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July 1997 Global specifications and nonquasilocality of projections of Gibbs measures
R. Fernández, C.-E. Pfister
Ann. Probab. 25(3): 1284-1315 (July 1997). DOI: 10.1214/aop/1024404514


We study the question of whether the quasilocality (continuity, almost Markovianness) property of Gibbs measures remains valid under a projection on a sub-$\sigma$-algebra. Our method is based on the construction of global specifications, whose projections yield local specifications for the projected measures. For Gibbs measures compatible with monotonicity preserving local specifications, we show that the set of configurations where quasilocality is lost is an event of the tail field. This set is shown to be empty whenever a strong uniqueness property is satisfied, and of measure zero when the original specification admits a single Gibbs measure. Moreover, we provide a criterion for nonquasilocality (based on a quantity related to the surface tension). We apply these results to projections of the extremal measures of the Ising model. In particular, our nonquasilocality criterion allows us to extend and make more complete previous studies of projections to a sublattice of one less dimension (Schonmann example).


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R. Fernández. C.-E. Pfister. "Global specifications and nonquasilocality of projections of Gibbs measures." Ann. Probab. 25 (3) 1284 - 1315, July 1997.


Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0895.60096
MathSciNet: MR1457620
Digital Object Identifier: 10.1214/aop/1024404514

Primary: 60G60 , 60J99 , 60K35
Secondary: 82B05 , 82B20 , 82B28

Keywords: decimation processes , discontinuity of conditional probabilities , Gibbs measures , global Markov property , Ising model , monotonicity preserving specifications , Nonquasilocality , projections of measures , Random fields

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • July 1997
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