{
  "id": "2311.03670",
  "version": "v1",
  "published": "2023-11-07T02:42:04.000Z",
  "updated": "2023-11-07T02:42:04.000Z",
  "title": "Vertex-removal stability and the least positive value of harmonic measures",
  "authors": [
    "Zhenhao Cai",
    "Gady Kozma",
    "Eviatar B. Procaccia",
    "Yuan Zhang"
  ],
  "comment": "34 pages, 9 figures",
  "categories": [
    "math.PR",
    "math.CA"
  ],
  "abstract": "We prove that for $\\mathbb{Z}^d$ ($d\\ge 2$), the vertex-removal stability of harmonic measures (i.e. it is feasible to remove some vertex while changing the harmonic measure by a bounded factor) holds if and only if $d=2$. The proof mainly relies on geometric arguments, with a surprising use of the discrete Klein bottle. Moreover, a direct application of this stability verifies a conjecture of Calvert, Ganguly and Hammond [9] for the exponential decay of the least positive value of harmonic measures on $\\mathbb{Z}^2$. Furthermore, the analogue of this conjecture for $\\mathbb{Z}^d$ with $d\\ge 3$ is also proved in this paper, despite vertex-removal stability no longer holding.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-07T02:42:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic measure",
      "positive value",
      "discrete klein bottle",
      "despite vertex-removal stability",
      "direct application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}