The computation of generalized percolation critical polynomials by the deletion-contraction algorithm

TL;DR

This paper extends the concept of critical polynomials for bond percolation thresholds to arbitrary periodic lattices, using a computer-implemented deletion-contraction algorithm to compute generalized polynomials for larger bases, improving approximation accuracy.

Contribution

It introduces a computational method to calculate generalized critical polynomials for larger bases on various lattices, enhancing the accuracy of percolation threshold predictions.

Findings

Polynomials computed for bases up to 36 bonds.

Predictions are within 10^{-5} to 10^{-7} of numerical values.

The method applies to all Archimedean lattices except kagome.

Abstract

Although every exactly known bond percolation critical threshold is the root in $[0,1]$ of a lattice-dependent polynomial, it has recently been shown that the notion of a critical polynomial can be extended to any periodic lattice. The polynomial is computed on a finite subgraph, called the base, of an infinite lattice. For any problem with exactly known solution, the prediction of the bond threshold is always correct, regardless of the base chosen. For unsolved problems, the polynomial is referred to as the generalized critical polynomial and provides an approximation that becomes more accurate with increasing number of bonds in the base, appearing to approach the exact answer. The polynomials are computed using the deletion-contraction algorithm, which quickly becomes intractable by hand for more than about 18 bonds. Here, I present generalized critical polynomials calculated with a computer program for bases of up to 36 bonds for all the Archimedean lattices, except the kagome which was considered in an earlier work. The polynomial estimates are generally within $10^{-5}$ to $10^{-7}$ of the numerical values, but the prediction for the $(4,8^2)$ lattice, though not exact, is not ruled out by simulations.