On an Effective Class of Ballistic Random Walks in Mixing Random Environments

We prove ballistic behaviour as well as an annealed functional central limit theorem for random walks in mixing random environments (RWRE). The ballistic hypothesis will be an effective polynomial condition as the one introduced by Berger \textit{et al.} \cite{BDR14}. The novel idea therein was the construction of several renormalization steps \textit{at the same time} and not only one as usual, turning out more flexibility for seed estimates. We shall follow a closer path as the one in \cite{BDR14} for our proof, indeed we introduce a new mixing effective criterion which will be implied by the polynomial condition. All of these constructions will dispose us to prove in a mixing framework the RWRE conjecture concerning the equivalence between each condition $(T^\gamma)|\ell$, for $\gamma\in (0,1)$ and $\ell \in \mathbb S^{d-1}$. This work complements the previous work \cite{Gue17} and completes the answer about the meaning of condition $(T')|\ell$ in a mixing setting, an open question appearing in \cite{CZ01}.