The torus plateau for the high-dimensional Ising model

TL;DR

This paper analyzes the high-dimensional Ising model near criticality on a finite torus, revealing a plateau in the two-point function and non-Gaussian behavior of the average spin, using advanced probabilistic techniques.

Contribution

It establishes the existence of a plateau in the two-point function and non-Gaussian limits for the average spin in high-dimensional finite-volume Ising models near criticality.

Findings

Two-point function exhibits a plateau at large volumes.

Susceptibility scales as r^{d/2} near criticality.

Non-Gaussian limit for the average spin at criticality.

Abstract

We consider the Ising model on a $d$-dimensional discrete torus of volume $r^d$, in dimensions $d>4$ and for large $r$, in the vicinity of the infinite-volume critical point $\beta_c$. We prove that for $\beta=\beta_c- {\rm const}\, r^{-d/2}$ (with a suitable constant) the susceptibility is bounded above and below by multiples of $r^{d/2}$. Additionally, again for $\beta=\beta_c- {\rm const}\, r^{-d/2}$, the two-point function has a ``plateau'': it decays like $|x|^{-(d-2)}$ when $|x|$ is small relative to the volume, but for larger $|x|$, it levels off to a constant value of order $r^{-d/2}$. We also prove that at $\beta=\beta_c- {\rm const}\, r^{-d/2}$ the renormalised coupling constant is nonzero, which implies a non-Gaussian limit for the average spin. The proof relies on near-critical estimates for the infinite-volume two-point function obtained recently by Duminil-Copin and Panis, and builds upon a strategy proposed by Papathanakos. The random current representation of the Ising model plays a central role in our analysis.