Limit theorem for subdiffusive random walk in Dirichlet random environment in dimension $d \ge 3$

Abstract

We consider random walk in Dirichlet random environment in ${\mathbf{Z}^d, d\ge 3}$, which corresponds to the case where the environment is constructed from i.i.d. transition probabilities at each vertex with a Dirichlet distribution with parameters $(\alpha_i)_{1 \le i \le 2d}$. Dirichlet environments are weakly elliptic and the walk can be slowdowned by local traps whose strength are governed by a parameter $\kappa$. In this paper we prove a stable limit theorem when the walk is ballistic but subdiffusive, i.e. when ${\kappa \in (1,2)}$. This completes the result of Poudevigne (arXiv:1909.03866) who proved a sub-ballistic stable limit theorem when $\kappa\in (0,1)$. Contrary to Poudevigne, we have to assume Sznitman's condition $\mathbf{(T)}$ to prove the limit theorem since we work at the level of fluctuations and need a better control on renewal times.