Central limit theorem for a tagged particle in asymmetric simple exclusion

We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the tagged particle at time t depends on the initial configuration, through the number of empty sites in the interval [ 0 , ( p − q ) α t ] divided by α , on the hyperbolic time scale and on a longer time scale, namely N 4 / 3 .