Limit Theorem for sub-ballistic random walks in Dirichlet environment in dimension $d\geq 3$

Abstract

We look at random walks in Dirichlet environment. It was known that in dimension $d\geq 3$, if the walk is sub-ballistic, the displacement of the walk is polynomial of order $\kappa$ for some explicit $\kappa$. We show that the walk, after renormalization, actually converges to a $\kappa$-stable completely asymmetric Levy Process.