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.
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Bauerschmidt, R., Brydges, D.C. & Slade, G. Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk. Commun. Math. Phys. 338, 169–193 (2015). https://doi.org/10.1007/s00220-015-2353-5
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DOI: https://doi.org/10.1007/s00220-015-2353-5