{
  "id": "0910.4927",
  "version": "v2",
  "published": "2009-10-26T16:56:24.000Z",
  "updated": "2010-06-18T10:30:49.000Z",
  "title": "Maximal Displacement for Bridges of Random Walks in a Random Environment",
  "authors": [
    "Nina Gantert",
    "Jonathon Peterson"
  ],
  "comment": "Revised version, 19 pages, 1 figure To appear in: AIHP Prob. & Stat",
  "categories": [
    "math.PR"
  ],
  "abstract": "It is well known that the distribution of simple random walks on $\\bf{Z}$ conditioned on returning to the origin after $2n$ steps does not depend on $p= P(S_1 = 1)$, the probability of moving to the right. Moreover, conditioned on $\\{S_{2n}=0\\}$ the maximal displacement $\\max_{k\\leq 2n} |S_k|$ converges in distribution when scaled by $\\sqrt{n}$ (diffusive scaling). We consider the analogous problem for transient random walks in random environments on $\\bf{Z}$. We show that under the quenched law $P_\\omega$ (conditioned on the environment $\\omega$), the maximal displacement of the random walk when conditioned to return to the origin at time $2n$ is no longer necessarily of the order $\\sqrt{n}$. If the environment is nestling (both positive and negative local drifts exist) then the maximal displacement conditioned on returning to the origin at time $2n$ is of order $n^{\\kappa/(\\kappa+1)}$, where the constant $\\kappa>0$ depends on the law on environment. On the other hand, if the environment is marginally nestling or non-nestling (only non-negative local drifts) then the maximal displacement conditioned on returning to the origin at time $2n$ is at least $n^{1-\\varepsilon}$ and at most $n/(\\ln n)^{2-\\varepsilon}$ for any $\\varepsilon>0$. As a consequence of our proofs, we obtain precise rates of decay for $P_\\omega(X_{2n}=0)$. In particular, for certain non-nestling environments we show that $P_\\omega(X_{2n}=0) = \\exp\\{-Cn -C'n/(\\ln n)^2 + o(n/(\\ln n)^2) \\}$ with explicit constants $C,C'>0$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-06-18T10:30:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37"
    ],
    "keywords": [
      "random environment",
      "local drifts",
      "maximal displacement",
      "simple random walks",
      "transient random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.4927G"
    }
  }
}