Communications in Mathematical Physics

, Volume 337, Issue 2, pp 817–877 | Cite as

Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis

  • Roland Bauerschmidt
  • David C. Brydges
  • Gordon Slade


We prove that the susceptibility of the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^d}\), in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent \({\frac{1}{4}}\) for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk two-point function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.


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© Springer-Verlag Berlin Heidelberg 2015

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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