Gaussian deconvolution and the lace expansion for spread-out models

We present a new proof of $\mathrm{\mid }x{\mathrm{\mid }}^{-(d-2)}$ 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).