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Discrete Riemann surfaces and the Ising model. (English) Zbl 1043.82005
Summary: We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl’s lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
39A12 Discrete version of topics in analysis
58J99 Partial differential equations on manifolds; differential operators
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