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

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