@article{1f0786687c9e4c33936ebde58dfdbb57,
title = "Marginal triviality of the scaling limits of critical 4D Ising and φ44 models",
abstract = "We prove that the scaling limits of spin fluctuations in four-dimensional Ising-type models with nearest-neighbor ferromagnetic interaction at or near the critical point are Gaussian. A similar statement is proven for the λφ4 fields over R4 with a lattice ultraviolet cutoff, in the limit of infinite volume and vanishing lattice spacing. The proofs are enabled by the models{\textquoteright} random current representation, in which the correlation functions{\textquoteright} deviation from Wick{\textquoteright}s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model{\textquoteright}s lattice scale. Guided by the analogy with random walk intersection amplitudes, the analysis focuses on the improvement of the so-called tree diagram bound by a logarithmic correction term, which is derived here through multi-scale analysis.",
keywords = "Critical behavior, Field theory, Ising model, Marginal dim, Scaling limits",
author = "Michael Aizenman and Hugo Duminil-Copin",
note = "Funding Information: Proof. For the first one, sum (A.33) for E being the full event and vertices in Bx and By, and use (A.10). For the second one, do the same with (A.32) instead. □ Acknowledgments. The work of M. Aizenman on this project was supported in part by the NSF grant DMS-1613296, and that of H. Duminil-Copin by the NCCR SwissMAP, the Swiss NSF and an IDEX Chair from Paris-Saclay. This project has received funding from the European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement No. 757296). The joint work was advanced through mutual visits to Princeton and Geneva University, sponsored by a Princeton-Unige partnership grant. We thank S. Goswami, A. Raoufi, P.-F. Rodriguez, and F. Severo for stimulating discussions, and M. Oulamara, R. Panis, P. Wilde-mann, and an anonymous referee for careful reading of the paper. Publisher Copyright: {\textcopyright} 2021. Department of Mathematics, Princeton University.",
year = "2021",
month = jul,
doi = "10.4007/annals.2021.194.1.3",
language = "English (US)",
volume = "194",
pages = "163--235",
journal = "Annals of Mathematics",
issn = "0003-486X",
publisher = "Princeton University Press",
number = "1",
}