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 Rassoul-Agha and Seppäläinen (2009) and Berger and Zeitouni in Berger and Zeitouni (2008) 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 E(τ2(lnτ)1+m)<+∞ for m>1+1/γ, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.