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.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On ${mathbb Z}^d$, RWDE are parameterized b
y a 2d-uplet of positive reals called weights. In this paper, we characterize for $dge 3$ the weights for which there exists an absolutely continuous invariant probability for the process viewed from the particle. We can deduce from this result and from [27] a complete description of the ballistic regime for $dge 3$.
We consider random walks in random Dirichlet environment (RWDE) which is a special type of random walks in random environment where the exit probabilities at each site are i.i.d. Dirichlet random variables. On $Z^d$, RWDE are parameterized by a $2d$-
uplet of positive reals. We prove that for all values of the parameters, RWDE are transient in dimension $dge 3$. We also prove that the Green function has some finite moments and we characterize the finite moments. Our result is more general and applies for example to finitely generated symmetric transient Cayley graphs. In terms of reinforced random walks it implies that directed edge reinforced random walks are transient for $dge 3$.
