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 \(\beta =\beta _c- \textrm{const}\, r^{-d/2}\) (with a suitable constant) the susceptibility is bounded above and below by multiples of \(r^{d/2}\) . Additionally, again for \(\beta =\beta _c- \textrm{const}\, r^{-d/2}\) , 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 \(\beta =\beta _c- \textrm{const}\, r^{-d/2}\) 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.
