{
  "id": "2210.07859",
  "version": "v1",
  "published": "2022-10-14T14:30:10.000Z",
  "updated": "2022-10-14T14:30:10.000Z",
  "title": "Biased Random Walk on Spanning Trees of the Ladder Graph",
  "authors": [
    "Nina Gantert",
    "Achim Klenke"
  ],
  "comment": "26 pages, 9 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a two-sided infinite horizontal ladder and pick a random spanning tree with a certain edge weight $c$ for the (vertical) rungs. Now take a random walk on that spanning tree with a bias $\\beta>1$ to the right. We give an explicit formula for the speed of that random walk and show that it is a continuous, unimodal function of $\\beta$ that is positive if and only if $\\beta < \\beta_c^{(1)}$ for an explicit critical value $\\beta_c^{(1)}$ depending on $c$. We also show a central limit theorem for $\\beta< \\beta_c^{(2)} $, for an explicit critical value $\\beta_c^{(2)}$. We see that $\\beta_c^{(2)}$ is smaller than the value of $\\beta$ which achieves the maximal value of the speed. Finally, we confirm the Einstein relation for the unbiased model ($\\beta=1$) by proving a Central Limit Theorem and computing the variance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-14T14:30:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60G50",
      "60F05"
    ],
    "keywords": [
      "biased random walk",
      "ladder graph",
      "central limit theorem",
      "explicit critical value",
      "two-sided infinite horizontal ladder"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}