Discrete complex analysis on isoradial graphs

Abstract

We study discrete complex analysis and potential theory on a large family of planar graphs, the so-called isoradial ones. Along with discrete analogues of several classical results, we prove uniform convergence of discrete harmonic measures, Green's functions and Poisson kernels to their continuous counterparts. Among other applications, the results can be used to establish universality of the critical Ising and other lattice models.

35 pages, 4 figures. Several changes: Sections 2.4-2.6 expanded; Theorem 3.10 added; Section 3.4 rewritten

References in corpus (3)

• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Approximation of conformal mappings by circle patterns
• Discrete Polynomials and Discrete Holomorphic Approximation

Cited by in corpus (8)

• Exact bosonization of the Ising model
• Conformal invariance of lattice models
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• Connection probabilities and RSW-type bounds for the FK Ising model
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• The energy density in the planar Ising model
• Holomorphic Spinor Observables in the Critical Ising Model
• Discrete Dirac operators on Riemann surfaces and Kasteleyn matrices