Polynomial sequences for bond percolation critical thresholds

TL;DR

This paper introduces a polynomial-based method to estimate bond percolation thresholds on specific lattices, achieving high-precision results and discussing potential limitations for certain graphs like the kagome lattice.

Contribution

It develops a polynomial approximation approach for bond percolation thresholds, providing highly accurate estimates and insights into the algebraic nature of these thresholds.

Findings

Estimated thresholds closely match numerical results with deviations around 10^{-5}

Method produces polynomials whose roots approximate the bond thresholds

Discussion on the potential non-polynomial nature of thresholds for some lattices

Abstract

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4,6,12) and (3^4,6) lattices using the linearity approximation described in (Scullard and Ziff, J. Stat. Mech. P03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, p_c(4,6,12)=0.69377849... and p_c(3^4,6)=0.43437077..., compared with Parviainen's numerical results of p_c \approx 0.69373383 and p_c \approx 0.43430621 . These deviations are of the order 10^{-5}, as is standard for this method, although they are outside Parviainen's typical standard error of 10^{-7}. Deriving thresholds in this way for a given lattice leads to a polynomial with integer coefficients, the root in [0,1] of which gives the estimate for the bond threshold. I show how the method can be refined, leading to a sequence of higher order polynomials making predictions that likely converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients.