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

Christophe Sabot; Laurent Tournier

Annales de l'I.H.P. Probabilités et statistiques (2011)

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

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topSabot, 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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