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

Abstract

We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\mathbb{Z}$$^{4}$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order $\textit{p}$ (for any $\textit{p}$ > 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.The work of RB was supported in part by the Simons Foundation. The work of GS and BCW was supported in part by NSERC of Canada. We thank the referees for useful suggestions