A by now famous result by Baik, Deift and Johansson, concerning the limit distribution of the length of a longest increasing subsequence of a random permutation, shows that the flux of Hammersley's interacting particle system at time t at location t 2/3. In fact, they derive the full limit distribution, but they use purely analytic methods, based on an integral representation of the exact distribution at time t. In this talk we will show the order t 2/3 using only probabilistic methods. As a consequence of this approach, we also show that a second class particle in the Hammersley process has superdiffusive behaviour, in the sense that the order of fluctation around its expected value at time t is t 2/3. In fact, a key ingredient of our approach is an identity linking the variance of the flux to the fluctation of a second class particle. Furthermore, we continue the sources and sinks approach to the Hammersley process.
This is joint work with Piet Groeneboom.