Bond percolation on isoradial graphs: criticality and universality

TL;DR

This paper proves the criticality and universality of bond percolation on isoradial graphs, extending known results to a broader class of planar graphs using the star-triangle transformation.

Contribution

It establishes the criticality and universality of bond percolation on isoradial graphs, generalizing previous lattice-specific results through the star-triangle transformation.

Findings

Percolation is critical on isoradial graphs.

Critical exponents are constant across the class of isoradial graphs.

Universality holds for models including Penrose tilings.

Abstract

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 2j-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.