{
  "id": "1805.01615",
  "version": "v1",
  "published": "2018-05-04T05:35:53.000Z",
  "updated": "2018-05-04T05:35:53.000Z",
  "title": "Uniform spanning forests associated with biased random walks on Euclidean lattices",
  "authors": [
    "Zhan Shi",
    "Vladas Sidoravicius",
    "He Song",
    "Longmin Wang",
    "Kainan Xiang"
  ],
  "comment": "16 pages. Comments are welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "The uniform spanning forest measure ($\\mathsf{USF}$) on a locally finite, infinite connected graph $G$ with conductance $c$ is defined as a weak limit of uniform spanning tree measure on finite subgraphs. Depending on the underlying graph and conductances, the corresponding $\\mathsf{USF}$ is not necessarily concentrated on the set of spanning trees. Pemantle~\\cite{PR1991} showed that on $\\mathbb{Z}^d$, equipped with the unit conductance $ c=1$, $\\mathsf{USF}$ is concentrated on spanning trees if and only if $d \\leq 4$. In this work we study the $\\mathsf{USF}$ associated with conductances induced by $\\lambda$--biased random walk on $\\mathbb{Z}^d$, $d \\geq 2$, $0 < \\lambda < 1$, i.e. conductances are set to be $c(e) = \\lambda^{-|e|}$, where $|e|$ is the graph distance of the edge $e$ from the origin. Our main result states that in this case $\\mathsf{USF}$ consists of finitely many trees if and only if $d = 2$ or $3$. More precisely, we prove that the uniform spanning forest has $2^d$ trees if $d = 2$ or $3$, and infinitely many trees if $d \\geq 4$. Our method relies on the analysis of the spectral radius and the speed of the $\\lambda$--biased random walk on $\\mathbb{Z}^d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-04T05:35:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "biased random walk",
      "euclidean lattices",
      "conductance",
      "uniform spanning tree measure",
      "main result states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}