Hammersley’s process with sources and sinks

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 one, 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 in Aldous and Diaconis (1995), 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 1-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.