Averages of determinants of Laplacians over moduli spaces for large genus
Yuxin He, Yunhui Wu
TL;DR
This work analyzes the determinants of Laplacians on random hyperbolic surfaces in the Weil-Petersson model for large genus. It ties the regularized determinant to Selberg zeta data through $\log \det(\Delta_X)=4\pi(g-1)E+\gamma_0-\int_{0}^{1}\frac{S_X(t)}{t}dt-\int_{1}^{\infty}\frac{S_X(t)-1}{t}dt$, and develops WP-volume and geodesic-count machinery to study the distribution of $\frac{\log \det(\Delta_X)}{4\pi(g-1)}$ as $g\to\infty$. Under Naud’s concentration result for $\log Z_0'(1)$, the paper proves that the $\beta$-th moments converge to $E^{\beta}$ for $\beta\in[1,2)$ and that the mean deviation decays with a positive rate $g^{-\delta}$ (with $\delta$ explicitly obtainable, e.g. $\delta>0.1$). The combination of Mirzakhani’s volume bounds, geodesic counting, and sharp decompositions of the Selberg trace data yields a precise probabilistic picture of Laplacian determinants on large-genus random surfaces, with implications for understanding spectral invariants in geometric topology and mathematical physics.
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 $\log \det(Δ_{X})$ 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 $\left|\frac{\log \det(Δ_{X})}{4π(g-1)}-E \right|$ over $\mathcal{M}_g$ has rate of decay $g^{-δ}$ for some uniform constant $δ\in (0,1)$; (2) the expected value of $\left|\frac{\log \det(Δ_{X})}{4π(g-1)}\right|^β$ over $\mathcal{M}_g$ approaches to $E^β$ whenever $β\in [1,2)$.