Determinants of Laplacians on random hyperbolic surfaces

We investigate the behaviour of the regularized determinant of the Laplace-Beltrami operator on compact hyperbolic surfaces when the genus goes to infinity. We show that for all popular models of random surfaces, with high probability as the genus goes to infinity, the determinant has an exponential growth with a universal exponent. Limit results for some moments of the logarithm of the determinant are then derived.