Renewal properties of the d = 1 Ising model

We consider the d = 1 Ising model with Kac potentials at inverse temperature β > 1 where the mean field predicts a phase transition with two possible equilibrium magnetizations ± mβ, mβ > 0. We show that when the Kac scaling parameter γ is sufficiently small, typical spin configurations are described (via a coarse graining) by an infinite sequence of successive plus and minus intervals where the empirical magnetization is “close” to mβ, and respectively, - mβ. We prove that the corresponding marginal of the unique DLR measure is a renewal process.