  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.
