Critical frontier of the Potts and percolation models in triangular-type and kagome-type lattices I: Closed-form expressions

Abstract

We consider the Potts model and the related bond, site, and mixed site-bond percolation problems on triangular-type and kagome-type lattices, and derive closed-form expressions for the critical frontier. For triangular-type lattices the critical frontier is known, usually derived from a duality consideration in conjunction with the assumption of a unique transition. Our analysis, however, is rigorous and based on an established result without the need of a uniqueness assumption, thus firmly establishing all derived results. For kagome-type lattices the exact critical frontier is not known. We derive a closed-form expression for the Potts critical frontier by making use of a homogeneity assumption. The closed-form expression is new, and we apply it to a host of problems including site, bond, and mixed site-bond percolation on various lattices. It yields exact thresholds for site percolation on kagome, martini, and other lattices, and is highly accurate numerically in other applications when compared to numerical determination.

22 pages, 13 figures

References in corpus (7)

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• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices
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Cited by in corpus (11)

• Critical manifold of the kagome-lattice Potts model
• Transfer matrix computation of critical polynomials for two-dimensional Potts models
• Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
• The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
• Polynomial sequences for bond percolation critical thresholds
• The percolation critical polynomial as a graph invariant
• Potts and percolation models on bowtie lattices
• Percolation on Hypergraphs with Four-Edges
• The computation of generalized percolation critical polynomials by the deletion-contraction algorithm
• Critical temperatures of the three- and four-state Potts models on the kagome lattice
• Critical manifold of the Potts model: Exact results and homogeneity approximation