Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk

Roland Bauerschmidt, David C. Brydges, Gordon Slade

Research output: Contribution to journalArticlepeer-review


We prove (Formula Presented.) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on (Formula Presented.), in the upper critical dimension d = 4. This is a statement that the critical exponent η exists and is equal to zero. Results of this nature have been proved previously for dimensions d≥5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

Original languageEnglish (US)
Pages (from-to)169-193
Number of pages25
JournalCommunications In Mathematical Physics
Issue number1
StatePublished - Aug 1 2015

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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