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Quenched Sub-Exponential Tail Estimates for One-Dimensional Random Walk in Random Environment.
- Source :
- Communications in Mathematical Physics; May1998, Vol. 194 Issue 1, p177-190, 14p
- Publication Year :
- 1998
-
Abstract
- Suppose that the integers are assigned i.i.d. random variables {ω<subscript>x</subscript>} (taking values in the unit interval), which serve as an environment. This environment defines a random walk { X <subscript> n </subscript>} (called a RWRE) which, when at x, moves one step to the right with probability ω<subscript>x</subscript>, and one step to the left with probability 1- ω<subscript> x </subscript>. Solomon (1975) determined the almost-sure asymptotic speed v <subscript>α</subscript> (=rate of escape) of a RWRE. Greven and den Hollander (1994) have proved a large deviation principle for X <subscript> n </subscript> / n, conditional upon the environment, with deterministic rate function. For certain environment distributions where the drifts 2 ω<subscript> x </subscript>-1 can take both positive and negative values, their rate function vanisheson an interval (0, v <subscript>α</subscript>). 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 <subscript> n </subscript> / n is smaller than v, v∈ (0, v <subscript>α</subscript>), 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 . [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00103616
- Volume :
- 194
- Issue :
- 1
- Database :
- Complementary Index
- Journal :
- Communications in Mathematical Physics
- Publication Type :
- Academic Journal
- Accession number :
- 49955602
- Full Text :
- https://doi.org/10.1007/s002200050354