A simple proof of exponential decay for the near-critical planar Ising model

TL;DR

This paper provides a straightforward proof that the two-point correlation function in the near-critical planar Ising model decays exponentially, using high temperature expansion and random current techniques.

Contribution

It introduces a simple, elementary proof of exponential decay in the near-critical Ising model, combining multiple representations and a novel insight about loop structures.

Findings

Exponential decay of the two-point function near criticality.

Effective use of random current and cluster representations.

Identification of loop structures in the near-critical measure.

Abstract

For the Ising model defined on $a\mathbb{Z}^2$ at critical temperature with external field $a^{15/8}h$, we give a simple and elementary proof that its truncated two-point function decays exponentially. The proof combines the high temperature expansion, random-cluster and random current representations. A new input in the proof is that, in the near-critical sourceless single current measure, there are many loops formed by a path on $a\mathbb{Z}^2$ with diameter of order $1$, together with two external edges that connect the path's endpoints to the ghost.