Abstract

We give conditions on a real-valued function $F$ on $\mathbb{Z}^d$, for $d>2$, which ensure that the solution $G$ to the convolution equation $(F*G)(x) = \delta_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our results were obtained by Hara in the 2000s, using intricate Fourier analysis. We give a new, very simple proof using Hölder's inequality and basic Fourier theory in $L^p$ space. Our motivation comes from critical phenomena in equilibrium statistical mechanics, where the convolution equation is provided by the lace expansion and the deconvolution $G$ is a critical two-point function. Our results significantly simplify existing proofs of critical $|x|^{-(d-2)}$ decay in high dimensions for self-avoiding walk, Ising and $\varphi^4$ models, percolation, and lattice trees and lattice animals. This is based on a joint work with Gordon Slade (arXiv:2310.07635).