The Sine$_\beta$ operator

Abstract

We show that Sine$_\beta$, the bulk limit of the Gaussian $\beta$-ensembles is the spectrum of a self-adjoint random differential operator \[ f\to 2 {R_t^{-1}} \left[ \begin{array}{cc} 0 &-\tfrac{d}{dt} \tfrac{d}{dt} &0 \end{array} \right] f, \qquad f:[0,1)\to \mathbb R^2, \] 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-P\'olya conjecture and de Brange's 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 Schr\"odinger 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\^o's classification of affine Brownian motions with the classification of limits of random matrix ensembles.