Quenched Sub-Exponential Tail Estimates for One-Dimensional Random Walk in Random Environment

Abstract:

Suppose that the integers are assigned i.i.d. random variables {ωx} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X n } (called a RWRE) which, when at x, moves one step to the right with probability ωx, and one step to the left with probability 1- ω x . Solomon (1975) determined the almost-sure asymptotic speed v α (=rate of escape) of a RWRE. Greven and den Hollander (1994) have proved a large deviation principle for X n /n, conditional upon the environment, with deterministic rate function. For certain environment distributions where the drifts 2 ω x -1 can take both positive and negative values, their rate function vanisheson an interval (0,v α). We find the rate of decay on this interval and prove it is a stretched exponential of appropriate exponent, that is the absolute value of the log of the probability that the empirical mean X n /n is smaller than v, v∈ (0,v α), behaves roughly like a fractional power of n. The annealed estimates of Dembo, Peres and Zeitouni (1996) play a crucial role in the proof. We also deal with the case of positive and zero drifts, and prove there a quenched decay of the form .

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Received: 13 December 1996 / Accepted: 3 October 1997

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Gantert, N., Zeitouni, O. Quenched Sub-Exponential Tail Estimates for One-Dimensional Random Walk in Random Environment . Comm Math Phys 194, 177–190 (1998). https://doi.org/10.1007/s002200050354

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Keywords

  • Random Environment
  • Tail Estimate