科研之友
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摘要
We show that , the bulk limit of the Gaussian -ensembles is the spectrum of a self-adjoint random differential operator where is the positive definite matrix representation of hyperbolic Brownian motion with variance in logarithmic time. The result connects the Montgomery-Dyson conjecture about the process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Plya 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 -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 -ensembles that has so far been missing in the literature. In particular, we connect It's classification of affine Brownian motions with the classification of limits of random matrix ensembles.
- 出版日期2017-7
