April 7, 2006

This is joint work with Piet Groeneboom.

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 variance of the flux of Hammersley's interacting particle system at time t at location t is of order 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.

Finding the order of the variance of the flux along a characteristic has been an open problem for many interacting particle systems since the mid eighties, and we are hoping that our probabilistic approach might be applicable to other systems than Hammersley. In fact, it has already been succesfully applied to TASEP by Balazs, Cator & Seppalainen (2006).
 

For nonequilibrium steady states, we identify observables whose fluctuations satisfy a general symmetry and for which a new reciprocity relation can be shown. Unlike the situation in recently discussed fluctuation theorems, these observables are time-reversal symmetric. That is essential for exploiting the fluctuation symmetry beyond linear response theory. Besides time-reversal, a crucial role is played by the reversal of the driving fields, that further resolves the space-time action. In particular, the time-symmetric part in the space-time action determines second order effects of the nonequilibrium driving.