Hammersley’s process with sources and sinks

Abstract

We show that, for a stationary version of Hammersley's process, with Poisson "sources" on the positive x-axis, and Poisson "sinks" on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley's process. This allows us to show that Hammersley's process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley's process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t, t)/t converges to 2, as t → ∞, where L(t, t) is the length of a longest North-East path from (0, 0) to (t, t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke's theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.

Busemann functions and equilibrium measures in last passage percolation models

“…This does seem to be the most important contribution of the interacting fluid representation: once we have a good candidate for ν α , we can check it by showing that it is invariant under the evolution of the interacting fluid. In fact, even in the results for the classical Hammersley case found in Aldous and Diaconis [1] and Cator and Groeneboom [5,6], this is where the interacting particle process proves its worth.…”

“…1 we can see that ν * ((0, t]) = 10, whereas ν * ((0, t/2]) = 4. We can compare ν * to the process of sinks in the classical Hammersley process with sources and sinks, as defined in [5]. We then have the following lemma.…”

Behavior of a second class particle in Hammersley's process

“…Burke's Theorem for Hammersley's process (see Cator and Groeneboom (2005)) shows that this process is again a stationary Hammersley process, if we start with a stationary process. It is not hard to see from Figure 2 that the trajectories of X and X ′ correspond to longest paths from (x, t) to (0, 0) in the reversed process.…”

“…In Cator and Groeneboom (2005), a connection was made between the continuous time Hammersley process and the behavior of second class particles, which are well studied in the literature on discrete interacting particle systems such as TASEP; see for example Liggett (1999). For the Hammersley process, it is natural to consider two types of second class particles: the usual one, where one adds an extra particle at the origin, and a dual second class particle, which corresponds to adding an extra sink at the origin (or removing the leftmost particle).…”

“…All three Poisson processes are assumed to be independent. To know the particle configuration at time s, we intersect a line at time s with the space-time paths.In Cator and Groeneboom (2005), a connection was made between the continuous time Hammersley process and the behavior of second class particles, which are well studied in the literature on discrete interacting particle systems such as TASEP; see for example Liggett (1999). For the MSC 2000 subject classifications.…”