A simple convergence proof for the lace expansion

TL;DR

This paper provides a straightforward proof using the lace expansion to show that the critical two-point function for weakly self-avoiding walk on high-dimensional integer lattices decays as a power law, specifically |x|^{-(d-2)}.

Contribution

It introduces a simple, elementary proof of the decay of the critical two-point function for weakly self-avoiding walk using the lace expansion and basic Fourier analysis.

Findings

Critical two-point function decays as |x|^{-(d-2)} for d>4

Proof employs elementary Fourier analysis and Riemann–Lebesgue Lemma

Simplifies previous complex proofs of decay behavior

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.