Universal asymptotic correlation functions for point group C_{6v} and an observation for triangular lattice Q-state...

We investigate a universal curve in asymptotic correlation functions of off-critical systems that possess C_{6v} symmetry following the argument for C_{4v} symmetry in our previous paper [Phys. Rev. E 102, 032141 (2020)2470-004510.1103/PhysRevE.102.032141]. Unlike the C_{4v} case, a minimal asymptotic form exists, which contains only two free parameters: the normalization constant and the modulus of the universal curve. We perform large-scale Monte Carlo simulations of the triangular lattice Q-state Potts model above the transition temperature. For Q=1, 2, 3, and 4, we successfully obtain numerical evidence that the minimal form gives the leading asymptotic behavior. We also discuss the possibility that the corrections to the minimal form are expressed using this form as a building block. From the minimal form with optimized parameters, we derive the equilibrium crystal shape of the honeycomb lattice Potts model, which is given by an algebraic curve of genus 1 and is universal among models with C_{6v}. Although the curve differs from those obtained in the C_{4v} case, the latter curves also have genus 1. We indicate that the birational equivalence concept can play an important role in comparing asymptotic forms for different point group symmetries, for example, C_{6v} and C_{4v}.