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

TL;DR

This work studies exponential decay of correlations in the 2D near-critical Ising model with a small external field on the lattice $aZ^2$, establishing a mass gap with the bound $<\sigma_x;\sigma_y>^f_{\Lambda_N,a,h} \le C_0 a^{1/4}|x-y|^{-1/4} e^{-C_1 h^{8/15}|x-y|}$. It provides an elementary proof by combining the random current, random-cluster, and high-temperature expansions, and introduces a new ingredient about loops in near-critical sourceless single-current measure that connect to the ghost. The key step reduces the two-point bound to a probability of no connection to the ghost in a double-current measure, and shows that the forced $x$ to $y$ path must cross many disjoint rectangles with a positive loop density, yielding a binomial tail and exponential decay. The result offers a simpler approach to the mass gap near criticality and complements previous loop-ensemble and current-based methods, with implications for the Ising spectrum in the scaling limit.

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.

Figures (2)

• Figure 1: The curve illustrates the path from $x$ to $y$ forced by the source constraint in $\mathbf{P}^{xy}_{\Lambda,a,h}$. Each dotted curve represents a loop from $\mathbf{P}^{\emptyset}_{\Lambda,a,h}$ (only the path on $a\mathbb{Z}^2$ is shown and the two marked endpoints of the path are connected to $\mathfrak{g}$ directly).
• Figure 2: An illustration of the events $E(R)$ (left) and $H(R)$ (right). Here $T:=[0,10]\times[0,3]$, $R:=[2,8]\times [0,3]$, $S:=[1,9]\times[1,2]$, $Q_1:=[1,2)\times [1,2]$ and $Q_8:=(8,9]\times[1,2]$. The dotted circuit is a dual open circuit in $T^a\setminus S^a$. $S^a$ has a left-right crossing. Both $Q^a_1$ and $Q^a_8$ have a top-bottom crossing. In the crossing cluster of $S^a$ only the two marked points in $Q^a_1$ and $Q^a_8$ are connected to the ghost directly by one external edge.

Theorems & Definitions (9)

• Theorem 1
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof