Positive association in the fractional fuzzy Potts model

Abstract

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 $\phi_{p,q}$, 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 H\"{a}ggstr\"{o}m [Ann. Appl. Probab. 9 (1999) 1149--1159] and H\"{a}ggstr\"{o}m and Schramm [Stochastic Process. Appl. 96 (2001) 213--242].