Abstract
We prove that random walks in i.i.d. random environments which oscillate in a given direction have velocity zero with respect to that direction. This complements existing results thus giving a general law of large numbers under the only assumption of a certain zero-one law, which is known to hold if the dimension is two.
Highlights
Remark: The version of this result which is quoted in [3, Theorem 3.2.2] assumes for some other purpose that the environment is uniformly elliptic, i.e. that P-a.s. ω(z, z + e) > ε for some uniform “ellipticity constant” ε > 0
There is no general law of large numbers under the assumption of (1) only which would state the existence of a deterministic v towards which Xn/n converges P0-a.s
There are two problems to be solved in order to derive such a law from Theorem A: (i) Show that v = 0 if 0 < P0[A ] < 1
Summary
Remark: The version of this result which is quoted in [3, Theorem 3.2.2] assumes for some other purpose that the environment is uniformly elliptic, i.e. that P-a.s. ω(z, z + e) > ε for some uniform “ellipticity constant” ε > 0. Given such an environment ω and some x ∈ Zd, the so-called quenched probability measure Px,ω on the path space (Zd)N is characterized by Using a certain renewal structure, Sznitman and Zerner (see [2] and [3, Theorem 3.2.2]) proved the following law of large numbers.
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