{
  "id": "2105.13486",
  "version": "v1",
  "published": "2021-05-27T22:47:36.000Z",
  "updated": "2021-05-27T22:47:36.000Z",
  "title": "A direct comparison between the mixing time of the interchange process with \"few\" particles and independent random walks",
  "authors": [
    "Jonathan Hermon",
    "Richard Pymar"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the interchange process with $k$ particles (${\\rm IP}(k)$) on $n$-vertex hypergraphs in which each hyperedge $e$ rings at rate $r_e$. When $e$ rings, the particles occupying it are permuted according to a random permutation from some arbitrary law, where our only assumption is that ${\\rm IP}(2)$ has uniform stationary distribution. We show that $t_{\\rm mix}^{{\\rm IP}(k)}(\\epsilon)=O_{b}(t_{\\rm mix}^{{\\rm IP}(2)}(\\epsilon/k))$, where $t_{\\rm mix}^{{\\rm IP}(i)}(\\epsilon)$ is the $\\epsilon$ total-variation mixing time of ${\\rm IP}(i)$, provided that $kn^{-2}Rt_{\\rm mix}^{{\\rm IP}(2)}(\\epsilon/k)=O((\\epsilon/k)^b)$ for some $b>0$, where $R=\\sum_e r_e|e|(|e|-1)$ is $n(n-1)$ times the particle-particle interaction rate at equilibrium. This has some consequences concerning the validity in this regime of conjectures of Oliveira about comparison of the $\\epsilon$ mixing time of ${\\rm IP}(k)$ to that of $k$ independent particles, each evolving according to ${\\rm IP}(1)$, denoted ${\\rm RW}(k)$, and of Caputo about comparison of the spectral-gap of ${\\rm IP}(k)$ to that of a single particle ${\\rm IP}(1)={\\rm RW}(1)$. We also show that $t_{\\rm mix}^{\\mathrm{IP}(k)}(\\epsilon) \\asymp t_{\\rm mix}^{{\\rm RW}(1)}(\\epsilon)\\asymp t_{{\\rm mix}}^{{\\rm RW}(k)}(\\epsilon k/4)$ for all $k\\lesssim n^{1-\\Omega(1)}$ and all $\\epsilon\\le\\frac 1k\\wedge\\frac 14$ for vertex-transitive graphs of constant degree, as well as for general graphs satisfying a mild (\"transience-like\") heat-kernel condition. In the case where the particles occupying a hyperedge $e$ are permuted uniformly at random when $e$ rings we obtain results bounding the spectral gap of ${\\rm IP}(k)$ in terms of that ${\\rm RW}(1)$. The proof does not use Morris' chameleon process. It can be seen as a rigorous and direct way of arguing that when the number of particles is fairly small, the system behaves similarly to $k$ independent particles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-27T22:47:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J27",
      "60K35",
      "82C22"
    ],
    "keywords": [
      "independent random walks",
      "interchange process",
      "direct comparison",
      "independent particles",
      "uniform stationary distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}