Abstract

We prove that all Gibbs measures of the $q$-state Potts model on ${\mathbb{Z}}^2$ are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular, all Gibbs measures are invariant under translations. This statement is new at points of first-order phase transition, that is, at $T=T_c(q)$ when $q>4$. In this case the structure of Gibbs measures is the most complex in the sense that there exist $q+1$ distinct extremal measures.

Most of the work is devoted to the FK-percolation model on ${\mathbb{Z}}^2$ with $q\ge 1$, where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are nonquantitative and follow the spirit of the seminal works of Aizenman (1980) and Higuchi (1981), which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. Similarly, a priori, the image of a Gibbs measure for the Potts model under the Edwards–Sokal procedure might not be a Gibbs measure for FK-percolation. We rule this out using a self-duality argument.

Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop $O(n)$ model in the range of parameters for which its spin representation is positively associated.

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