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 $\log \det(\Delta_{X})$ of Laplacian as a function on $\mathcal{M}_g$ and show that there exists a universal constant $E>0$ such that as $g\to \infty$, (1) the expected value of $\left|\frac{\log \det(\Delta_{X})}{4\pi(g-1)}-E \right|$ over $\mathcal{M}_g$ has rate of decay $g^{-\delta}$ for some uniform constant $\delta \in (0,1)$; (2) the expected value of $\left|\frac{\log \det(\Delta_{X})}{4\pi(g-1)}\right|^\beta$ over $\mathcal{M}_g$ approaches to $E^\beta$ whenever $\beta \in [1,2)$.