Critical correlation functions for the 4-dimensional weakly self-avoiding walk and n-component $|\varphi|^4$ model

Abstract

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice $\mathbb{Z}^4$, for the weakly coupled $n$-component $|\varphi|^4$ spin model for all $n \geq 1$, and for the continuous-time weakly self-avoiding walk. For the $|\varphi|^4$ model, we prove that the critical two-point function has $|x|^{-2}$ (Gaussian) decay asymptotically, for $n \ge 1$. We also determine the asymptotic decay of the critical correlations of the squares of components of $\varphi$, including the logarithmic corrections to Gaussian scaling, for $n \geq 1$. The above extends previously known results for $n = 1$ to all $n \ge 1$, and also observes new phenomena for $n > 1$, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually- and self-avoiding walks, for all $p \ge 1$, including the logarithmic corrections. This extends a previously known result for $p = 1$, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the $n = 0$ case of the $|\varphi|^4$ model, and provides a unified treatment of both models, and of all the above results.