{
  "id": "1401.7296",
  "version": "v2",
  "published": "2014-01-28T18:55:51.000Z",
  "updated": "2016-01-04T17:36:54.000Z",
  "title": "The Simple Exclusion Process on the Circle has a diffusive Cutoff Window",
  "authors": [
    "Hubert Lacoin"
  ],
  "comment": "37 pages, 3 Figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we investigate the mixing time of the simple exclusion process on the circle with $N$ sites, with a number of particle $k(N)$ tending to infinity, both from the worst initial condition and from a typical initial condition. We show that the worst-case mixing time is asymptotically equivalent to $(8\\pi^2)^{-1}N^2\\log k$, while the cutoff window, is identified to be $N^2$. Starting from a typical condition, we show that there is no cutoff and that the mixing time is of order $N^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-28T18:55:51.000Z",
      "abstract": "In this paper, we investigate the mixing time of the simple exclusion process on the circle with $N$ sites, with a number of particle $k(N)$ tending to infinity. For $k\\le N/2$ We show that the mixing time is asymptotically equivalent to $(8\\pi^2)^{-1}N^2\\log k$, while the cutoff window which the time need for the distance to equilibrium to drop from $1-\\varepsilon$ to $\\varepsilon$ for $\\varepsilon>0$ is identified to be $N^2$. We also prove that starting for most initial conditions, a time $O(N^2)$ is sufficient for mixing. Some portion of the proof remains valid in higher dimension.",
      "comment": "33 pages, 3 Figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-01-04T17:36:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple exclusion process",
      "diffusive cutoff window",
      "mixing time",
      "proof remains valid",
      "initial conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.7296L"
    }
  }
}