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• 作者：Roland Bauerschmidt (1)
  David C. Brydges (2)
  Gordon Slade (2)
  
• 关键词：Renormalisation group ; Critical phenomena ; Logarithmic corrections ; Susceptibility ; Specific heat ; Scaling limit ; 82B28 ; 82B27 ; 82B20 ; 60K35
• 刊名：Journal of Statistical Physics
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：157
• 期：4-5
• 页码：692-742
• 全文大小：865 KB
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• 作者单位：Roland Bauerschmidt (1)
  David C. Brydges (2)
  Gordon Slade (2)
  
  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA
  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada
  
• ISSN：1572-9613

文摘

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.