Averages of determinants of Laplacians over moduli spaces for large genus

Let ${\mathcal{M}}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. We view the regularized determinant $\log \det ({\mathrm{\Delta }}_X)$ of Laplacian as a function on ${\mathcal{M}}_g$ and show that there exists a universal constant $E>0$ such that as $g\to \mathrm{\infty }$, (1) the expected value of $\mid \frac{\log \det ({\mathrm{\Delta }}_X)}{4\pi (g-1)}-E\mid$ over ${\mathcal{M}}_g$ has rate of decay $g^{-\delta }$ for some uniform constant $\delta \in (0,1)$; (2) the expected value of ${\mid \frac{\log \det ({\mathrm{\Delta }}_X)}{4\pi (g-1)}\mid }^{\beta }$ over ${\mathcal{M}}_g$ approaches to $E^{\beta }$ whenever $\beta \in [1,2)$.