{
  "id": "2207.09401",
  "version": "v1",
  "published": "2022-07-19T16:51:31.000Z",
  "updated": "2022-07-19T16:51:31.000Z",
  "title": "Properties of the gradient squared of the discrete Gaussian free field",
  "authors": [
    "Alessandra Cipriani",
    "Rajat S. Hazra",
    "Alan Rapoport",
    "Wioletta M. Ruszel"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study the properties of the centered (norm of the) gradient squared of the discrete Gaussian free field in $U_{\\epsilon}=U/\\epsilon\\cap \\mathbb{Z}^d$, $U\\subset \\mathbb{R}^d$ and $d\\geq 2$. The covariance structure of the field is a function of the transfer current matrix and this relates the model to a class of systems (e.g. height-one field of the Abelian sandpile model or pattern fields in dimer models) that have a Gaussian limit due to the rapid decay of the transfer current. Indeed, we prove that the properly rescaled field converges to white noise in an appropriate local Besov-H\\\"older space. Moreover, under a different rescaling, we determine the $k$-point correlation function and cumulants on $U_{\\epsilon}$ and in the continuum limit as $\\epsilon\\to 0$. This result is related to the analogue limit for the height-one field of the Abelian sandpile (\\citet{durre}), with the same conformally covariant property in $d=2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-19T16:51:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G60",
      "60K35",
      "82B20",
      "82B41"
    ],
    "keywords": [
      "discrete gaussian free field",
      "height-one field",
      "transfer current matrix",
      "point correlation function",
      "gaussian limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}