Gaussian deconvolution and the lace expansion

TL;DR

This work develops a general Gaussian deconvolution framework for the convolution equation $(F*G)(x)=\delta_{0,x}$ on $\mathbb{Z}^d$, $d>2$, proving that under symmetry, decay, and moment conditions on $F$, the solution has leading behavior $G(x) \sim \lambda C_{\mu}(x)$ with explicit error $O(|x|^{-(d-2+s)})$ for admissible $s$. The core method isolates the leading term using a pair of moments vanishing via a carefully chosen $\lambda$ and $\mu$, and then bounds the remainder by Fourier-analysis of $\hat f=\hat C\hat E\hat G$, exploiting weak derivatives to manage non-summable derivatives and, in a refinement, fractional derivatives to obtain sharper decay. The results extend and simplify previous Gaussian lemmas (Hara08) and apply to critical two-point functions arising in the lace expansion for self-avoiding walk, Ising/\varphi^4 models, percolation, LTLA, and related systems in high dimensions, also delivering improved error estimates. The framework accommodates inhomogeneous equations and, via a companion spread-out model analysis (LS24b), broadens applicability beyond nearest-neighbor setups. Overall, the approach yields conceptually transparent proofs of $|x|^{-(d-2)}$ decay in a broad lace-expansion context with quantitative error control and improved bounds, contributing to rigorous understanding of mean-field critical behaviour.

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 $(F*G)(x) = δ_{0,x}$ has Gaussian decay $|x|^{-(d-2)}$ for large $|x|$. Precursors of our results were obtained in the 2000s, using intricate Fourier analysis. In 2022, a very simple deconvolution theorem was proved, but its applicability was limited. We extend the 2022 theorem to remove its limitations while maintaining its simplicity -- our main tools are Hölder's inequality, weak derivatives, 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 $G$ 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. We also improve previous error estimates.

Theorems & Definitions (38)

• Theorem 1.2: Gaussian deconvolution
• Corollary 1.3
• proof
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof : Proof of Theorem \ref{['thm:gaussian_lemma_int']} assuming Proposition \ref{['prop:f_NN']}