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We prove $|x|^{-2}$ decay of the critical two-point function for thecontinuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the uppercritical dimension $d=4$. This is a statement that the critical exponent $\eta$exists and is equal to zero. Results of this nature have been proved previouslyfor dimensions $d \geq 5$ using the lace expansion, but the lace expansion doesnot apply when $d=4$. The proof is based on a rigorous renormalisation groupanalysis of an exact representation of the continuous-time weakly self-avoidingwalk as a supersymmetric field theory. Much of the analysis applies more widelyand has been carried out in a previous paper, where an asymptotic formula forthe susceptibility is obtained. Here, we show how observables can beincorporated into the analysis to obtain a pointwise asymptotic formula for thecritical two-point function. This involves perturbative calculations similar tothose familiar in the physics literature, but with error terms controlledrigorously.

Critical two-point function of the 4-dimensional weakly self-avoiding walk