The torus plateau for the high-dimensional Ising model

We consider the Ising model on a d-dimensional discrete torus of volume rᵈ, in dimensions $d>4$ and for large r, in the vicinity of the infinite-volume critical point βc. We prove that for β=βc- const\, r-d/2 (with a suitable constant) the susceptibility is bounded above and below by multiples of rd/2, and that the two-point function has a ``plateau'' in the sense that 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 β=βc- 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.