Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $|\varphi|^4$ spins

TL;DR

This paper investigates the correlation length in 4D $| abla|^4$ spin models and self-avoiding walks, revealing a logarithmic correction to mean-field scaling with a specific power depending on the number of components.

Contribution

It extends the rigorous renormalisation group analysis to include correlation length corrections for all $n \\ge 0$ in 4D models, identifying a universal logarithmic correction.

Findings

Logarithmic correction to mean-field scaling with power 1/2(n+2)/(n+8)

Analysis valid for all $n \\ge 0$ and $p>0$

Improved renormalisation group method applied

Abstract

We study the 4-dimensional $n$-component $|\varphi|^4$ spin model for all integers $n \ge 1$, and the 4-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case $n=0$ interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order $p$, and prove the existence of a logarithmic correction to mean-field scaling, with power $\frac 12\frac{n+2}{n+8}$, for all $n \ge 0$ and $p>0$. The proof is based on an improvement of a rigorous renormalisation group method developed previously.