A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under $(T)_γ$

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_γ$ condition of Sznitman we reduce the moment condition to ${\Bbb E}(τ^2(\ln τ)^{1+m})<+\infty$ for $m>1+1/γ$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

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