Random walks in random Dirichlet environment are transient in dimension $d\ge 3$

Abstract

We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${\mathbb{Z}}^d$, RWDE are parameterized by a $2d$-uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $d\ge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $d\ge 3$.

New version published at PTRF with an analytic proof of lemma 1

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Cited by in corpus (8)

• Localization for Linearly Edge Reinforced Random Walks
• Reversed Dirichlet environment and directional transience of random walks in Dirichlet random environment
• Random Dirichlet environment viewed from the particle in dimension $d\ge 3$
• Reinforced random walk
• Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model
• Ellipticity criteria for ballistic behavior of random walks in random environment
• Random Walks in Dirichlet Random Environments on $\mathbb{Z}$ with Bounded Jumps
• A zero-one law for random walks in random environments on ${\mathbb{Z}}^2$ with bounded jumps