Using Symmetry to Improve Percolation Threshold Bounds

Abstract

We show that symmetry, represented by a graph's automorphism group, can be used to greatly reduce the computational work for the substitution method. This allows application of the substitution method over larger regions of the problem lattices, resulting in tighter bounds on the percolation threshold p c . We demonstrate the symmetry reduction technique using bond percolation on the (3, 12 2 ) lattice, where we improve the bounds on p c from (0.738598, 0.744900) to (0.739399, 0.741757), a reduction of more than 62% in width, from 0.006302 to 0.002358.

Self-dual Planar Hypergraphs and Exact Bond Percolation Thresholds

“…The remarkable fact that allows exact bond percolation threshold values to be obtained is that it is possible to choose the parameters p and q so that the two probability measures are exactly equal. (Note that in cases with more boundary vertices, where the probability measures cannot be made equal, the concept of stochastic ordering of probability measures may be used to determine mathematically rigorous bounds for percolation thresholds, using the substitution method [12,13,22,24,25,26,27].) By the duality relationship between G and G * , we have that for each configuration of open and closed edges, the following five statements hold: While these statements are intuitively clear by drawing diagrams, the proofs of these statements rely on duality.…”

Predictions of bond percolation thresholds for the kagomé and Archimedean(3,122)lattices

“…This work is largely carried out by Wierman and co-workers [19,20], using a technique called substitution. The method is such that continual refinements are possible and the most current rigorous bounds are [21]:…”

A Disproof of Tsallis' Bond Percolation Threshold Conjecture for the Kagome Lattice

“…The substitution method compares probabilities of connections between vertices in the percolation model on an unsolved lattice with those on a solved reference lattice. Previous attempts to disprove the Tsallis conjecture, by May and Wierman [17,18], used the hexagonal lattice bond model as the reference model, since the hexagonal lattice has some structural similarity with the kagome lattice. Since then, a class of new exact solutions has been established.…”

“…Our bounds are obtained by the substitution method, using stochastic ordering to compare an unsolved percolation model to an exactly-solved model. In addition to the kagome lattice bond model, it has been applied to several other percolation models [17,18,33,34,35,36]. The initial motivation for the development of the substitution method was to understand the approach of Ottavi, who proposed bounds of 0.522372 p c 0.528924 for the kagome lattice bond percolation threshold.…”