{
  "id": "2310.07685",
  "version": "v1",
  "published": "2023-10-11T17:35:55.000Z",
  "updated": "2023-10-11T17:35:55.000Z",
  "title": "Moderate Deviations for the Capacity of the Random Walk range in dimension four",
  "authors": [
    "Arka Adhikari",
    "Izumi Okada"
  ],
  "comment": "52 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen \\cite{BCR} concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, \\begin{equation*} \\label{eq:maxineq} \\inf_{f: \\|\\nabla f\\|_{L^2}<\\infty} \\frac{\\|f\\|^{1/2}_{L^2} \\|\\nabla f\\|^{1/2}_{L^2}}{ [\\int_{(\\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \\text{d}x \\text{d}y]^{1/4}}. \\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-11T17:35:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60G50"
    ],
    "keywords": [
      "random walk range",
      "moderate deviation results",
      "dimensional analog",
      "deviation statistics",
      "generalized gagliardo-nirenberg inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}