  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
$\beta$, and compute its asymptotics for large gaps. Moreover, the stochastic
differential equation version of the Brownian carousel exhibits a phase
transition at beta=2.
