Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis

Abstract

In a recent paper (arXiv:0911.2514), one of us (FYW) considered the Potts model and bond and site percolation on two general classes of two-dimensional lattices, the triangular-type and kagome-type lattices, and obtained closed-form expressions for the critical frontier with applications to various lattice models. For the triangular-type lattices Wu's result is exact, and for the kagome-type lattices Wu's expression is under a homogeneity assumption. The purpose of the present paper is two-fold: First, an essential step in Wu's analysis is the derivation of lattice-dependent constants $A,B,C$ for various lattice models, a process which can be tedious. We present here a derivation of these constants for subnet networks using a computer algorithm. Secondly, by means of a finite-size scaling analysis based on numerical transfer matrix calculations, we deduce critical properties and critical thresholds of various models and assess the accuracy of the homogeneity assumption. Specifically, we analyze the $q$-state Potts model and the bond percolation on the 3-12 and kagome-type subnet lattices $(n\times n):(n\times n)$, $n\le 4$, for which the exact solution is not known. To calibrate the accuracy of the finite-size procedure, we apply the same numerical analysis to models for which the exact critical frontiers are known. The comparison of numerical and exact results shows that our numerical determination of critical thresholds is accurate to 7 or 8 significant digits. This in turn infers that the homogeneity assumption determines critical frontiers with an accuracy of 5 decimal places or higher. Finally, we also obtained the exact percolation thresholds for site percolation on kagome-type subnet lattices $(1\times 1):(n\times n)$ for $1\le n\le 6$.

31 pages,8 figures

References in corpus (13)

• Determination of the bond percolation threshold for the Kagome lattice
• Estimation of Bond Percolation Thresholds on the Archimedean Lattices
• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices
• Universal condition for critical percolation thresholds of kagome-like lattices
• New critical frontiers for the Potts and percolation models
• Critical surfaces for general inhomogeneous bond percolation problems
• 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
• New sum rule identities and duality relation for the Potts $n$-point correlation function
• Exact characterization of O(n) tricriticality in two dimensions
• Critical line of an n-component cubic model
• Critical properties of a dilute O($n$) model on the kagome lattice
• Crossing bonds in the random-cluster model

Cited by in corpus (14)

• Simultaneous analysis of three-dimensional percolation models
• Critical manifold of the kagome-lattice Potts model
• Transfer matrix computation of critical polynomials for two-dimensional Potts models
• Effect of Dimensionality on the Percolation Thresholds of Various $d$-Dimensional Lattices
• The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices
• The percolation critical polynomial as a graph invariant
• Polynomial sequences for bond percolation critical thresholds
• Critical frontier of the Potts and percolation models in triangular-type and kagome-type lattices I: Closed-form expressions
• Random site percolation on honeycomb lattices with complex neighborhoods
• 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