Positive association in the fractional fuzzy Potts model

A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph G obtained in two steps: first a subgraph of G is chosen according to a random cluster measure Φ_p,q, and then a spin (±1) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever q ≥1, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of Häggström [Ann. Appl. Probab. 9 (1999) 1149-1159] and Häggström and Schramm [Stochastic Process. Appl. 96 (2001) 213-242].