  On a periodic planar graph whose edge weights satisfy a certain simple
geometric condition, the discrete Laplacian and d-bar operators have the
property that their determinants and inverses only depend on the local geometry
of the graph.
  We obtain explicit expressions for the logarithms of the (normalized)
determinants, as well as the inverses of these operators.
  We relate the logarithm of the determinants to the volume plus mean curvature
of an associated hyperbolic ideal polyhedron.
  In the associated dimer and spanning tree models, for which the determinants
of d-bar and the Laplacian respectively play the role of the partition
function, this allows us to compute the entropy and correlations in terms of
the local geometry.
  In addition, we define a continuous family of special discrete analytic
functions, which, via convolutions gives a general process for constructing
discrete analytic functions and discrete harmonic functions on critical planar
graphs.
