Positive association in the fractional fuzzy Potts model

Jeff Kahn; Nicholas Weininger

doi:10.1214/009117907000000042

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arXiv · 0711.3136

Positive association in the fractional fuzzy Potts model

Jeff Kahn·Nicholas Weininger

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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].

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Jeff Kahn, Nicholas Weininger. 2007-11-20. Positive association in the fractional fuzzy Potts model. https://doi.org/10.1214/009117907000000042

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