Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment

Jonathon Peterson

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 3, page 685-709
  • ISSN: 0246-0203

Abstract

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We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP>0 and there exists an s∈(1, 2) such that the annealed law of n−1/s(Xn−nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {tk} and {tk'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence tk' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0, 1).

How to cite

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Peterson, Jonathon. "Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment." Annales de l'I.H.P. Probabilités et statistiques 45.3 (2009): 685-709. <http://eudml.org/doc/78039>.

@article{Peterson2009,
abstract = {We consider a nearest-neighbor, one-dimensional random walk \{Xn\}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP&gt;0 and there exists an s∈(1, 2) such that the annealed law of n−1/s(Xn−nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences \{tk\} and \{tk'\} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence tk' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0, 1).},
author = {Peterson, Jonathon},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk; random environment},
language = {eng},
number = {3},
pages = {685-709},
publisher = {Gauthier-Villars},
title = {Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment},
url = {http://eudml.org/doc/78039},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Peterson, Jonathon
TI - Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 3
SP - 685
EP - 709
AB - We consider a nearest-neighbor, one-dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient with speed vP&gt;0 and there exists an s∈(1, 2) such that the annealed law of n−1/s(Xn−nvP) converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {tk} and {tk'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence tk, but the quenched limiting distribution along the subsequence tk' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameter s∈(0, 1).
LA - eng
KW - random walk; random environment
UR - http://eudml.org/doc/78039
ER -

References

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  7. [7] S. M. Kozlov and S. A. Molchanov. Conditions for the applicability of the central limit theorem to random walks on a lattice (Russian). Dokl. Akad. Nauk SSSR 278 (1984) 531–534. Zbl0603.60020MR764989
  8. [8] J. Peterson. Limiting distributions and large devations for random walks in random enviroments. Ph.D. thesis, University of Minnesota, 2008. Available at arXiv:math0810.0257v1. 
  9. [9] J. Peterson and O. Zeitouni. Quenched limits for transient, zero-speed one-dimensional random walk in random environment. Ann. Probab. 37 (2009) 143–188. Zbl1179.60070MR2489162
  10. [10] F. Solomon. Random walks in random environments. Ann. Probab. 3 (1975) 1–31. Zbl0305.60029MR362503
  11. [11] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics 189–312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. Zbl1060.60103MR2071631

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