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

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.