Behavior of a second class particle in Hammersley's process

Abstract

In the case of a rarefaction fan in a non-stationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particles touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane. IntroductionIn Hammersley (1972), a discrete interacting particle process is introduced to study the behavior of the length of longest increasing subsequences of random permutations. In Aldous and Diaconis (1995), this discrete process is generalized to a continuous time interacting particle process on the real line, and they use the ergodic decomposition theorem to show local convergence to a Poisson process, when moving out along a ray. In this paper, we will consider Hammersley's process with sources and sinks, as introduced in Groeneboom (2002). For an extensive description of this process, we refer to Cator and Groeneboom (2005), since our results will be partly based on results derived in that paper. Here we will suffice with a brief description, based on Figure 1.We consider the space-time paths of particles that started on the x-axis as sources, distributed according to a Poisson distribution and we consider the t-axis as a time axis. In the positive quadrant we have a Poisson process of what we call α-points (denoted in Figure 1 by ×). At the time an α-point appears, the particle immediately to the right of it jumps to the location of the α-point. Finally, we have a Poisson process of sinks on the t-axis. Each sink makes the leftmost particle disappear. 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. Primary: 60C05,60K35, secondary 60F05.

Busemann functions and the speed of a second class particle in the rarefaction fan

“…Part of the results concerning the distributional behavior of second class particles and competition interfaces in the rarefaction regime were already known [4,9,10,11,12,13]. The genuine contributions are (2.5), (2.8) and (2.17), and how it can be used to compute the distribution of the asymptotic speed, as soon as we have a good candidate for the equilibrium measure.…”

“…We will also consider the Hammersley interacting particle process [1], where the situation concerning the second class particle in a rarefaction fan is very similar to TASEP: almost sure existence of the asymptotic speed is proved by Coletti and Pimentel in [9] (they even prove this for more general objects than second class particles), but the distribution of the second class particle in a Leandro P. R. Pimentel was supported by grant numbers 613.000.605 and 040.11.146 from the Netherlands Organisation for Scientific Research (NWO). rarefaction fan was only determined for a very specific family of initial conditions, involving Poisson processes ( [4,9]).…”

“…rarefaction fan was only determined for a very specific family of initial conditions, involving Poisson processes ( [4,9]).…”

Jeu de taquin dynamics on infinite Young tableaux and second class particles

“…Second-class particles have also been studied recently in connection with Hammersley's process, an interacting particle system that is also related to the RSK algorithm and Ulam's problem on longest increasing subsequences. In this setting, a result on the trajectory of second-class particles analogous to Theorem 7.3 was proved by Coletti and Pimentel (2007); see also Groeneboom (2005, 2006), Cator and Dobrynin (2006) for related results, and Cator and Pimentel (2013) for a recent work considerably generalizing the results of Coletti and Pimentel.…”

Limiting speed of a second class particle in ASEP

“…See Gonçalves [2014] for a related result in the totally asymmetric constant rate zero-range process. Cator and Dobrynin [2006] studied the Hammersely procss and obtained an exciting limit theorem for the second class particle at the rarefaction fan. Ferrari et al [2009a] proved the analogue of the main result of Ferrari and Kipnis [1995] in ASEP.…”