Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces—a central limit theorem and almost sure RMT statistics

Abstract We study smooth linear spectral statistics of twisted Laplacians on random n -covers of a fixed compact hyperbolic surface X . We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy window around a fixed energy level when averaged over the space of all degree n covers of X . The second is the centered energy variance of a typical surface, a quantity similar to the normal energy variance. In the first case, we show a central limit theorem. Specifically, we show that the distribution of such fluctuations tends to a Gaussian with variance given by the corresponding quantity for the Gaussian Orthogonal/Unitary Ensemble (GOE/GUE). In the second case, we show that the centered energy variance of a typical random n -cover is that of the GOE/GUE. In both cases, we consider a double limit where first we let n —the covering degree—go to ∞ then let L → ∞ where 1/ L is the window length. A fundamental component of our proofs are the results we prove in [11] which concern the random cover model for random surfaces.