The Sine$_beta$ operator

We show that Sine$_beta$, the bulk limit of the Gaussian $beta$-ensembles is the spectrum of a self-adjoint random differential operator [ fto 2 {R_t^{-1}} left[ begin{array}{cc} 0 &-tfrac{d}{dt} tfrac{d}{dt} &0 end{array} right] f, qquad f:[0,1)to mathbb R^2, ] where $R_t$ is the positive definite matrix representation of hyperbolic Brownian motion with variance $4/beta$ in logarithmic time. 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 Branges attempt to prove the Riemann hypothesis. We identify the Brownian carousel as the Sturm-Liouville phase function of this operator. We provide similar operator representations for several other finite dimensional random ensembles and their limits: finite unitary or orthogonal ensembles, Hua-Pickrell ensembles and their limits, hard-edge $beta$-ensembles, as well as the Schrodinger point process. In this more general setting, hyperbolic Brownian motion is replaced by a random walk or Brownian motion on the affine group. Our approach provides a unified framework to study $beta$-ensembles that has so far been missing in the literature. In particular, we connect It^os classification of affine Brownian motions with the classification of limits of random matrix ensembles.