Gaussian deconvolution and the lace expansion

• 成果类型: Article; Early Access
• 署名作者: Liu, Yucheng; Slade, Gordon
• 署名单位: University of British Columbia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01350-9
• 发表日期: 2024
• 关键词: self-avoiding walk; critical-behavior; lattice trees; percolation; exponents

摘要

We give conditions on a real-valued function F on Z(d), for d>2, which ensure that the solution G to the convolution equation (F & lowast;G)(x)=delta(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 H & ouml;lder'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 phi(4) models, percolation, and lattice trees and lattice animals. We also improve previous error estimates.