{
  "id": "0903.4465",
  "version": "v1",
  "published": "2009-03-25T20:47:44.000Z",
  "updated": "2009-03-25T20:47:44.000Z",
  "title": "Ballisticity conditions for random walk in random environment",
  "authors": [
    "Alexander Drewitz",
    "Alejandro F. Ramírez"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions $d\\ge 2$. In 2002, Sznitman introduced for each $\\gamma\\in (0,1)$ the ballisticity conditions $(T)_\\gamma$ and $(T'),$ the latter being defined as the fulfilment of $(T)_\\gamma$ for all $\\gamma\\in (0,1).$ He proved that $(T')$ implies ballisticity and that for each $\\gamma\\in (0.5,1),$ $(T)_\\gamma$ is equivalent to $(T')$. It is conjectured that this equivalence holds for all $\\gamma\\in (0,1).$ Here we prove that for $\\gamma\\in (\\gamma_d,1),$ where $\\gamma_d$ is a dimension dependent constant taking values in the interval $(0.366,0.388),$ $(T)_\\gamma$ is equivalent to $(T').$ This is achieved by a detour along the effective criterion, the fulfilment of which we establish by a combination of techniques developed by Sznitman giving a control on the occurrence of atypical quenched exit distributions through boxes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-25T20:47:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "82D30"
    ],
    "keywords": [
      "random walk",
      "random environment",
      "ballisticity conditions",
      "dimension dependent constant",
      "implies ballisticity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.4465D"
    }
  }
}