Finite-order correlation length of the \(|\varphi|^4\) spin model in four dimensions

The correlation length of order \(p\) for the \(|\varphi|^4\) spin model (a continuous-spin version of the O(n) model) is a normalization of the \(p\)-th moment of its two-point function. We will outline the proof (based on a renormalisation group method of Bauerschmidt, Brydges, and Slade) that, in the upper-critical dimension 4, this quantity undergoes mean-field scaling with a logarithmic correction as the critical point for this model is approached from above (for sufficiently weak coupling). Via a supersymmetric integral representation, this result also extends to the weakly self-avoiding walk with a contact attraction, for which the correlation length of order \(p\) is closely related to the mean \(p\)-th displacement of the walk. This is joint work with Roland Bauerschmidt, Gordon Slade, and Alexandre Tomberg.