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.