{
  "id": "2310.07635",
  "version": "v1",
  "published": "2023-10-11T16:29:07.000Z",
  "updated": "2023-10-11T16:29:07.000Z",
  "title": "Gaussian deconvolution and the lace expansion",
  "authors": [
    "Yucheng Liu",
    "Gordon Slade"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "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 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\\\"older'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. We also improve previous error estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-11T16:29:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "60K35",
      "82B27",
      "82B41",
      "82B43"
    ],
    "keywords": [
      "lace expansion",
      "gaussian deconvolution",
      "significantly simplify existing proofs",
      "convolution equation",
      "intricate fourier analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}