{
  "id": "2405.04335",
  "version": "v1",
  "published": "2024-05-07T14:04:32.000Z",
  "updated": "2024-05-07T14:04:32.000Z",
  "title": "The tail distribution function of the partition function for directed polymer in the weak disorder phase",
  "authors": [
    "Stefan Junk",
    "Hubert Lacoin"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate the upper tail distribution of the partition function of the directed polymer in a random environment on $\\mathbb Z^d$ in the weak disorder phase. We show that the distribution of the infinite volume partition function $W^{\\beta}_{\\infty}$ displays a power-law decay, with an exponent $p^*(\\beta)\\in [1+\\frac{2}{d},\\infty)$. We also prove that the distribution of the suprema of the point-to-point and point-to-line partition functions display the same behavior. On the way to these results, we prove a technical estimate of independent interest: the $L^p$-norm of the partition function at the time when it overshoots a high value $A$ is comparable to $A$. We use this estimate to extend the validity of many recent results that were proved under the assumption that the environment is upper bounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-07T14:04:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K37",
      "82B26",
      "82B27",
      "82B44"
    ],
    "keywords": [
      "weak disorder phase",
      "tail distribution function",
      "directed polymer",
      "infinite volume partition function",
      "point-to-line partition functions display"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}