A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising Model

TL;DR

This paper introduces a new, more natural finite-volume approach to the Aizenman-Higuchi theorem for the 2D Ising model, providing quantitative results and potential applicability to other systems.

Contribution

It offers an optimal finite-volume, quantitative analogue of the classical theorem with a clearer conceptual framework and broader applicability.

Findings

Derived an optimal finite-volume, quantitative version of the theorem

Provided a more natural proof scheme that enhances understanding

Potentially applicable to other systems where classical methods fail

Abstract

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model are convex combinations of the two pure phases. We present here a new approach to this result, with a number of advantages: (i) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (ii) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (iii) this new approach might be applicable to systems for which the classical method fails.