## Annals of Probability

- Ann. Probab.
- Volume 33, Number 3 (2005), 879-903.

### Hammersley’s process with sources and sinks

Eric Cator and Piet Groeneboom

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 33, Number 3 (2005), 879-903.

**Dates**

First available in Project Euclid: 6 May 2005

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1115386713

**Digital Object Identifier**

doi:10.1214/009117905000000053

**Mathematical Reviews number (MathSciNet)**

MR2135307

**Zentralblatt MATH identifier**

1066.60011

**Subjects**

Primary: 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Longest increasing subsequence Ulam’s problem Hammersley’s process local Poisson convergence totally asymmetric simple exclusion processes (TASEP) second-class particles Burke’s theorem

#### Citation

Cator, Eric; Groeneboom, Piet. Hammersley’s process with sources and sinks. Ann. Probab. 33 (2005), no. 3, 879--903. doi:10.1214/009117905000000053. https://projecteuclid.org/euclid.aop/1115386713