{
  "id": "math/0703434",
  "version": "v3",
  "published": "2007-03-14T20:11:22.000Z",
  "updated": "2009-01-20T13:35:25.000Z",
  "title": "Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction",
  "authors": [
    "Francesco Caravenna",
    "Jean-Dominique Deuschel"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/08-AOP395 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2008, Vol. 36, No. 6, 2388-2433",
  "doi": "10.1214/08-AOP395",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a random field $\\varphi:\\{1,...,N\\}\\to\\mathbb{R}$ as a model for a linear chain attracted to the defect line $\\varphi=0$, that is, the x-axis. The free law of the field is specified by the density $\\exp(-\\sum_iV(\\Delta\\varphi_i))$ with respect to the Lebesgue measure on $\\mathbb{R}^N$, where $\\Delta$ is the discrete Laplacian and we allow for a very large class of potentials $V(\\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\\varepsilon\\ge0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\\varepsilon$ of the pinning reward varies: both in the pinning ($a=\\mathrm{p}$) and in the wetting ($a=\\mathrm{w}$) case, there exists a critical value $\\varepsilon_c^a$ such that when $\\varepsilon>\\varepsilon_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\\varepsilon<\\varepsilon_c^a$ (delocalization). The two critical values are nontrivial and distinct: $0<\\varepsilon_c^{\\mat hrm{p}}<\\varepsilon_c^{\\mathrm{w}}<\\infty$, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\\varepsilon=\\varepsilon_c^{\\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\\varepsilon=\\varepsilon_c^{\\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-01-20T13:35:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60F05",
      "82B41"
    ],
    "keywords": [
      "laplacian interaction",
      "wetting transition",
      "defect line",
      "markov renewal theory description",
      "pinning model"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......3434C"
    }
  }
}