Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field

TL;DR

This paper establishes precise asymptotic decay rates for the truncated two-point function in the positive field Ising model, using advanced probabilistic and combinatorial techniques.

Contribution

It introduces a novel application of modern Ornstein-Zernike theory to derive sharp asymptotics for the Ising model with positive field.

Findings

Decay correction term is proportional to  ext{||x||}^{-(d-1)/2}

Provides rigorous proof of exponential decay with precise asymptotics

Extends the understanding of correlation decay in the Ising model

Abstract

We prove that the correction to exponential decay of the truncated two points function in the homogeneous positive field Ising model is $c\|x\|^{-(d-1)/2}$. The proof is based on the development in the random current representation of a "modern" Ornstein-Zernike theory, as developed by Campanino, Ioffe and Velenik.