Renewal properties of the d = 1 Ising model

Abstract

We consider the [Formula: see text] Ising model with Kac potentials at inverse temperature [Formula: see text] where the mean field predicts a phase transition with two possible equilibrium magnetizations [Formula: see text], [Formula: see text]. We show that when the Kac scaling parameter [Formula: see text] 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 [Formula: see text], and respectively, [Formula: see text]. We prove that the corresponding marginal of the unique DLR measure is a renewal process.