Four-Dimensional Weakly Self-avoiding Walk with Contact Self-attraction

Roland Bauerschmidt, Gordon Slade, Benjamin C. Wallace

Research output: Contribution to journalArticlepeer-review


We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on Z4, for sufficiently small attraction. We prove that the susceptibility and correlation length of order p (for any p> 0) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of | x| - 2. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction.

Original languageEnglish (US)
Pages (from-to)317-350
Number of pages34
JournalJournal of Statistical Physics
Issue number2
StatePublished - Apr 1 2017


  • Collapse transition
  • Renormalisation group
  • Weakly self-avoiding walk

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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