A SHARP TRANSITION FOR THE 2-DIMENSIONAL ISING PERCOLATION

We show that the percolation transition for the two-dimensional Ising model is sharp. Namely, we show that for every reciprocal temperature beta > 0, there exists a critical value h(c)(beta) of external magnetic field h such that the following two statements hold. If h > h(c)(beta), then the percolation probability (i.e., the probability that the origin is in the infinite cluster of + spins) with respect to the Gibbs state mu(beta,h) for the parameter (beta,h) is positive. If h < h(c)(beta), then the connectivity function tau(beta,h)(+) origin is connected by +spins to X with respect to mu(beta,h)) decays exponentially as \X\ --> infinity. We also show that the percolation probability is continuous in (P, h) except on the half line {(beta, 0); beta greater than or equal to beta(c)}.