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

Abstract

We prove $|x{|}^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on ${\mathbb{Z}}^d$, 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 $d\ge 5$ 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.

26 pages, revised version, will appear in Commun. Math. Phys

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