Ornstein-Zernike behavior for Ising models with infinite-range interactions

TL;DR

This paper proves Ornstein-Zernike asymptotics for the two-point function of the Ising model with infinite-range interactions above the critical temperature, extending previous results to more general, non-finite-range models.

Contribution

It provides the first proof of Ornstein-Zernike behavior for a nontrivial model with infinite-range interactions, including various self-repulsive models.

Findings

Established OZ asymptotics for infinite-range Ising models

Extended results to a broad class of self-repulsive models

Applicable to Green functions of polymer models like self-avoiding walk

Abstract

We prove Ornstein-Zernike behavior for the large-distance asymptotics of the two-point function of the Ising model above the critical temperature under essentially optimal assumptions on the interaction. The main contribution of this work is that the interactions are not assumed to be of finite range. To the best of our knowledge, this is the first proof of OZ asymptotics for a nontrivial model with infinite-range interactions. Our results actually apply to the Green function of a large class of "self-repulsive in average" models, including a natural family of self-repulsive polymer models that contains, in particular, the self-avoiding walk, the Domb-Joyce model and the killed random walk. We aimed at a pedagogical and self-contained presentation.