A uniqueness condition for Gibbs measures,with application to the 2-dimensional Ising antiferromagnet

A uniqueness condition for Gibbs measures is given. This condition is stated in terms of (absence of) a certain type of percolation involving two independent realisations. This result can be applied in certain concrete situations by comparison with ordinary percolation. In this way we prove that the Ising antiferromagnet on a square lattice has a unique Gibbs measure if (4–|h)<1/2ln(P_c/(1–P_c)), whereh denotes the external magnetic field, the inverse temperature, andP_c the critical probability for site percolation on that lattice. SinceP_c is larger than 1/2, this extends a result by Dobrushin, Kolafa and Shlosman (whose proof was computer-assisted).The research which led to this paper started while the author was at Cornell University, partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University