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 ($\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q\geq1$, 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].

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