{
  "id": "1911.02300",
  "version": "v1",
  "published": "2019-11-06T10:47:31.000Z",
  "updated": "2019-11-06T10:47:31.000Z",
  "title": "Mean number and correlation function of critical points of isotropic Gaussian fields",
  "authors": [
    "Jean-Marc Azais",
    "Céline Delmas"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let X = {X(t) : t $\\in$ R N } be an isotropic Gaussian random field with real values. In a first part we study the mean number of critical points of X with index k using random matrices tools. We obtain an exact expression for the probability density of the eigenvalue of rank k of a N-GOE matrix. We deduce some exact expressions for the mean number of critical points with a given index. In a second part we study attraction or repulsion between these critical points. A measure is the correlation function. We prove attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. The attraction between critical points that occurs when the dimension is greater than two is due to critical points with adjacent indexes. A strong repulsion between maxima and minima is observed. The correlation function between maxima (or minima) depends on the dimension of the ambient space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-06T10:47:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical points",
      "correlation function",
      "mean number",
      "isotropic gaussian fields",
      "isotropic gaussian random field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}