The Sineβ operator

• 成果类型: Article
• 署名作者: Valko, Benedek; Virag, Balint
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of Toronto
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0709-x
• 发表日期: 2017
• 页码: 275-327
• 关键词: level-spacing distributions; SCHRODINGER-OPERATORS; RANDOM MATRICES; zeros; UNIVERSALITY; models

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.