{
  "id": "1704.04943",
  "version": "v1",
  "published": "2017-04-17T12:12:34.000Z",
  "updated": "2017-04-17T12:12:34.000Z",
  "title": "Two point function for critical points of a random plane wave",
  "authors": [
    "Dmitry Beliaev",
    "Valentina Cammarota",
    "Igor Wigman"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important geometric observable: critical points. We first compute one-point function for the critical point process, in particular we compute the expected number of critical points inside any open set. After that we compute the short-range asymptotic behaviour of the two-point function. This gives an unexpected result that the second factorial moment of the number of critical points in a small disc scales as the fourth power of the radius.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-17T12:12:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical point",
      "random plane wave",
      "generic compact riemanian manifolds",
      "small disc scales",
      "second factorial moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}