UBC Probability Seminar: Yucheng Liu

We give conditions on a real-valued function F on ℤd, for d>2, which ensure that the solution G to the convolution equation (F∗G)(x)=δ0,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 H\"older's inequality and basic Fourier theory in Lp 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 φ4 models, percolation, and lattice trees and lattice animals. This is based on a joint work with Gordon Slade (arXiv:2310.07635).