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Continuum limits of random matrices and the Brownian carousel

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Abstract

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine  β , a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane.

The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine  β is continuous in the gap size and β, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at β=2.

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Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53705, USA

    Benedek Valkó

  2. Department of Mathematics, University of Toronto, Toronto, ON M5S, 2E4, Canada

    Bálint Virág

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  1. Benedek Valkó
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  2. Bálint Virág
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Correspondence to Bálint Virág.

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Valkó, B., Virág, B. Continuum limits of random matrices and the Brownian carousel. Invent. math. 177, 463–508 (2009). https://doi.org/10.1007/s00222-009-0180-z

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  • DOI: https://doi.org/10.1007/s00222-009-0180-z

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