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 be described as the spectrum of a random self adjoint operator. We show that this is indeed the case: in fact 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.

(Joint with Balint Virag)