Random matrices, differential operators and carousels

Benedek Valko, University of Wisconsin
Fine Hall 214

The Sine_\beta process is the bulk limit process of the Gaussian beta-ensembles. We show that this process can be obtained as the spectrum of a self-adjoint random differential operator. The result connects the Montgomery-Dyson conjecture about the Sine_2 process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Polya conjecture, and de Brange’s approach of possibly proving the Riemann hypothesis. Our proof relies on the Brownian carousel representation of the Sine_beta process and a connection between hyperbolic carousels and first order differential operators acting on R^2 valued functions. [Joint with B. Virag (Toronto).]