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

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