Determinants of Laplacians on converging hyperbolic surfaces

TL;DR

This paper investigates how the determinants of Laplacians on hyperbolic surfaces behave as the surfaces grow and converge locally, revealing that under certain conditions, their normalized logarithms tend to a universal constant.

Contribution

It establishes a link between the geometric convergence of hyperbolic surfaces and the asymptotic behavior of their Laplacian determinants, introducing conditions involving short geodesics.

Findings

Normalized log determinants converge to a constant

Convergence depends on short geodesic length sums

Results depend only on the law of the limiting surface

Abstract

Let $S_k$ be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of $S_k$ converges to a constant depending only the law of the limiting surface.