A bond percolation critical probability determination based on the star-triangle transformation

Abstract

The bond percolation critical probability of a planar graph with square and triangular faces, obtained by inserting a diagonal in every other face of the square lattice, is the root of 1-p-6p2+6p3-p5=0 in (0, 1), which is approximately 0.404 518. The proof uses the star-triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical.