{
  "id": "1608.03265",
  "version": "v1",
  "published": "2016-08-10T19:42:03.000Z",
  "updated": "2016-08-10T19:42:03.000Z",
  "title": "Pinning of a renewal on a quenched renewal",
  "authors": [
    "Kenneth S. Alexander",
    "Quentin Berger"
  ],
  "comment": "53 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $\\sigma$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $\\sigma$ have infinite mean. The \"polymer\" -- of length $\\sigma_N$ -- is given by another renewal $\\tau$, whose law is modified by the Boltzmann weight $\\exp(\\beta\\sum_{n=1}^N \\mathbf{1}_{\\{\\sigma_n\\in \\tau\\}})$. Our assumption is that $\\tau$ and $\\sigma$ have gap distributions with power-law-decay exponents $1+\\alpha$ and $1+\\tilde \\alpha$ respectively, with $\\alpha\\geq 0,\\tilde \\alpha>0$. There is a localization phase transition: above a critical value $\\beta_c$ the free energy is positive, meaning that $\\tau$ is \\emph{pinned} on the quenched renewal $\\sigma$. We consider the question of relevance of the disorder, that is to know when $\\beta_c$ differs from its annealed counterpart $\\beta_c^{\\rm ann}$. We show that $\\beta_c=\\beta_c^{\\rm ann}$ whenever $ \\alpha+\\tilde \\alpha \\geq 1$, and $\\beta_c=0$ if and only if the renewal $\\tau\\cap\\sigma$ is recurrent. On the other hand, we show $\\beta_c>\\beta_c^{\\rm ann}$ when $ \\alpha+\\frac32\\, \\tilde \\alpha <1$. We give evidence that this should in fact be true whenever $ \\alpha+\\tilde \\alpha<1$, providing examples for all such $ \\alpha,\\tilde \\alpha$ of distributions of $\\tau,\\sigma$ for which $\\beta_c>\\beta_c^{\\rm ann}$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($\\sigma_N=\\tau_N$), and one in which the polymer length is $\\tau_N$ rather than $\\sigma_N$. In both cases we show the critical point is the same as in the original model, at least when $ \\alpha>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-10T19:42:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K05",
      "60K35",
      "60K37",
      "82B27",
      "82B44"
    ],
    "keywords": [
      "quenched renewal",
      "pinning model",
      "localization phase transition",
      "nonzero potential values",
      "renewal times"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}