{
  "id": "2310.12843",
  "version": "v1",
  "published": "2023-10-19T15:55:06.000Z",
  "updated": "2023-10-19T15:55:06.000Z",
  "title": "Local behavior of critical points of isotropic Gaussian random fields",
  "authors": [
    "Paul Marriott",
    "Weinan Qi",
    "Yi Shen"
  ],
  "comment": "49 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we examine isotropic Gaussian random fields defined on $\\mathbb R^N$ satisfying certain conditions. Specifically, we investigate the type of a critical point situated within a small vicinity of another critical point, with both points surpassing a given threshold. It is shown that the Hessian of the random field at such a critical point is equally likely to have a positive or negative determinant. Furthermore, as the threshold tends to infinity, almost all the critical points above the threshold are local maxima and the saddle points with index $N-1$. Consequently, we conclude that the closely paired critical points above a high threshold must comprise one local maximum and one saddle point with index $N-1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-19T15:55:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G60",
      "60G70"
    ],
    "keywords": [
      "isotropic gaussian random fields",
      "critical point",
      "local behavior",
      "local maximum",
      "saddle point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}