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

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 -