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The torus plateau for the high-dimensional Ising model (2405.17353v2)

Published 27 May 2024 in math-ph, math.MP, and math.PR

Abstract: 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 $\beta_c$. 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.

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