Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

Federico Camia,Charles M Newman,Jianping Jiang

doi:10.1002/cpa.21884

Abstract

AbstractWe consider the Ising model at its critical temperature with external magnetic field ha15/8 on the square lattice with lattice spacing a. We show that the truncated two‐point function in this model decays exponentially with a rate independent of a as a ↓ 0. As a consequence, we show exponential decay in the near‐critical scaling limit Euclidean magnetization field. For the lattice model with a = 1, the mass (inverse correlation length) is of order h8/15 as h ↓ 0; for the Euclidean field, it equals exactly Ch8/15 for some C. Although there has been much progress in the study of critical scaling limits, results on near‐critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near‐critical scaling limits. Thus we provide the first substantial application of measure ensembles. © 2020 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Highlights

In this paper we obtain the first proof of exponential decay for the important Euclidean field theory that is a near-critical scaling limit of the planar Ising model at the critical temperature, with an external magnetic field
As a corollary of our main results, we provide a rigorous proof of the power-law behavior of the correlation length for the planar Ising model at the critical temperature, as the external magnetic field tends to zero
Key to our arguments is the use of conformal measure ensembles, introduced in [16], where they were called cluster area measures, and constructed for percolation and the FK (Fortuin-Kasteleyn)-Ising model in [8]

Summary

Introduction

In this paper we obtain the first proof of exponential decay (or equivalently, a mass gap lower bound) for the important Euclidean field theory that is a near-critical scaling limit of the planar Ising model at the critical temperature, with an external magnetic field. As a corollary of our main results, we provide a rigorous proof of the power-law behavior of the correlation length for the planar Ising model at the critical temperature, as the external magnetic field tends to zero. We show exponential decay for the near-critical Ising model on aZ2 with H = a15/8h Speaking, this means (see Theorem 1 below) that there is a lower bound on m (H) behaving like H8/15 as H ↓ 0. The remaining step consists in showing that the probability of Acx is not affected much by the boundary condition in B(x, 2|x − y|/3), which follows from Proposition 2

Preliminary definitions and results

Exponential decay on the lattice