For Weil-Petersson random hyperbolic surfaces of large genus, the normalized log-determinant of the Laplacian converges in L^1 with polynomial rate, and its beta-moments converge to E^beta for beta below 2 but diverge for beta at least 2.
The spectrum of hyperbolic surfaces
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Averages of determinants of Laplacians over moduli spaces for large genus
For Weil-Petersson random hyperbolic surfaces of large genus, the normalized log-determinant of the Laplacian converges in L^1 with polynomial rate, and its beta-moments converge to E^beta for beta below 2 but diverge for beta at least 2.