Random walks on FKG-horizontally oriented lattices

We study the asymptotic behavior of the simple random walk on oriented versions of Z. The considered lattices are not directed on the vertical axis but unidirectional on the horizontal one, with centered random orientations which are positively correlated. We prove that the simple random walk is transient and also prove a functional limit theorem in the space D([0,∞[, R) of cadlag functions, with an unconventional normalization.