{
  "id": "1305.2034",
  "version": "v2",
  "published": "2013-05-09T08:52:02.000Z",
  "updated": "2014-03-20T10:41:03.000Z",
  "title": "Hausdorff and packing spectra, large deviations, and free energy for branching random walks in $\\R^d$",
  "authors": [
    "Najmeddine Attia",
    "Julien Barral"
  ],
  "comment": "56 pages. This version contains now 2 figures, as well as a result on the growth of minimal supporting tree for the free energy. The paper is accepted for publication in Comm. Math. Phys",
  "categories": [
    "math-ph",
    "math.MG",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Consider an $\\R^d$-valued branching random walk (BRW) on a supercritical Galton Watson tree. Without any assumption on the distribution of this BRW we compute, almost surely and simultaneously, the Hausdorff and packing dimensions of the level sets $E(K)$ of infinite branches in the boundary of the tree (endowed with its standard metric) along which the averages of the BRW have a given closed connected set of limit points $K$. This goes beyond multifractal analysis, which only considers those level sets when $K$ ranges in the set of singletons $\\{\\alpha\\}$, $\\alpha\\in\\R^d$. We also give a $0$-$\\infty$ law for the Hausdorff and packing measures of the level sets $E(\\{\\alpha\\})$, and compute the free energy of the associated logarithmically correlated random energy model in full generality. Moreover, our results complete the previous works on multifractal analysis by including the levels $\\alpha$ which do not belong to the range of the gradient of the free energy. This covers in particular a situation until now badly understood, namely the case where a first order phase transition occurs. As a consequence of our study, we can also describe the whole singularity spectrum of Mandelbrot measures, as well as the associated free energy function (or $L^q$-spectrum), when a first order phase transition occurs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-20T10:41:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A78",
      "60G57",
      "82B44"
    ],
    "keywords": [
      "free energy",
      "branching random walk",
      "first order phase transition occurs",
      "large deviations",
      "packing spectra"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-014-2087-9",
      "journal": "Communications in Mathematical Physics",
      "year": 2014,
      "month": "Oct",
      "volume": 331,
      "number": 1,
      "pages": 139
    },
    "note": {
      "typesetting": "TeX",
      "pages": 56,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014CMaPh.331..139A"
    }
  }
}