Discrete Riemann Surfaces and the Ising Model

By C. Mercat. 2001

280 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e1f27967e80c90ecbf2e4fa0f5ee8bc0c83358f

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Citations in the corpus

Brane tilings and their applications (2008, background): “The concept of isoradial embedding is known in mathematics since long ago [11, 71, 72], but their physical meaning is first proposed in [70].”
Finite elements in computational electromagnetism (2002, unknown): “We remind readers that the discrete formulation of electrodynamics, based on cochains and discrete Hodge operators, is only one example of a discrete ‘lattice model’, which plays an increasingly important role in physics (Dezin 1995, Mercat 2001).”
Discrete analytic functions: An exposition (2004, background): “As we mentioned in the introduction, discrete analytic functions and holomorphic forms on general maps were introduced by Mercat [18].”
Geometric representations of graphs (2014, background): “Mercat [136] defined critical maps: these are maps which admit a critical analytic function.”
Convergence of the Cotangent Formula: An Overview (2008, background): “In particular, we show that mean curvature vectors converge in the sense of distributions, but fail to converge in L2.”
A discrete uniformization theorem for polyhedral surfaces (2013, methodology): “Other different versions of discrete Riemann mapping theorem and discrete conformality have appeared in the work by Thurston [ 38], Cannon [ 12], Schramm [ 36], Glickenstein [ 18], Hersonsky [ 21], Mercat [30] and others.”
Dimer models and the special McKay correspondence (2009, background): “ir of such zigzag paths on a consistent dimer model in Figure 5.6. Figure 5.6: A consistent non-isoradial dimer model 5.3 Isoradial dimer models The following notion is due to Duﬃn [Duf68] and Mercat [Mer01]: Deﬁnition 5.2. A dimer model is isoradial if one can choose an embedding of the graph into ”
Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, methodology): “It was introduced in [47], although similar objects appeared earlier in Kadanoff and Ceva [22] (without complex weight and boundary problem discussions) and in Mercat [33] (again without discussion of boundary problem and covariance).”
On the convergence of metric and geometric properties of polyhedral surfaces (2007, methodology): “Combinatorial approaches were introduced by Mercat [13] and Bobenko et.”
Variational principles for circle patterns and Koebe’s theorem (2002, background): “rtices corresponding to vertices of Σ. Quad-graphs turn out to be crucial for deﬁning discrete integrable systems [BS02] (in particular those related to circle patterns) and discrete Riemann surfaces [Mer01]. Suppose we are given a positive radius rf for each face f of Σ and an angle θe ∈ (0,π) for ”
Integrable systems on quad-graphs (2001, methodology): “ts neighbors connected with v 0 by the edges e1 = (v 0,v 1),...en = (v 0,vn) ∈ E(G), then the face from F(G∗) corresponding to v 0 is deﬁned by its boundary e∗ 1 ∪...∪ ∗n (cf. Fig. 2). Now, following [M], we introduce a new complex, the doubleD, constructed from G, G∗ (notice that our terminology he”
Numerical Computation of Surface Conformal Mappings (2012, methodology): “Another approach of discrete holomorphy was introduced in [52] by discretization of the Cauchy-Riemann equation.”
Quivers, tilings, branes and rhombi (2005, background): “This might be done in the context of discrete Riemann surfaces [41].”