{
  "id": "2206.13834",
  "version": "v1",
  "published": "2022-06-28T09:06:32.000Z",
  "updated": "2022-06-28T09:06:32.000Z",
  "title": "Complexity of Gaussian random fields with isotropic increments: critical points with given indices",
  "authors": [
    "Antonio Auffinger",
    "Qiang Zeng"
  ],
  "comment": "Submission arXiv:2007.07668 was updated and split into two articles. This submission corresponds to the second part of arXiv:2007.07668v2. 50 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the landscape complexity of the Hamiltonian $X_N(x) +\\frac\\mu2 \\|x\\|^2,$ where $X_{N}$ is a smooth Gaussian process with isotropic increments on $\\mathbb R^{N}$. This model describes a single particle on a random potential in statistical physics. We derive asymptotic formulas for the mean number of critical points of index $k$ with critical values in an open set as the dimension $N$ goes to infinity. In a companion paper, we provide the same analysis without the index constraint.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-28T09:06:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian random fields",
      "isotropic increments",
      "critical points",
      "smooth gaussian process",
      "landscape complexity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}