{
  "id": "1402.0969",
  "version": "v2",
  "published": "2014-02-05T08:34:11.000Z",
  "updated": "2014-05-19T07:12:27.000Z",
  "title": "Invariant Coupling of Determinantal Measures on Sofic Groups",
  "authors": [
    "Russell Lyons",
    "Andreas Thom"
  ],
  "comment": "39 pages, no figures; v2 final version",
  "doi": "10.1017/etds.2014.70",
  "categories": [
    "math.PR",
    "math.GR",
    "math.OA"
  ],
  "abstract": "To any positive contraction $Q$ on $\\ell^2(W)$, there is associated a determinantal probability measure ${\\mathbf P}^Q$ on $2^W$, where $W$ is a denumerable set. Let $\\Gamma$ be a countable sofic finitely generated group and $G = (\\Gamma, \\mathsf{E})$ be a Cayley graph of $\\Gamma$. We show that if $Q_1$ and $Q_2$ are two $\\Gamma$-equivariant positive contractions on $\\ell^2(\\Gamma)$ or on $\\ell^2(\\mathsf{E})$ with $Q_1 \\le Q_2$, then there exists a $\\Gamma$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination ${\\bf P}^{Q_1} \\preccurlyeq {\\bf P}^{Q_2}$. In particular, this applies to the wired and free uniform spanning forests, which was known before only when $\\Gamma$ is residually amenable. In the case of spanning forests, we also give a second more explicit proof, which has the advantage of showing an explicit way to create the free uniform spanning forest as a limit over a sofic approximation. Another consequence of our main result is to prove that all determinantal probability measures ${\\bf P}^Q$ as above are ${\\bar d}$-limits of finitely dependent processes. Thus, when $\\Gamma$ is amenable, ${\\bf P}^Q$ is isomorphic to a Bernoulli shift, which was known before only when $\\Gamma$ is abelian. We also prove analogous results for sofic unimodular random rooted graphs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-19T07:12:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "determinantal measures",
      "sofic groups",
      "free uniform spanning forest",
      "invariant coupling",
      "sofic finitely generated group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.0969L"
    }
  }
}