{
  "id": "1906.03167",
  "version": "v1",
  "published": "2019-06-07T15:36:24.000Z",
  "updated": "2019-06-07T15:36:24.000Z",
  "title": "Random walk on the simple symmetric exclusion process",
  "authors": [
    "Marcelo R. Hilário",
    "Daniel Kious",
    "Augusto Teixeira"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $\\rho \\in [0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $\\rho$ except for at most two values $\\rho_-, \\rho_+ \\in [0, 1]$. The asymptotic speed we obtain in our LLN is a monotone function of $\\rho$. Also, $\\rho_-$ and $\\rho_+$ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is $1/2$ and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-07T15:36:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "60G55"
    ],
    "keywords": [
      "simple symmetric exclusion process",
      "random walker",
      "independent simple symmetric random walks",
      "functional central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}