Critical surfaces for general inhomogeneous bond percolation problems

Abstract

We present a method of general applicability for finding exact or accurate approximations to bond percolation thresholds for a wide class of lattices. To every lattice we sytematically associate a polynomial, the root of which in $[0,1]$ is the conjectured critical point. The method makes the correct prediction for every exactly solved problem, and comparison with numerical results shows that it is very close, but not exact, for many others. We focus primarily on the Archimedean lattices, in which all vertices are equivalent, but this restriction is not crucial. Some results we find are kagome: $p_c=\mathrm{0.524430...}$, $(3,12^2):p_c=\mathrm{0.740423...}$, $(3^3,4^2):p_c=\mathrm{0.419615...}$, $(3,4,6,4):p_c=\mathrm{0.524821...}$, $(4,8^2):p_c=\mathrm{0.676835...}$, $(3^2,4,3,4)$: $p_c=\mathrm{0.414120...}$ . The results are generally within $10^{-5}$ of numerical estimates. For the inhomogeneous checkerboard and bowtie lattices, errors in the formulas (if they are not exact) are less than $10^{-6}$.

Submitted to J. Stat. Mech

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• Bond percolation thresholds on Archimedean lattices from critical polynomial roots
• Crossover from Isotropic to Directed Percolation
• Polynomial sequences for bond percolation critical thresholds
• The percolation critical polynomial as a graph invariant
• Dimer Covering and Percolation Frustration
• 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 on high dimensional $D_n$ and dense packing lattices
• Percolation thresholds of randomly rotating patchy particles on Archimedean lattices
• Potts critical frontiers of inhomogeneous and asymmetric bow-tie lattices