Determinants of Laplacians on random hyperbolic surfaces

TL;DR

This paper studies how the regularized determinant of the Laplace-Beltrami operator behaves on random hyperbolic surfaces as their genus increases, revealing universal exponential growth patterns.

Contribution

It demonstrates that for various models of random hyperbolic surfaces, the determinant exhibits universal exponential growth with high probability as genus increases.

Findings

Determinant grows exponentially with genus

Universal growth rate across different models

Limit results for moments of the logarithm

Abstract

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.