Annals of Probability

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

Eric Cator and Leandro P. R. Pimentel

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Abstract

In this paper we will show how the results found in [Probab. Theory Related Fields 154 (2012) 89–125], about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process or the Hammersley interacting particle process. The method will be to use the well-known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well-known connection between second class particles and competition interfaces.

Article information

Source
Ann. Probab., Volume 41, Number 4 (2013), 2401-2425.

Dates
First available in Project Euclid: 3 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1372859756

Digital Object Identifier
doi:10.1214/11-AOP709

Mathematical Reviews number (MathSciNet)
MR3112921

Zentralblatt MATH identifier
1276.60108

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C21: Dynamic continuum models (systems of particles, etc.)

Keywords
TASEP second class particles rarefaction fan

Citation

Cator, Eric; Pimentel, Leandro P. R. Busemann functions and the speed of a second class particle in the rarefaction fan. Ann. Probab. 41 (2013), no. 4, 2401--2425. doi:10.1214/11-AOP709. https://projecteuclid.org/euclid.aop/1372859756


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References

  • [1] Aldous, D. and Diaconis, P. (1995). Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199–213.
  • [2] Amir, G., Angel, O. and Valkó, B. (2011). The TASEP speed process. Ann. Probab. 39 1205–1242.
  • [3] Balázs, M., Cator, E. and Seppäläinen, T. (2006). Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 1094–1132 (electronic).
  • [4] Cator, E. and Dobrynin, S. (2006). Behavior of a second class particle in Hammersley’s process. Electron. J. Probab. 11 670–685 (electronic).
  • [5] Cator, E. and Groeneboom, P. (2005). Hammersley’s process with sources and sinks. Ann. Probab. 33 879–903.
  • [6] Cator, E. and Groeneboom, P. (2006). Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 1273–1295.
  • [7] Cator, E. A. and Pimentel, L. P. R. (2012). Busemann functions and equilibrium measures in last-passage percolation. Probab. Theory Related Fields 154 89–125.
  • [8] Cohen, J. W. (1969). The Single Server Queue. North-Holland Series in Applied Mathematics and Mechanics 8. North-Holland, Amsterdam.
  • [9] Coletti, C. F. and Pimentel, L. P. R. (2007). On the collision between two PNG droplets. J. Stat. Phys. 126 1145–1164.
  • [10] Coupier, D. (2011). Multiple geodesics with the same direction. Available at http://arxiv.org/abs/1104.1321.
  • [11] Ferrari, P. A. and Kipnis, C. (1995). Second class particles in the rarefaction fan. Ann. Inst. Henri Poincaré Probab. Stat. 31 143–154.
  • [12] Ferrari, P. A., Martin, J. B. and Pimentel, L. P. R. (2009). A phase transition for competition interfaces. Ann. Appl. Probab. 19 281–317.
  • [13] Ferrari, P. A. and Pimentel, L. P. R. (2005). Competition interfaces and second class particles. Ann. Probab. 33 1235–1254.
  • [14] Mountford, T. and Guiol, H. (2005). The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15 1227–1259.
  • [15] Newman, C. M. (1995). A surface view of first-passage percolation. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) 1017–1023. Birkhäuser, Basel.
  • [16] Pimentel, L. P. R. (2007). Multitype shape theorems for first passage percolation models. Adv. in Appl. Probab. 39 53–76.
  • [17] Pyke, R. (1959). The supremum and infimum of the Poisson process. Ann. Math. Statist. 30 568–576.
  • [18] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41–53.
  • [19] Seppäläinen, T. (1998). Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields 4 1–26.
  • [20] Wüthrich, M. V. (2002). Asymptotic behaviour of semi-infinite geodesics for maximal increasing subsequences in the plane. In In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability 51 205–226. Birkhäuser, Boston, MA.