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

Abstract

For Π ⊂ 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 LΠ∩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 Π is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.