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

TL;DR

This paper derives rigorous closed-form expressions for the critical thresholds of Potts and percolation models on triangular and kagome lattices, providing new exact results and high-accuracy approximations for various lattice percolation problems.

Contribution

It provides the first rigorous derivation of the critical frontier for kagome-type lattices and extends known results for triangular lattices without relying on the assumption of a unique transition.

Findings

Exact critical thresholds for site percolation on kagome and related lattices.

New closed-form expression for Potts critical frontier on kagome-type lattices.

High-accuracy numerical estimates for various percolation problems.

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.