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

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.

arXiv comment43 pages, 6 figures. Moderate revisions including restatement of the main result

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