Disagreement Percolation in the Study of Markov Fields

Abstract

Recently, one of the authors (van den Berg) has obtained a uniqueness condition for Gibbs measures, in terms of disagreement percolation involving two independent realizations. In the present paper we study the dependence of Markov fields on boundary conditions by taking a more suitable coupling. This coupling leads to a new uniqueness condition, which improves the one mentioned above. We also compare it with the Dobrushin uniqueness condition. In the case of the Ising model, our coupling shares certain properties with the Fortuin-Kasteleyn representation: It gives an explicit expression of the boundary effect on a certain vertex in terms of percolation-like probabilities.