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Holomorphie discrète et modèle d'Ising

Abstract : My thesis generalizes the notion of criticality for the Ising model in two dimensions and defines a newtheory of discrete holomorphic functions on a cellular decomposition of a Riemann surface. The Ising model converges in the thermodynamical limit to a continuous conformal field theory, on square or triangular lattices near the critical temperature. We extend criticality to more general cellular decompositions. The key point is to double the decomposition, and to consider its dual. We define holomorphy with respect to a given metric by a straightforward discretization of the Cauchy-Riemmann equation. Classical theorems still hold and are proved essentially the same way (Dirichlet principle): harmonicity, explicit abelian differentials, poles, residue theorem. There are differences: pointwise product doesn't preserve holomorphy, poles of higher orders are {\sl multipoles} of order one; the dimension of the space of abelian differentials is twice the genus. We define the notion of semi-criticality, needed to define, given a discrete holomorphic function $f$ and a local map Z, a discrete closed $1$-form $fdZ$, it is holomorphic, on critical maps. This class contains lattices but much more. The continuous limit of a convergent sequence of discrete holomorphic functions on a converging sequence of semi-critical maps, is a holomorphic function. We apply this theory to the Ising model. In the case of square, triangular and hexagonal lattices, we prove that our criticality is equivalent to statistical criticality, lengths and coupling constants being related. On every map, we setup a discretized version of the Dirac equation without mass and the existence of a solution is equivalent to criticality. Using this solution, we contract the spinor into holomorphic and anti-holomorphic parts on any critical map, exhibiting the existence of the analogue of discrete conformal blocks.
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Contributor : Christian Mercat <>
Submitted on : Tuesday, October 22, 2002 - 9:44:25 AM
Last modification on : Friday, June 19, 2020 - 9:10:04 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 6:10:35 PM


  • HAL Id : tel-00001851, version 1



Christian Mercat. Holomorphie discrète et modèle d'Ising. Mathématiques [math]. Université Louis Pasteur - Strasbourg I, 1998. Français. ⟨tel-00001851⟩



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