{
  "id": "1409.7766",
  "version": "v1",
  "published": "2014-09-27T05:35:29.000Z",
  "updated": "2014-09-27T05:35:29.000Z",
  "title": "The Random Transposition Dynamics on Random Regular Graphs and the Gaussian Free Field",
  "authors": [
    "Shirshendu Ganguly",
    "Soumik Pal"
  ],
  "comment": "42 pages, 6 figures. arXiv admin note: text overlap with arXiv:1406.7043 by other authors",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider $d$ many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-) graph of degree $2d$. We consider the following dynamics. The dimension of each permutation grows by coupled Chinese Restaurant Processes, while in time each permutation evolves according to the random transposition chain. Asymptotically in the size of the graph one observes a remarkable evolution of short cycles and linear eigenvalue statistics in dimension and time. In dimension, it was shown in Johnson and Pal (2014), that cycle counts are described by a Poisson field of Yule processes. Here, we give a Poisson random surface description in dimension and time of the limiting cycle counts for every $d$. As $d$ grows to infinity, the fluctuation of the limiting cycle counts, across dimension, converges to the Gaussian Free Field. When time is run infinitesimally slowly, this field is preserved by a stationary Gaussian dynamics. The laws of these processes are similar to eigenvalue fluctuations of the minor process of a real symmetric Wigner matrix whose coordinates evolve as i.i.d. stationary stochastic processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-27T05:35:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian free field",
      "random regular graphs",
      "random transposition dynamics",
      "limiting cycle counts",
      "real symmetric wigner matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.7766G"
    }
  }
}