# Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment

• Volume: 47, Issue: 1, page 1-8
• ISSN: 0246-0203

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## Abstract

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We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ ℓ →n +∞ for some ℓ) with positive probability. In dimension 2, this result is strenghened to an almost sure directional transience thanks to the 0–1 law from [Ann. Probab.29 (2001) 1716–1732]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Random walks in random Dirichlet environment are transient in dimension d ≥ 3 (2009), Preprint]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used initially.

## How to cite

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Sabot, Christophe, and Tournier, Laurent. "Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment." Annales de l'I.H.P. Probabilités et statistiques 47.1 (2011): 1-8. <http://eudml.org/doc/241940>.

@article{Sabot2011,
abstract = {We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ ℓ →n +∞ for some ℓ) with positive probability. In dimension 2, this result is strenghened to an almost sure directional transience thanks to the 0–1 law from [Ann. Probab.29 (2001) 1716–1732]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Random walks in random Dirichlet environment are transient in dimension d ≥ 3 (2009), Preprint]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used initially.},
author = {Sabot, Christophe, Tournier, Laurent},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random walk; random environment; Dirichlet distribution; directional transience; time reversal},
language = {eng},
number = {1},
pages = {1-8},
publisher = {Gauthier-Villars},
title = {Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment},
url = {http://eudml.org/doc/241940},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Sabot, Christophe
AU - Tournier, Laurent
TI - Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 1
SP - 1
EP - 8
AB - We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of ℤd or, equivalently, oriented edge reinforced random walks on ℤd. The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of ℤd. We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it satisfies Xn ⋅ ℓ →n +∞ for some ℓ) with positive probability. In dimension 2, this result is strenghened to an almost sure directional transience thanks to the 0–1 law from [Ann. Probab.29 (2001) 1716–1732]. Our proof relies on the property of stability of Dirichlet environment by time reversal proved in [Random walks in random Dirichlet environment are transient in dimension d ≥ 3 (2009), Preprint]. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used initially.
LA - eng
KW - random walk; random environment; Dirichlet distribution; directional transience; time reversal
UR - http://eudml.org/doc/241940
ER -

## References

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6. [6] C. Sabot. Random walks in random Dirichlet environment are transient in dimension d≥3. Preprint, 2009. Available at arXiv:0905.3917v1[math.PR].
7. [7] A.-S. Sznitman. Topics in random walks in random environment. In School and Conference on Probability Theory. ICTP Lect. Notes XVII 203–266 (electronic). Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004. Zbl1060.60102MR2198849
8. [8] L. Tournier. Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 (2009) 431–451 (electronic). Zbl1192.60113MR2480548
9. [9] M. Zerner. The zero–one law for planar random walks in i.i.d. random environments revisited. Electron. Comm. Probab. 12 (2007) 326–335 (electronic). Zbl1128.60090MR2342711
10. [10] M. Zerner and F. Merkl. A zero–one law for planar random walks in random environment. Ann. Probab. 29 (2001) 1716–1732. Zbl1016.60093MR1880239

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