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, on $Z^d$, $d\ge 3$, we explicitly compute the critical integrability exponent. Our result is more general and applies forexample to finitely generated transient Cayley graphs. In terms of reinforced random walks it implies that linearly edge-oriented 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 $d\ge 3$. 2008. ⟨hal-00341904v2⟩