Discrete complex analysis on isoradial graphs

Dmitry Chelkak, Stanislav Smirnov

Research output: Contribution to journalArticlepeer-review

74 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1590-1630
Number of pages41
JournalAdvances in Mathematics
Issue number3
Publication statusPublished - 20 Oct 2011
Externally publishedYes


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


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