A constructive mixing condition for 2-D Gibbs measures with random interactions

Abstract

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.