{
  "id": "1609.07557",
  "version": "v1",
  "published": "2016-09-24T02:50:03.000Z",
  "updated": "2016-09-24T02:50:03.000Z",
  "title": "A characterization of $L_{2}$ mixing and hypercontractivity via hitting times and maximal inequalities",
  "authors": [
    "Jonathan Hermon",
    "Yuval Peres"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "There are several works characterizing the total-variation mixing time of a reversible Markov chain in term of natural probabilistic concepts such as stopping times and hitting times. In contrast, there is no known analog for the $L_{2}$ mixing time, $\\tau_{2}$ (while there are sophisticated analytic tools to bound $ \\tau_2$, in general they do not determine $\\tau_2$ up to a constant factor and they lack a probabilistic interpretation). In this work we show that $\\tau_2$ can be characterized up to a constant factor using hitting times distributions. We also derive a new extremal characterization of the Log-Sobolev constant, $c_{\\mathrm{LS}}$, as a weighted version of the spectral gap. This characterization yields a probabilistic interpretation of $c_{\\mathrm{LS}}$ in terms of a hitting time version of hypercontractivity. As applications of our results, we show that (1) for every reversible Markov chain, $\\tau_2$ is robust under addition of self-loops with bounded weights, and (2) for weighted nearest neighbor random walks on trees, $\\tau_2 $ is robust under bounded perturbations of the edge weights.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-24T02:50:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J27",
      "60J10",
      "05C81"
    ],
    "keywords": [
      "hitting time",
      "maximal inequalities",
      "characterization",
      "reversible markov chain",
      "hypercontractivity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}