On the Ising model with random boundary condition

ABSTRACT. The infinite-volume limit behavior of the 2d Ising model under pos-sibly strong random boundary conditions is studied. The model exhibits chaotic size-dependence at low temperatures and we prove that the ‘+ ’ and ‘- ’ phases are the only almost sure limit Gibbs measures, assuming that the limit is taken along a sparse enough sequence of squares. In particular, we give a multi-scale perturbative argument to show that in a sufficiently large volume typical spin configuration under a typical boundary condition contains no interfaces. 1