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

Christophe Sabot (1)

1 ICJ - Institut Camille Jordan

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 $\Z^d$, 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 $d\ge 3$.

Mots clés: Random walk in random environment; Dirichlet distribution; Reinforced random walks; Max-Flow Min-Cut theorem

Domaines: Probabilités [math.PR]

Christophe Sabot. Random walks in random Dirichlet environment are transient in dimension $d\ge 3$. Probability Theory and Related Fields, 2011, 151 (1-2), pp.297-317. ⟨10.1007/s00440-010-0300-0⟩. ⟨hal-00341904v3⟩