Structure of Gibbs measures for planar FK-percolation and Potts models

Abstract

We prove that all Gibbs measures of the q-state Potts model on 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 Z 2 with q ≥ 1, where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman [1] and Higuchi [32], 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. The result for the Potts model is then derived using the Edwards-Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards-Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.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.

Uniqueness of the infinite tree in low-dimensional random forests

“…Indeed, is is notable that no abstract theory of Gibbs measures has previously been developed for these models despite their broad popularity. For example, in Glazman and Manolescu's work on the structure of the set of Gibbs measures for the random cluster model on Z 2 [22], the authors consider only an (a priori ) special class of Gibbs measures in which infinite clusters are always considered to be connected at infinity. As discussed in [22,Remark 1.5], considering only this restricted class of Gibbs measures has various downsides, including that this class is not (a priori ) preserved under planar duality.…”

“…For example, in Glazman and Manolescu's work on the structure of the set of Gibbs measures for the random cluster model on Z 2 [22], the authors consider only an (a priori ) special class of Gibbs measures in which infinite clusters are always considered to be connected at infinity. As discussed in [22,Remark 1.5], considering only this restricted class of Gibbs measures has various downsides, including that this class is not (a priori ) preserved under planar duality. Our definition of Gibbs measures for models of this form is given strong justification by the fact that it coincides with the set of all possible limits of the models in finite-volume, with arbitrary boundary conditions, and is more general than that of [22].…”

“…As discussed in [22,Remark 1.5], considering only this restricted class of Gibbs measures has various downsides, including that this class is not (a priori ) preserved under planar duality. Our definition of Gibbs measures for models of this form is given strong justification by the fact that it coincides with the set of all possible limits of the models in finite-volume, with arbitrary boundary conditions, and is more general than that of [22]. The two notions can be shown to coincide for the random cluster model in the translation-invariant case, but it is currently unclear whether the two notions will coincide without the assumption of translation-invariance.…”