Polynomial sequences for bond percolation critical thresholds

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.

submitted to Journal of Statistical Mechanics

References in corpus (14)

• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Percolation thresholds on 2D Voronoi networks and Delaunay triangulations
• Estimation of Bond Percolation Thresholds on the Archimedean Lattices
• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices
• Critical surfaces for general bond percolation problems
• Universal condition for critical percolation thresholds of kagome-like lattices
• Rigorous confidence intervals for critical probabilities
• Critical surfaces for general inhomogeneous bond percolation problems
• Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
• Percolation in Networks with Voids and Bottlenecks
• Critical frontier of the Potts and percolation models in triangular-type and kagome-type lattices I: Closed-form expressions
• Exact factorization of correlation functions in 2-D critical percolation
• Equality of bond percolation critical exponents for pairs of dual lattices
• Universal crossing probability in anisotropic systems

Cited by in corpus (8)

• Critical manifold of the kagome-lattice Potts model
• Critical polynomials in the nonplanar and continuum percolation models
• Bond percolation thresholds on Archimedean lattices from critical polynomial roots
• The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
• The percolation critical polynomial as a graph invariant
• Universal Critical Behavior of Percolation in Orientationally Ordered Janus Particles and Other Anisotropic Systems
• The computation of generalized percolation critical polynomials by the deletion-contraction algorithm
• Percolation thresholds of randomly rotating patchy particles on Archimedean lattices