May 27

10:30 am

## Hammersley\'s Process with Sources and Sinks

### Piet Groeneboom

Seminar

Hammersley (1972) initiated a very interesting "hydrodynamical" approach to the study of the behavior of the lengths of longest increasing subsequences of random permutations. In the nineties Aldous and Diaconis (1995) introduced a modified version of the interacting particle process, studied in Hammersley (1972), and used this modification in a proof of the fact that the length of a longest increasing subsequence of a (uniform) random permutation of length n, divided by sqrt{n}, converges in probability to 2. The key to the proof in Aldous and Diaconis (1995) is the local convergence to a Poisson process of their version of Hammersley's process, for the proof of which they use the ergodic decomposition theorem. I will discuss a further extension of Hammersley's process, in which "sources and sinks" are introduced in such a way that the process becomes stationary. This leads to very simple proofs of the results in Aldous and Diaconis (1995), avoiding the ergodic decomposition theorem and the introduction of mixed Poisson processes. It also establishes a link with the theory of totally asymmetric simple exclusion processes (TASEP) and their shock waves. The theory of the lengths of longest increasing subsequences of random permutations and its relation to other models has been studied intensively by both physicists and mathematicians in the past 10 years and some remarkable connections with the theory of eigenvalues of random matrices have been established.

My talk is based on joint work with Eric Cator.

**References**

Aldous and Diaconis (1995). Hammersley's interacting particle process and longest increasing subsequences}. Probab. Th. Relat. Fields 103, 199-213.

Cator and Groeneboom (2003). Hammersley's process with sources and sinks. To appear in Annals of Probability.

Hammersley (1972). A few seedlings of research. Proc. 6th Berkeley Symp. Math. Statist. and Probability, 1, 345-394.