A simple convergence proof for the lace expansion

Abstract

We use the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk on ${\mathbb{Z}}^d$ has decay $|x{|}^{-(d-2)}$ in dimensions $d>4$. The proof uses elementary Fourier analysis and the Riemann--Lebesgue Lemma.

An improvement of Section 3 of arXiv:2006.06532v1, which has been divided into two independent papers. 9 pages. Minor changes in this version. To be published in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques