{
  "id": "2506.20147",
  "version": "v1",
  "published": "2025-06-25T05:59:22.000Z",
  "updated": "2025-06-25T05:59:22.000Z",
  "title": "Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics",
  "authors": [
    "Xi Geng",
    "Sheng Wang",
    "Weijun Xu"
  ],
  "comment": "57 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability one, the solution $u$ to PAM with constant initial data has pointwise growth asymptotics \\[ u(t,x)\\sim e^{L^{*}t^{5/3}+o(t^{5/3})} \\] as $t \\rightarrow +\\infty$. Both the power $t^{5/3}$ on the exponential and the exact value of $L^*$ are different from their counterparts in the Euclidean situation. They are determined through an explicit optimisation procedure. Our proof relies on certain fine localisation techniques, which also reveals a stronger non-Euclidean localisation mechanism.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-25T05:59:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic anderson model",
      "hyperbolic space",
      "stronger non-euclidean localisation mechanism",
      "time-independent gaussian potential",
      "exact quenched asymptotic growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 57,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}