Gaussian deconvolution and the lace expansion for spread-out models

Abstract

We present a new proof of $|x{|}^{-(d-2)}$ decay of critical two-point functions for spread-out statistical mechanical models on ${\mathbb{Z}}^d$ above the upper critical dimension, based on the lace expansion and assuming appropriate diagrammatic estimates. Applications include spread-out models of the Ising model and self-avoiding walk in dimensions $d>4$, and spread-out percolation for $d>6$. The proof is based on an extension of the new Gaussian deconvolution theorem we obtained in a recent paper. It provides a technically simpler and conceptually more transparent approach than the method of Hara, van der Hofstad and Slade (2003).

21 pages. Minor edits. To appear in Ann. Inst. H. Poincaré Probab. Statist