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-Pólya 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 Schrödinger 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 Itô's classification of affine Brownian motions with the classification of limits of random matrix ensembles.

51 pages, 2 figures

References in corpus (2)

• From Random Matrices to Stochastic Operators
• Diffusion at the random matrix hard edge

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