Discrete and zeta-regularized determinants of the Laplacian on polygonal domains with Dirichlet boundary conditions

TL;DR

This paper investigates the asymptotic behavior of the discrete Laplacian's determinant on polygonal domains with Dirichlet conditions, connecting it to the zeta-regularized continuum Laplacian determinant, including non-simply connected cases.

Contribution

It extends known asymptotic results to polygonal domains with complex topology, linking discrete and continuum Laplacian determinants via zeta-regularization.

Findings

Asymptotic expansion of the discrete Laplacian determinant for large domain scaling.

Extension of results to non-simply connected domains with monodromy conditions.

Connection established between discrete and continuum Laplacian determinants.

Abstract

For $\Pi \subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $L\Pi \cap \mathbb{Z}^2$ with Dirichlet boundary conditions has an asymptotic expansion for large $L$ involving the zeta-regularized determinant of the associated continuum Laplacian. When $\Pi$ is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.