Elsevier

Advances in Mathematics

Volume 228, Issue 3, 20 October 2011, Pages 1590-1630
Advances in Mathematics

Discrete complex analysis on isoradial graphs

https://doi.org/10.1016/j.aim.2011.06.025Get rights and content
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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, Greenʼs 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.

MSC

39A12
52C20
60G50

Keywords

Discrete harmonic functions
Discrete holomorphic functions
Discrete potential theory
Isoradial graphs
Random walk

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