The torus plateau for the high-dimensional Ising model

2024/05/27 by Liu, Yucheng, Panis, Romain, Slade, Gordon · 1 citation
#60K35 #82B20 #82B27 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2405.17353

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 β_c. We prove that for β=β_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 β=β_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 β=β_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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