{
  "id": "1907.12822",
  "version": "v1",
  "published": "2019-07-30T10:18:29.000Z",
  "updated": "2019-07-30T10:18:29.000Z",
  "title": "Random walk through a fertile site",
  "authors": [
    "Michel Bauer",
    "P. L. Krapivsky",
    "Kirone Mallick"
  ],
  "comment": "19 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study non-interacting random walkers (RWs) on homogeneous hyper-cubic lattices with one special fertile site where RWs can reproduce at rate $\\mu$. We show that the total number $\\mathcal{N}(t)$ and the density of RWs at any site grow exponentially with time in low dimensions, $d=1$ and $d=2$; above the lower critical dimension, $d>d_c=2$, the number of RWs may remain finite forever for any $\\mu$, and surely remains finite when $\\mu\\leq \\mu_d$. We determine the critical multiplication rate $\\mu_d$ and show that the average number of RWs grows exponentially if $\\mu>\\mu_d$. The distribution $P_N(t)$ of the total number of RWs remains broad when $d\\leq 2$, and also when $d>2$ and $\\mu>\\mu_d$. We derive explicit expressions for the first moments of $\\mathcal{N}(t)$ and establish a recurrence that allows, in principle, to compute an arbitrary moment. In the critical regime, $\\langle \\mathcal{N}\\rangle$ grows as $\\sqrt{t}$ for $d=3$, $t/\\ln t$ for $d=4$ and $t$ (for $d>4$). Higher moments grow anomalously, $\\langle \\mathcal{N}^m\\rangle\\sim \\langle \\mathcal{N}\\rangle^{2m-1}$, instead of the normal growth, $\\langle \\mathcal{N}^m\\rangle\\sim \\langle \\mathcal{N}\\rangle^{m}$, valid in the exponential phase. The distribution of the number of RWs in the critical regime is asymptotically stationary and universal, viz. it is independent of the spatial dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T10:18:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "total number",
      "study non-interacting random walkers",
      "critical regime",
      "higher moments grow",
      "remain finite forever"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}