{
  "id": "1804.04608",
  "version": "v1",
  "published": "2018-04-12T16:18:12.000Z",
  "updated": "2018-04-12T16:18:12.000Z",
  "title": "Cutoff for the mean-field zero-range process",
  "authors": [
    "Mathieu Merle",
    "Justin Salez"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the mixing time of the unit-rate zero-range process on the complete graph, in the regime where the number $n$ of sites tends to infinity while the density of particles per site stabilizes to some limit $\\rho>0$. We prove that the worst-case total-variation distance to equilibrium drops abruptly from $1$ to $0$ at time $n\\left(\\rho+\\frac{1}{2}\\rho^2\\right)$. More generally, we determine the mixing time from an arbitrary initial configuration. The answer turns out to depend on the largest initial heights in a remarkably explicit way. The intuitive picture is that the system separates into a slowly evolving solid phase and a quickly relaxing liquid phase. As time passes, the solid phase {dissolves} into the liquid phase, and the mixing time is essentially the time at which the system becomes completely liquid. Our proof combines meta-stability, separation of timescale, fluid limits, propagation of chaos, entropy, and a spectral estimate by Morris (2006).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-12T16:18:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J27",
      "37A25",
      "82C22"
    ],
    "keywords": [
      "mean-field zero-range process",
      "mixing time",
      "worst-case total-variation distance",
      "largest initial heights",
      "arbitrary initial configuration"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}