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July 1997 A constructive mixing condition for 2-D Gibbs measures with random interactions
J. van den Berg
Ann. Probab. 25(3): 1316-1333 (July 1997). DOI: 10.1214/aop/1024404515


For spin systems with random, finite-range interactions, we define an analog of the usual weak mixing property, which we call “weak mixing in expectation” (WME). This property implies (almost sure) uniqueness of the Gibbs measure.

We concentrate on the two-dimensional case, for which we present finite-volume conditions which are sufficient for WME. We also show the reverse: if the system is WME, then the condition is satisfied for some (sufficiently large) volume. Simultaneously, we obtain an extension (to random interactions) of the result by Martinelli, Olivieri and Schonmann that weak mixing implies strong mixing.

Our method is based on a rescaled version of the disagreement percolation approach of van den Berg and Maes, combined with ideas and techniques of Gielis and Maes, and Martinelli, Olivieri and Schonmann. However, apart from some general results on coupling, stated in Section 2, this paper is self-contained.


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J. van den Berg. "A constructive mixing condition for 2-D Gibbs measures with random interactions." Ann. Probab. 25 (3) 1316 - 1333, July 1997.


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

zbMATH: 0895.60097
MathSciNet: MR1457621
Digital Object Identifier: 10.1214/aop/1024404515

Primary: 60K35
Secondary: 82B20 , 82B43 , 82B44

Keywords: Gibbs measures , Mixing , percolation , random interactions

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • July 1997
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