Polynomial sequences for bond percolation critical thresholds

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.