Discrete Riemann Surfaces and the Ising Model |
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Authors: | Christian Mercat |
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Institution: | (1) Université Louis Pasteur, Strasbourg, France. E-mail: mercat@math.u-strasbg.fr, FR |
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Abstract: | 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.
Received: 23 May 2000/ Accepted: 21 November 2000 |
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