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 in the 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, by the number of empty sites in the interval $[0,(p-q)\alpha t]$ divided by $\alpha$ in the hyperbolic and in a longer time scale, namely $N^{4/3}$.