The Annals of Probability

The oriented swap process

Omer Angel, Alexander Holroyd, and Dan Romik

Full-text: Open access


Particles labelled 1, …, n are initially arranged in increasing order. Subsequently, each pair of neighboring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behavior of this process as n→∞. We prove that the space–time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non-differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time (2+o(1))n. The finishing times of individual particles converge to deterministic limits, with fluctuations asymptotically governed by the Tracy–Widom distribution.

Article information

Ann. Probab. Volume 37, Number 5 (2009), 1970-1998.

First available in Project Euclid: 21 September 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60C05: Combinatorial probability

Sorting network exclusion process second-class particle permutahedron interacting particle system


Angel, Omer; Holroyd, Alexander; Romik, Dan. The oriented swap process. Ann. Probab. 37 (2009), no. 5, 1970--1998. doi:10.1214/09-AOP456.

Export citation


  • [1] Amir, G., Angel, O. and Valko, B. The TASEP speed process. Preprint. Available at arXiv:0811.3706.
  • [2] Angel, O., Holroyd, A. E., Romik, D. and Virág, B. (2007). Random sorting networks. Adv. Math. 215 839–868.
  • [3] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178.
  • [4] Benjamini, I., Berger, N., Hoffman, C. and Mossel, E. (2005). Mixing times of the biased card shuffling and the asymmetric exclusion process. Trans. Amer. Math. Soc. 357 3013–3029 (electronic).
  • [5] Borodin, A., Okounkov, A. and Olshanski, G. (2000). Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 481–515 (electronic).
  • [6] Ferrari, P. A., Goncalves, P. and Martin, J. B. Crossing probabilities in asymmetric exclusion processes. Preprint. Available at
  • [7] Ferrari, P. A. and Kipnis, C. (1995). Second class particles in the rarefaction fan. Ann. Inst. H. Poincaré Probab. Statist. 31 143–154.
  • [8] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476.
  • [9] Kipnis, C., Olla, S. and Varadhan, S. R. S. (1989). Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure Appl. Math. 42 115–137.
  • [10] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [11] 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.
  • [12] Rezakhanlou, F. (1991). Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys. 140 417–448.
  • [13] Rost, H. (1981). Nonequilibrium behaviour of a many particle process: Density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58 41–53.
  • [14] Tracy, C. A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.