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

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 $\mathrm{\mid }\phi {\mathrm{\mid }}^4$ 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 $(n+2)\mathrm{/}(n+8)$. 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 $\mathrm{\mid }\phi {\mathrm{\mid }}^4$ model, as well as scaling limits of the spin field close to the critical point.