  We study the fluctuations of smooth linear statistics of Laplace eigenvalues
of compact hyperbolic surfaces lying in short energy windows, when averaged
over the moduli space of surfaces of a given genus. The average is taken with
respect to the Weil-Petersson measure. We show that first taking the large
genus limit, then a short window limit, the distribution tends to a Gaussian.
The variance was recently shown to be given by the corresponding quantity for
the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also
consistent with those in Random Matrix Theory, as conjectured in the physics
literature for a fixed surface.
