Renewal properties of the $d=1$ Ising model

We consider the $d=1$ Ising model with Kac potentials at inverse temperature $\beta >1$ where mean field predicts a phase transition with two possible equilibrium magnetization $\pm m_{\beta }$, $m_{\beta }>0$. We show that when the Kac scaling parameter $\gamma$ 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 $m_{\beta }$ and respectively $-m_{\beta }$. We prove that the corresponding marginal of the unique DLR measure is a renewal process.