{
  "id": "1508.04791",
  "version": "v1",
  "published": "2015-08-19T20:47:06.000Z",
  "updated": "2015-08-19T20:47:06.000Z",
  "title": "The intermediate disorder regime for a directed polymer model on a hierarchical lattice",
  "authors": [
    "Tom Alberts",
    "Jeremy Clark",
    "Sasa Kocic"
  ],
  "comment": "41 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number $b\\in \\mathbb{N}$ and a segment number $s\\in \\mathbb{N}$. When $b\\leq s$ previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature $\\beta$, and thus weak disorder reigns only for $\\beta=0$ (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature $\\beta\\equiv \\beta_{n}$ vanishes at an appropriate rate as the size $n$ of the system grows. Our analysis requires separate treatment for the cases $b<s$ and $b=s$. In the case $b<s$ we prove that when the inverse temperature is taken to be of the form $\\beta_{n}=\\widehat{\\beta} (b/s)^{n/2}$ for $\\widehat{\\beta}>0$, the normalized partition function of the system converges weakly as $n \\to \\infty$ to a distribution $\\mathbf{L}(\\widehat{\\beta})$ depending continuously on the parameter $\\widehat{\\beta}$. In the case $b=s$ we find a critical point in the behavior of the model when the inverse temperature is scaled as $\\beta_{n}=\\widehat{\\beta}/n$; for an explicitly computable critical value $\\kappa_{b} > 0$ the variance of the normalized partition function converges to zero with large $n$ when $\\widehat{\\beta}\\leq \\kappa_{b}$ and grows without bound when $\\widehat{\\beta}>\\kappa_{b}$. Finally, we prove a central limit theorem for the normalized partition function when $\\widehat{\\beta}\\leq \\kappa_{b}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-19T20:47:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60F05"
    ],
    "keywords": [
      "intermediate disorder regime",
      "directed polymer model",
      "inverse temperature",
      "hierarchical lattice",
      "weak disorder reigns"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150804791A"
    }
  }
}