Journal of Statistical Physics

, Volume 157, Issue 4–5, pp 692–742 | Cite as

Scaling Limits and Critical Behaviour of the \(4\)-Dimensional \(n\)-Component \(|\varphi |^4\) Spin Model

  • Roland Bauerschmidt
  • David C. Brydges
  • Gordon Slade


We consider the \(n\)-component \(|\varphi |^4\) spin model on \({\mathbb {Z}}^4\), for all \(n \ge 1\), with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent \(\frac{n+2}{n+8}\) for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for \(n =1,2,3\); double logarithmic scaling for \(n=4\); and is bounded when \(n>4\). In addition, for the model defined on the \(4\)-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the \(|\varphi |^4\) model.


Renormalisation group Critical phenomena Logarithmic corrections Susceptibility Specific heat Scaling limit 

Mathematics Subject Classification

82B28 82B27 82B20 60K35 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Roland Bauerschmidt
    • 1
  • David C. Brydges
    • 2
  • Gordon Slade
    • 2
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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