Four-dimensional weakly self-avoiding walk with contact self-attraction

TL;DR

This paper analyzes the critical behavior of a four-dimensional weakly self-avoiding walk with small contact self-attraction, showing it exhibits mean-field-like scaling with logarithmic corrections, similar to the case without attraction.

Contribution

It extends the rigorous renormalisation group analysis to include contact self-attraction, demonstrating that small attraction does not alter the critical behavior of the walk.

Findings

Susceptibility and correlation length have logarithmic corrections.

Critical two-point function behaves as a multiple of |x|^{-2}.

Small contact self-attraction does not change the critical behavior.

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 $p$ (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.