{
  "id": "1812.10776",
  "version": "v1",
  "published": "2018-12-27T17:25:42.000Z",
  "updated": "2018-12-27T17:25:42.000Z",
  "title": "Einstein relation for random walk in a one-dimensional percolation model",
  "authors": [
    "Nina Gantert",
    "Matthias Meiners",
    "Sebastian Müller"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias $\\lambda > 0$, then its asymptotic linear speed $\\overline{\\mathrm{v}}$ is continuous in the variable $\\lambda > 0$ and differentiable for all sufficiently small $\\lambda > 0$. In the paper at hand, we complement this result by proving that $\\overline{\\mathrm{v}}$ is differentiable at $\\lambda = 0$. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at $\\lambda = 0$ equals the diffusivity of the unbiased walk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-27T17:25:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B43",
      "60K37"
    ],
    "keywords": [
      "random walk",
      "one-dimensional percolation model",
      "einstein relation",
      "conditional bond percolation model",
      "infinite ladder graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}