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

TL;DR

This paper proves that random walks in random Dirichlet environments are transient in dimensions three and higher, providing insights into their behavior and properties of the Green function.

Contribution

It establishes transience of RWDE in dimensions d≥3 for all parameters and characterizes the moments of the Green function, extending to Cayley graphs.

Findings

RWDE are transient in d≥3 for all parameters

Green function has finite moments and their characterization

Directed edge reinforced random walks are transient in d≥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 $\Z^d$, 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 $d\ge 3$.