On the decay of correlations in the random field Ising model

TL;DR

This paper provides the first quantitative analysis of correlation decay in the 2D random field Ising model, introducing a novel method that enhances understanding of phase uniqueness and decay rates.

Contribution

It offers the first quantitative version of the Aizenman-Wehr theorem and introduces a new method for proving decay of correlations in the model.

Findings

Quantitative decay rates of correlations established

New method for correlation decay proved effective

Enhanced understanding of phase uniqueness in the model

Abstract

In a celebrated 1990 paper, Aizenman and Wehr proved that the two-dimensional random field Ising model has a unique infinite volume Gibbs state at any temperature. The proof is ergodic-theoretic in nature and does not provide any quantitative information. This article proves the first quantitative version of the Aizenman-Wehr theorem. The proof introduces a new method for proving decay of correlations that may be interesting in its own right. A fairly detailed sketch of the main ideas behind the proof is also included.