Positive association in the fractional fuzzy Potts model

Details Positive association in the fractional fuzzy Potts model Positive association in the fractional fuzzy Potts model

2007-11-20

arxiv.org/abs/0711.3136v1

Annals of Probability 2007, Vol. 35, No. 6, 2038-2043

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000042 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000042

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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