Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk

We prove vertical bar x vertical bar(-2) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z(d), in the upper critical 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 previously for dimensions d >= 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.