Potts and percolation models on bowtie lattices

TL;DR

This paper derives the exact critical thresholds for Potts and percolation models on bowtie lattices, confirming known results and providing highly accurate numerical simulations for noninteger q and percolation thresholds.

Contribution

It presents the exact critical frontiers for Potts models on bowtie lattices and validates them with high-precision Monte Carlo simulations, including for noninteger q and percolation thresholds.

Findings

Exact critical thresholds for Potts models on bowtie lattices.

Monte Carlo simulations confirm theoretical predictions with high precision.

Percolation thresholds differ from isotropic lattices due to lattice anisotropy.

Abstract

We give the exact critical frontier of the Potts model on bowtie lattices. For the case of $q=1$, the critical frontier yields the thresholds of bond percolation on these lattices, which are exactly consistent with the results given by Ziff et al [J. Phys. A 39, 15083 (2006)]. For the $q=2$ Potts model on the bowtie-A lattice, the critical point is in agreement with that of the Ising model on this lattice, which has been exactly solved. Furthermore, we do extensive Monte Carlo simulations of Potts model on the bowtie-A lattice with noninteger $q$. Our numerical results, which are accurate up to 7 significant digits, are consistent with the theoretical predictions. We also simulate the site percolation on the bowtie-A lattice, and the threshold is $s_c=0.5479148(7)$. In the simulations of bond percolation and site percolation, we find that the shape-dependent properties of the percolation model on the bowtie-A lattice are somewhat different from those of an isotropic lattice, which may be caused by the anisotropy of the lattice.