{
  "id": "2102.02606",
  "version": "v1",
  "published": "2021-02-04T13:38:50.000Z",
  "updated": "2021-02-04T13:38:50.000Z",
  "title": "Mixing time for the asymmetric simple exclusion process in a random environment",
  "authors": [
    "Hubert Lacoin",
    "Shangjie Yang"
  ],
  "comment": "34 pages, 5 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the simple exclusion process in the integer segment $ [1, N]$ with $k\\le N/2$ particles and spatially inhomogenous jumping rates. A particle at site $x\\in [ 1, N]$ jumps to site $x-1$ (if $x\\ge 2$) at rate $1-\\omega_x$ and to site $x+1$ (if $x \\le N-1$) at rate $\\omega_x$ if the target site is not occupied. The sequence $\\omega=(\\omega_x)_{ x \\in \\mathbb{Z}}$ is chosen by IID sampling from a probability law whose support is bounded away from zero and one (in other words the random environment satisfies the uniform ellipticity condition). We further assume $\\mathbb{E}[ \\log \\rho_1 ]<0$ where $\\rho_1:= (1-\\omega_1)/\\omega_1$, which implies that our particles have a tendency to move to the right. We prove that the mixing time of the exclusion process in this setup grows like a power of $N$. More precisely, for the exclusion process with $N^{\\beta+o(1)}$ particles where $\\beta\\in [0,1)$, we have in the large $N$ asymptotic $$ N^{\\max\\left(1,\\frac {1}{\\lambda}, \\beta+ \\frac 1 {2\\lambda}\\right)+o(1)} \\le t_{\\mathrm{Mix}}^{N,k} \\le N^{C+o(1)}$$ where $\\lambda>0$ is such that $\\mathbb{E}[\\rho_1^{\\lambda}]=1$ ($\\lambda=\\infty$ if the equation has no positive root) and $C$ is a constant which depends on the distribution of $\\omega$. We conjecture that our lower bound is sharp up to sub-polynomial correction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-04T13:38:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60J27"
    ],
    "keywords": [
      "asymmetric simple exclusion process",
      "mixing time",
      "random environment satisfies",
      "uniform ellipticity condition",
      "sub-polynomial correction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}