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arXiv · 0712.2000

Continuum limits of random matrices and the Brownian carousel

Benedek Valko·Balint Virag
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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 beta siblings converge to Sine_beta, 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_beta 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 beta=2.

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Benedek Valko, Balint Virag. 2009-01-21. Continuum limits of random matrices and the Brownian carousel. https://doi.org/10.1007/s00222-009-0180-z

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