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We show that at any location away from the spectral edge, the eigenvalues ofthe Gaussian unitary ensemble and its general beta siblings converge toSine_beta, a translation invariant point process. This process has a geometricdescription in term of the Brownian carousel, a deterministic function ofBrownian motion in the hyperbolic plane.  The Brownian carousel, a description of the a continuum limit of randommatrices, provides a convenient way to analyze the limiting point processes. Weshow that the gap probability of Sine_beta is continuous in the gap size and$\beta$, and compute its asymptotics for large gaps. Moreover, the stochasticdifferential equation version of the Brownian carousel exhibits a phasetransition at beta=2.

Continuum limits of random matrices and the Brownian carousel