Averages of determinants of Laplacians over moduli spaces for large genus

Abstract

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 $\beta \in [1,2)$.

Journal of the London Mathematical Society, to appear, 21 pages, comments are welcome

References in corpus (6)

• Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-\epsilon$
• A random cover of a compact hyperbolic surface has relative spectral gap $\frac{3}{16}-\epsilon$
• Large genus asymptotics for lengths of separating closed geodesics on random surfaces
• Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus
• Arbitrarily small spectral gaps for random hyperbolic surfaces with many cusps
• Prime geodesic theorem and closed geodesics for large genus