ON WEAK MIXING IN LATTICE MODELS

For lattice models on Z(d), weak mixing is the property that the influ ence of the boundary condition on a finite decays exponentially with d istance from that region. For a wide class of models on Z(2), includin g all finite range models, we show that weak mixing is a consequence o f Gibbs uniqueness, exponential decay of an appropriate form of connec tivity, and a natural coupling property. In particular, on Z(2), the F ortuin-Kasteleyn random cluster model is weak mixing whenever uniquene ss holds and the connectivity decays exponentially, and the q-state Po tts model above the critical temperature is weak mixing whenever corre lations decay exponentially, a hypothesis satisfied if q is sufficient ly large. Ratio weak mixing is the property that uniformly over events A and B occurring On subsets Lambda and Gamma, respectively, of the l attice, \P(A boolean AND B)/P(A)P(B) - 1\ decreases exponentially in t he distance between Lambda and Gamma. We show that under mild hypothes es, for example finite range, weak mixing implies ratio weak mixing.