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

Abstract

For Π⊂R2, 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 LΠ∩Z2 with Dirichlet boundary conditions has an asymptotic expression for large L involving the zeta-regularized determinant of the associated continuum Laplacian. When Π is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.