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 φp,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].
"Positive association in the fractional fuzzy Potts model." Ann. Probab. 35 (6) 2038 - 2043, November 2007. https://doi.org/10.1214/009117907000000042