Absence of Phase Transition for Antiferromagnetic Potts Models via the Dobrushin Uniqueness Theorem

TL;DR

This paper proves that antiferromagnetic Potts models on various lattices do not undergo phase transitions at certain parameters, using the Dobrushin uniqueness theorem, and establishes bounds for exponential decay of correlations.

Contribution

It provides new bounds for the absence of phase transitions in antiferromagnetic Potts models on different lattices using the Dobrushin theorem.

Findings

Exponential decay of correlations for q > 2r at all temperatures.

Improved bounds for specific 2D lattices: square, triangular, hexagonal, Kagomé.

No phase transition occurs under these bounds.

Abstract

We prove that the $q$-state Potts antiferromagnet on a lattice of maximum coordination number $r$ exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever $q > 2r$. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay for $q \ge 7$), triangular lattice ($q \ge 11$), hexagonal lattice ($q \ge 4$), and Kagom\'e lattice ($q \ge 6$). The proofs are based on the Dobrushin uniqueness theorem.