INTERFACES AND TYPICAL GIBBS CONFIGURATIONS FOR ONE-DIMENSIONAL KAC POTENTIALS

We consider a one dimensional Ising spin system with a ferromagnetic Kac potential gammaJ(gamma\r\), J having compact support. We study the system in the limit, gamma --> 0, below the Lebowitz-Penrose critical temperature, where there are two distinct thermodynamic phases with different magnetizations. We prove that the empirical spin average in blocks of size deltagamma-1 (for any positive delta) converges, as gamma --> 0, to one of the two thermodynamic magnetizations, uniformly in the intervals of size gamma(-p), for any given positive p greater-than-or-equal-to 1. We then show that the intervals where the magnetization is approximately constant have lengths of the order of exp(cgamma-1), c > 0, and that, when normalized, they converge to independent variables with exponential distribution. We show this by proving large deviation estimates and applying the Ventsel and Friedlin methods to Gibbs random fields. Finally, if the temperature is low enough, we characterize the interface, namely the typical magnetization pattern in the region connecting the two phases.