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 Seppalainen 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.