Finite-order correlation length for 4-dimensional weakly self-avoiding walk and $\mathrm{\mid }\phi {\mathrm{\mid }}^4$ spins

We study the 4-dimensional $n$-component $\mathrm{\mid }\phi {\mathrm{\mid }}^4$ 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 $\frac{1}{2}\frac{n+2}{n+8}$, for all $n\ge 0$ and $p>0$. The proof is based on an improvement of a rigorous renormalisation group method developed previously.