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

On the Central Limit Theorem for linear eigenvalue statistics on random surfaces of large genus