The Laplacian and Dirac operators on critical planar graphs

  title={The Laplacian and Dirac operators on critical planar graphs},
  author={Richard W. Kenyon},
  journal={Inventiones mathematicae},
  • R. Kenyon
  • Published 2002
  • Mathematics, Physics
  • Inventiones mathematicae
Abstract.On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and Dirac 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… Expand
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