{
  "id": "2002.06899",
  "version": "v1",
  "published": "2020-02-17T11:52:51.000Z",
  "updated": "2020-02-17T11:52:51.000Z",
  "title": "One-dimensional polymers in random environments: stretching vs. folding",
  "authors": [
    "Quentin Berger",
    "Chien-Hao Huang",
    "Niccolo Torri",
    "Ran Wei"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article we study a \\emph{non-directed polymer model} on $\\mathbb Z$, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of ``rewards'' (or penalities) $\\beta \\omega_x -h$ sitting on the range of the random walk, where $(\\omega_x)_{x\\in \\mathbb Z}$ are i.i.d.\\ random variables (the disorder), and where $\\beta\\geq 0$ (disorder strength) and $h\\in \\mathbb{R}$ (external field) are two parameters. When $\\beta=0,h>0$, this corresponds to a random walk penalized by its range; when $\\beta>0, h=0$, this corresponds to the ``standard'' polymer model in random environment, except that it is non-directed. In this work, we allow the parameters $\\beta,h$ to vary according to the length of the random walk, and we study in detail the competition between the \\emph{stretching effect} of the disorder, the \\emph{folding effect} of the external field (if $h\\ge 0$), and the \\emph{entropy cost} of atypical trajectories. We prove a complete description of the (rich) phase diagram. For instance, in the case $\\beta>0, h=0$ of the non-directed polymer, if $\\omega_x$ ha a finite second moment, we find a transversal fluctuation exponent $\\xi=2/3$, and we identify the limiting distribution of the rescaled log-partition function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-17T11:52:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82D60",
      "60K37",
      "60G70"
    ],
    "keywords": [
      "random environment",
      "one-dimensional polymers",
      "one-dimensional simple random walk",
      "external field",
      "polymer model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}