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

摘要

In a recent paper (arXiv:0911.2514), one of us (FYW) considered the Pottsmodel and bond and site percolation on two general classes of two-dimensionallattices, the triangular-type and kagome-type lattices, and obtainedclosed-form expressions for the critical frontier with applications to variouslattice models. For the triangular-type lattices Wu's result is exact, and forthe kagome-type lattices Wu's expression is under a homogeneity assumption. Thepurpose of the present paper is two-fold: First, an essential step in Wu'sanalysis is the derivation of lattice-dependent constants $A, B, C$ for variouslattice models, a process which can be tedious. We present here a derivation ofthese constants for subnet networks using a computer algorithm. Secondly, bymeans of a finite-size scaling analysis based on numerical transfer matrixcalculations, we deduce critical properties and critical thresholds of variousmodels and assess the accuracy of the homogeneity assumption. Specifically, weanalyze the $q$-state Potts model and the bond percolation on the 3-12 andkagome-type subnet lattices $(n\times n):(n\times n)$, $n\leq 4$, for which theexact solution is not known. To calibrate the accuracy of the finite-sizeprocedure, we apply the same numerical analysis to models for which the exactcritical frontiers are known. The comparison of numerical and exact resultsshows that our numerical determination of critical thresholds is accurate to 7or 8 significant digits. This in turn infers that the homogeneity assumptiondetermines critical frontiers with an accuracy of 5 decimal places or higher.Finally, we also obtained the exact percolation thresholds for site percolationon kagome-type subnet lattices $(1\times 1):(n\times n)$ for $1\leq n \leq 6$.