Renormalisation group analysis of 4D spin models and self-avoiding walk

TL;DR

This paper reviews recent rigorous renormalisation group results on critical phenomena in four-dimensional spin models and self-avoiding walks, highlighting logarithmic corrections and scaling behaviors near criticality.

Contribution

It provides a rigorous analysis of critical behavior in 4D $| abla|^4$ models and self-avoiding walks, extending known results to supersymmetric cases and detailed scaling limits.

Findings

Susceptibility diverges with logarithmic correction

Critical two-point function analyzed

Scaling limits of the spin field established

Abstract

We give an overview of results on critical phenomena in 4 dimensions, obtained recently using a rigorous renormalisation group method. In particular, for the $n$-component $|\varphi|^4$ spin model in dimension 4, with small coupling constant, we prove that the susceptibility diverges with a logarithmic correction to the mean-field behaviour with exponent $(n+2)/(n+8)$. This result extends rigorously to $n=0$, interpreted as a supersymmetric version of the model that represents exactly the continuous-time weakly self-avoiding walk. We also analyse the critical two-point function of the weakly self-avoiding walk, the specific heat and pressure of the $|\varphi|^4$ model, as well as scaling limits of the spin field close to the critical point.