{
  "id": "1609.00989",
  "version": "v1",
  "published": "2016-09-04T20:40:37.000Z",
  "updated": "2016-09-04T20:40:37.000Z",
  "title": "Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails",
  "authors": [
    "Marek Biskup",
    "Wolfgang Koenig",
    "Renato Soares dos Santos"
  ],
  "comment": "66 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "We study the solutions $u=u(x,t)$ to the Cauchy problem on $\\mathbb Z^d\\times(0,\\infty)$ for the parabolic equation $\\partial_t u=\\Delta u+\\xi u$ with initial data $u(x,0)=1_{\\{0\\}}(x)$. Here $\\Delta$ is the discrete Laplacian on $\\mathbb Z^d$ and $\\xi=(\\xi(z))_{z\\in\\mathbb Z^d}$ is an i.i.d.\\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\\Delta+\\xi$ and the distance to the origin. The processes $t\\mapsto Z_t$ and $t \\mapsto \\tfrac1t \\log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $\\Delta+\\xi$ in large sets recently proved by the first two authors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-04T20:40:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H25",
      "82B44"
    ],
    "keywords": [
      "parabolic anderson model",
      "mass concentration",
      "doubly-exponential tails",
      "eigenvalue order statistics",
      "local dirichlet eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}