Using Symmetry to Improve Percolation Threshold Bounds

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 pc. We demonstrate the symmetry reduction technique using bond percolation on the (3, 12) lattice, where we improve the bounds on pc from (0.738598, 0.744900) to (0.739399, 0.741757), a reduction of more than 62% in width, from 0.006302 to 0.002358.