Renewal properties of the $d=1$ Ising model

TL;DR

This paper studies the one-dimensional Ising model with Kac potentials at low temperature, showing that its magnetization exhibits a renewal process structure with alternating phases near two equilibrium values.

Contribution

It demonstrates that the marginal of the DLR measure in this model forms a renewal process, revealing detailed renewal properties of the phase structure.

Findings

Magnetization alternates between phases near ±m_β

The process of phase intervals is a renewal process

Results hold for sufficiently small Kac scaling parameter γ

Abstract

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.