The percolation critical polynomial as a graph invariant

Abstract

Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0,1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction p_c=0.52440572..., which differs from the numerical value, p_c=0.52440503(5), by only 6.9 x 10^{-7}.

References in corpus (4)

• Critical surfaces for general bond percolation problems
• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices
• Universal condition for critical percolation thresholds of kagome-like lattices
• Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis

Cited by in corpus (12)

• Critical manifold of the kagome-lattice Potts model
• Critical polynomials in the nonplanar and continuum percolation models
• Transfer matrix computation of critical polynomials for two-dimensional Potts models
• Bond percolation thresholds on Archimedean lattices from critical polynomial roots
• On the growth constant for square-lattice self-avoiding walks
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Exact percolation probabilities for a square lattice: Site percolation on a plane, cylinder, and torus
• Duality with real-space renormalization and its application to bond percolation
• The computation of generalized percolation critical polynomials by the deletion-contraction algorithm
• Percolation thresholds of randomly rotating patchy particles on Archimedean lattices
• Transfer matrix computation of generalised critical polynomials in percolation
• Critical points of the random cluster model with Newman-Ziff sampling