Exact bond percolation thresholds in two dimensions

TL;DR

This paper extends exact bond percolation threshold solutions to a broader class of two-dimensional lattices using a triangle-triangle transformation, including new solutions for self-dual graphs.

Contribution

It introduces a method to find exact percolation thresholds for self-dual graphs decomposable into triangles, expanding the set of solvable lattice problems.

Findings

Extended exact thresholds to new classes of graphs

Generalized solutions for bow-tie lattice

Identified multiple classes of self-dual graphs

Abstract

Recent work in percolation has led to exact solutions for the site and bond critical thresholds of many new lattices. Here we show how these results can be extended to other classes of graphs, significantly increasing the number and variety of solved problems. Any graph that can be decomposed into a certain arrangement of triangles, which we call self-dual, gives a class of lattices whose percolation thresholds can be found exactly by a recently introduced triangle-triangle transformation. We use this method to generalize Wierman's solution of the bow-tie lattice to yield several new solutions. We also give another example of a self-dual arrangement of triangles that leads to a further class of solvable problems. There are certainly many more such classes.