The low-temperature phase of Kac-Ising models

TL;DR

This paper investigates the low-temperature phase of ferromagnetic Kac-Ising models in dimensions two and higher, demonstrating the existence of two disjoint translation-invariant Gibbs states under specific temperature and interaction range conditions.

Contribution

It introduces a novel analysis combining blocking procedures and contour representations to establish phase coexistence in Kac-Ising models at low temperatures.

Findings

Existence of two disjoint Gibbs states under certain temperature conditions.

Application of a contour representation suitable for long-range spin systems.

Use of a Peierls argument variant to prove phase coexistence.

Abstract

We analyse the low temperature phase of ferromagnetic Kac-Ising models in dimensions $d\geq 2$. We show that if the range of interactions is $\g^{-1}$, then two disjoint translation invariant Gibbs states exist, if the inverse temperature $\b$ satisfies $\b -1\geq \g^\k$ where $\k=\frac {d(1-\e)}{(2d+1)(d+1)}$, for any $\e>0$. The prove involves the blocking procedure usual for Kac models and also a contour representation for the resulting long-range (almost) continuous spin system which is suitable for the use of a variant of the Peierls argument.