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The Laplacian and Dirac operators on critical planar graphs. (English) Zbl 1038.58037
An isoradial embedding (also called critical embedding) of a planar graph is a regular polyhedral embedding in which all circumcircles have the same radius. Explicit expressions are presented for the logarithm of the normalized determinant of the discrete Laplacian and the discrete Dirac operator for these graphs. The definition of the Dirac operator is inspired by the dimer model from statistical mechanics where the Dirac operator is called Kasteleyn matrix and where the partition function is the determinant of this matrix. Also a general construction of discrete holomorphic functions and discrete harmonic functions is discussed.

MSC:
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J52 Determinants and determinant bundles, analytic torsion
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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