{
  "id": "1703.09731",
  "version": "v1",
  "published": "2017-03-28T18:04:34.000Z",
  "updated": "2017-03-28T18:04:34.000Z",
  "title": "Survival asymptotics for branching random walks in IID environments",
  "authors": [
    "Janos Englander",
    "Yuval Peres"
  ],
  "comment": "2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We first study a model, introduced recently in \\cite{ES}, of a critical branching random walk in an IID random environment on the $d$-dimensional integer lattice. The walker performs critical (0-2) branching at a lattice point if and only if there is no `obstacle' placed there. The obstacles appear at each site with probability $p\\in [0,1)$ independently of each other. We also consider a similar model, where the offspring distribution is subcritical. Let $S_n$ be the event of survival up to time $n$. We show that on a set of full $\\mathbb P_p$-measure, as $n\\to\\infty$, (i) Critical case: P^{\\omega}(S_n)\\sim\\frac{2}{qn}; (ii) Subcritical case: P^{\\omega}(S_n)= \\exp\\left[\\left( -C_{d,q}\\cdot \\frac{n}{(\\log n)^{2/d}} \\right)(1+o(1))\\right], where $C_{d,q}>0$ does not depend on the branching law. Hence, the model exhibits `self-averaging' in the critical case but not in the subcritical one. I.e., in (i) the asymptotic tail behavior is the same as in a \"toy model\" where space is removed, while in (ii) the spatial survival probability is larger than in the corresponding toy model, suggesting spatial strategies. We utilize a spine decomposition of the branching process as well as some known results on random walks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-28T18:04:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80"
    ],
    "keywords": [
      "branching random walk",
      "survival asymptotics",
      "iid environments",
      "critical case",
      "asymptotic tail behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}