The torus plateau for the high-dimensional Ising model

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 ${\hat{I}}_c^2$. We prove that for ${\hat{I}}^2={\hat{I}}_c^2-\mathrm{const}{r}^{-d/2}$ (with a suitable constant) the susceptibility is bounded above and below by multiples of $r^{d/2}$. Additionally, again for ${\hat{I}}^2={\hat{I}}_c^2-\mathrm{const}{r}^{-d/2}$, 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 ${\hat{I}}^2={\hat{I}}_c^2-\mathrm{const}{r}^{-d/2}$ 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.

30 pages, 3 figures. Minor edits