Aperiodic Non-Isomorphic Lattices with Equivalent Percolation and Random-Cluster Models

Abstract

We explicitly construct an uncountable class of infinite aperiodic plane graphs which have equal, and explicitly computable, bond percolation thresholds. Furthermore for both bond percolation and the random-cluster model all large scale properties, such as the values of the percolation threshold and the critical exponents, of the graphs are equal. This equivalence holds for all values of $p$ and all $q\in[0,\infty]$ for the random-cluster model.

The graphs are constructed by placing a copy of a rotor gadget graph or its reflection in each hyperedge of a connected self-dual 3-uniform plane hypergraph lattice. The exact bond percolation threshold may be explicitly determined as the root of a polynomial by using a generalised star-triangle transformation. Related randomly oriented models share the same bond percolation threshold value.