Percolation Techniques in Disorderedspin Systems : the Uniqueness Regimei

We consider lattice spin systems with short range but random and unbounded interactions. Statics : We give an elementary proof of uniqueness of Gibbs measures at high temperature or strong magnetic elds, and of the exponential decay of the corresponding quenched correlation functions. The analysis is based on the study of disagreement percolation (as initiated in van den Berg{Maes (1994)). Dynamics : We give criteria for ergodicity of spin ip dynamics and estimate the speed of convergence to the unique invariant measure. We nd for this convergence a stretched exponential in time for a class of \directed" dynamics (such as in the disordered Toom or Stavskaya model). For the general case, we show that the relaxation is faster than any power in time. No assumptions of reversibility are made. The methods are based on relating the problem to an oriented percolation problem (contact process) and (for the general case) using a slightly modiied version of the multiscale analysis of e.g. Klein (1993).