Probability Seminar: Speeds in (partially) asymmetric processes

Start a totally asymmetric simple exclusion process with a second class particle at 0, particles to its left and holes to its right. If Xt is the location at time t of the second class particle, then Xt / t converges a.s. to a uniform [-1,1] random variable. I will prove an analogous result for partially asymmetric exclusion process (with Balázs and Seppäläinen), and explain why this is interesting (with Amir and Valkó).