The percolation critical polynomial as a graph invariant

TL;DR

This paper introduces a generalized critical polynomial for percolation thresholds as a graph invariant, computable via deletion-contraction, providing highly accurate approximations for complex lattices like kagome.

Contribution

It demonstrates that the percolation critical polynomial can be viewed as a graph invariant similar to the Tutte polynomial, enabling precise computational approximations.

Findings

Computed critical polynomial for kagome lattice with up to 36 bonds

Achieved a critical probability estimate with a difference of only 6.9 x 10^{-7} from numerical value

Validated the polynomial as a highly accurate approximation method

Abstract

Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction p_c=0.52440572..., which differs from the numerical value, p_c=0.52440503(5), by only 6.9 x 10^{-7}.