{
  "id": "2107.12276",
  "version": "v1",
  "published": "2021-07-26T15:35:12.000Z",
  "updated": "2021-07-26T15:35:12.000Z",
  "title": "Maximum of the membrane model on regular trees",
  "authors": [
    "Alessandra Cipriani",
    "Biltu Dan",
    "Rajat Subhra Hazra",
    "Rounak Ray"
  ],
  "comment": "30 pages. Comments are welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "The discrete membrane model is a Gaussian random interface whose inverse covariance is given by the discrete biharmonic operator on a graph. In literature almost all works have considered the field as indexed over $\\mathbb{Z}^d$, and this enabled one to study the model using methods from partial differential equations. In this article we would like to investigate the dependence of the membrane model on a different geometry, namely trees. The covariance is expressed via a random walk representation which was first determined by Vanderbei (1984). We exploit this representation on $m$-regular trees and show that the infinite volume limit on the infinite tree exists when $m\\ge 3$. Further we determine the behavior of the maximum under the infinite and finite volume measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-26T15:35:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "82B20",
      "82B41",
      "60G70"
    ],
    "keywords": [
      "regular trees",
      "random walk representation",
      "discrete biharmonic operator",
      "discrete membrane model",
      "infinite volume limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}