Determinants of Laplacians on random hyperbolic surfaces

Abstract

For sequences (X_j) of random closed hyperbolic surfaces with volume Vol(X_j) tending to infinity, we prove that there exists a universal constant E > 0 such that for all ϵ > 0, the regularized determinant of the Laplacian satisfies

$${{\log \det ({\Delta _{{X_j}}})} \over {{\rm{Vol}}({X_j})}} \in [E -\epsilon ,E +\epsilon]$$

with high probability as j → +⋡. This result holds for various models of random surfaces, including the Weil–Petersson model.