Marginal triviality of the scaling limits of critical 4D Ising and φ44 models

Michael Aizenman, Hugo Duminil-Copin

Research output: Contribution to journalArticlepeer-review

45 Scopus citations


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’ random current representation, in which the correlation functions’ deviation from Wick’s law is expressed in terms of intersection probabilities of random currents with sources at distances that are large on the model’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.

Original languageEnglish (US)
Pages (from-to)163-235
Number of pages73
JournalAnnals of Mathematics
Issue number1
StatePublished - Jul 2021

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • Critical behavior
  • Field theory
  • Ising model
  • Marginal dim
  • Scaling limits


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