GOE statistics on the moduli space of surfaces of large genus

For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space $\mathcal{M}_{g}$ of all genus g surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the short window limit, we recover GOE statistics for the variance. The proof makes essential use of Mirzakhani’s integration formula.