Ballisticity conditions for random walk in random environment |
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Authors: | A Drewitz A F Ram??rez |
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Institution: | 1. Institut f??r Mathematik, Technische Universit?t Berlin, Sekr. MA 7-5, Str. des 17. Juni 136, 10623, Berlin, Germany 2. Facultad de Matem??ticas, Pontificia Universidad Cat??lica de Chile, Vicu?a Mackenna 4860 Macul, Santiago, Chile
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Abstract: | Consider a random walk in a uniformly elliptic i.i.d. random environment in dimensions d ?? 2. In 2002, Sznitman introduced for each ${\gamma\in (0, 1)}$ the ballisticity conditions (T) ?? and (T??), the latter being defined as the fulfillment of (T) ?? for all ${\gamma\in (0, 1)}$ . He proved that (T??) implies ballisticity and that for each ${\gamma\in (0.5, 1)}$ , (T) ?? 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 ?? d is a dimension dependent constant taking values in the interval (0.366, 0.388), (T) ?? is equivalent to (T??). This is achieved by a detour along the effective criterion, the fulfillment 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. |
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