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

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 Sepp\"al\"ainen 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 $(T)_{\gamma}$ condition of Sznitman we reduce the moment condition to ${\Bbb E}(\tau^2(\ln \tau)^{1+m})<+\infty$ for $m>1+1/\gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.