A note on disagreement percolation

We construct a coupling of two distinct Gibbs measures for Markov random fi elds with the same specifications, such that the existence of an infinite p ath of disagreements between the two configurations has probability 0. This shows that the independence assumption in the disagreement percolation met hod for proving Gibbsian uniqueness cannot be dropped without being replace d by other conditions. A similar counterexample is given for couplings of M arkov chains. (C) 2001 John Wiley & Sons, Inc.