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

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 unique, and we apply it to a host of problems including site, bond, and mixed site-bond percolations 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.