescaling) look like the bulk eigenvalue limit point process of the Gaussian Unitary Ensemble. It is natural to ask if this point process can we described as the spectrum of a random self adjoint operator. I will show that this is indeed the case: for any beta>0 the bulk limit of the Gaussian beta ensemble can be obtained as the spectrum of a self adjoint random differential operator. I will describe the operator and show how to derive it as the limit of operators corresponding to finite ensembles in the circular beta ensemble case.

(Joint with Balint Virag)
MSC Code: