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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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