Bond percolation on isoradial graphs: criticality and universality

• Published in: Probability theory and related fields Vol. 159; no. 1-2; pp. 273 - 327
• Main Authors: Grimmett, Geoffrey R., Manolescu, Ioan
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2014; Springer Nature B.V
• Subjects: Bonding; Critical point; Economics; Exponents; Finance; Graph theory; Graphs; Insurance; Lattices; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Percolation; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Quasicrystalline structure; Statistics for Business; Studies; Texts; Theoretical; Transformations; Transporting; Rhombic tiling; Isoradial graph; Inhomogeneous percolation; Yang–Baxter equation; Bond percolation; Arm exponent; Critical exponent; 82B43; Universality; Penrose tiling; Star–triangle transformation; 60K35; Box-crossing; Scaling relations
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-013-0507-y

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star–triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2 j -alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.