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

Rafael Leon Greenblatt

Published 2021-02-09Version 1

For $\Pi \subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of 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 $L$ in which the term of order 1 is the logarithm of the zeta-regularized determinant of the corresponding continuum Laplacian. When $\Pi$ is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.