Journal of Statistical Physics

, 137:936 | Cite as

Two Speed TASEP

  • Alexei Borodin
  • Patrik L. Ferrari
  • Tomohiro Sasamoto


We consider the TASEP on ℤ with two blocks of particles having different jump rates. We study the large time behavior of particles’ positions. It depends both on the jump rates and the region we focus on, and we determine the complete process diagram. In particular, we discover a new transition process in the region where the influence of the random and deterministic parts of the initial condition interact.

Slow particles may create a shock, where the particle density is discontinuous and the distribution of a particle’s position is asymptotically singular. We determine the diffusion coefficient of the shock without using second class particles.

We also analyze the case where particles are effectively blocked by a wall moving with speed equal to their intrinsic jump rate.


TASEP Airy processes Shock fluctuations Random matrices 


  1. 1.
    Adler, M., Ferrari, P.L., van Moerbeke, P.: Airy processes with wanderers and new universality classes. arXiv:0811.1863. Ann. Probab. (2008, to appear)
  2. 2.
    Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33, 1643–1697 (2006) CrossRefGoogle Scholar
  3. 3.
    Borodin, A., Ferrari, P.L.: Anisotropic growth of random surfaces in 2+1 dimensions. arXiv:0804.3035 (2008)
  4. 4.
    Borodin, A., Ferrari, P.L.: Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13, 1380–1418 (2008) MATHMathSciNetGoogle Scholar
  5. 5.
    Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process. Int. Math. Res. Papers 2007, rpm002 (2007) Google Scholar
  6. 6.
    Borodin, A., Ferrari, P.L., Prähofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129, 1055–1080 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Borodin, A., Ferrari, P.L., Prähofer, M., Sasamoto, T., Warren, J.: Maximum of Dyson Brownian motion and non-colliding systems with a boundary. arXiv:0905.3989 (2009)
  8. 8.
    Borodin, A., Ferrari, P.L., Sasamoto, T.: Transition between Airy1 and Airy2 processes and TASEP fluctuations. Commun. Pure Appl. Math. 61, 1603–1629 (2008) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Borodin, A., Kuan, J.: Random surface growth with a wall and Plancherel measures for O(∞). arXiv:0904.2607 (2009)
  10. 10.
    Borodin, A., Péché, S.: Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys. 132, 275–290 (2008) MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Burke, P.J.: The output of a queuing system. Oper. Res. 4, 699–704 (1956) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Coletti, C.F., Ferrari, P.A., Pimentel, L.P.R.: The variance of the shock in the HAD process. arXiv:0801.2526 (2008)
  13. 13.
    Derrida, B., Gerschenfeld, A.: Current fluctuations of the one dimensional symmetric exclusion process with a step initial condition. J. Stat. Phys. 136, 1–15 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Defosseux, M.: Orbit measures and interlaced determinantal point processes. C. R. Math. Acad. Sci. Paris 346, 783–788 (2008) MATHMathSciNetGoogle Scholar
  15. 15.
    Defosseux, M.: Orbit measures, random matrix theory and interlaced determinantal processes. arXiv:0810.1011 (2008)
  16. 16.
    Derrida, B., Janowsky, S.A., Lebowitz, J.L., Speer, E.R.: Exact solution of the totally asymmetric simple exclusion process: shock profiles. J. Stat. Phys. 73, 813–842 (1993) MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91, 81–101 (1992) MATHCrossRefGoogle Scholar
  18. 18.
    Ferrari, P.L.: The universal Airy1 and Airy2 processes in the totally asymmetric simple exclusion process. In: Baik, J., et al. (eds.) Integrable Systems and Random Matrices: In Honor of Percy Deift. Contemporary Math., pp. 321–332. American Mathematics Society, Providence (2008) Google Scholar
  19. 19.
    Ferrari, P.L.: Java animation of the TASEP with one slow particle.
  20. 20.
    Ferrari, P.L., Prähofer, M.: One-dimensional stochastic growth and Gaussian ensembles of random matrices. Markov Processes Relat. Fields 12, 203–234 (2006) (Proceedings of “Inhomogeneous Random Systems 2005”) MATHGoogle Scholar
  21. 21.
    Ferrari, P.L., Spohn, H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557–L561 (2005) CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Forrester, P.J., Nordenstam, E.: The anti-symmetric GUE Minor Process. arXiv:0804.3293 (2008)
  23. 23.
    Imamura, T., Sasamoto, T.: Dynamical properties of a tagged particle in the totally asymmetric simple exclusion process with the step-type initial condition. J. Stat. Phys. 128, 799–846 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Johansson, K.: Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier 55, 2129–2145 (2005) MATHMathSciNetGoogle Scholar
  25. 25.
    Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45, 3058–3085 (2004) MATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Katori, M., Tanemura, H.: Infinite systems of noncolliding generalized meanders and Riemann–Liouville differintegrals. Probab. Theory Relat. Fields 138, 113–156 (2007) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. arXiv:math.CA/9602214
  28. 28.
    Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: Sidoravicius, V. (ed.) In and Out of Equilibrium. Progress in Probability, vol. 51, pp. 185–204. Birkhäuser, Boston (2002) Google Scholar
  29. 29.
    Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on ℤd. Commun. Math. Phys. 140, 417–448 (1991) MATHCrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005) CrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Sasamoto, T.: Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques, J. Stat. Mech., P07007 (2007) Google Scholar
  32. 32.
    Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Physica A 369, 71–99 (2006) CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    Spohn, H.: Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics. Springer, Heidelberg (1991) MATHGoogle Scholar
  34. 34.
    Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996) MATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Tracy, C.A., Widom, H.: Nonintersecting Brownian excursions. Ann. Appl. Probab. 17, 953–979 (2007) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008) MATHCrossRefMathSciNetADSGoogle Scholar
  37. 37.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009) CrossRefMathSciNetADSGoogle Scholar
  38. 38.
    Warren, J., Windridge, P.: Some examples of dynamics for Gelfand Tsetlin patterns. Electron. J. Probab. 14, 1745–1769 (2009) MATHMathSciNetGoogle Scholar
  39. 39.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1999) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Alexei Borodin
    • 1
  • Patrik L. Ferrari
    • 2
  • Tomohiro Sasamoto
    • 3
    • 4
  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Bonn UniversityBonnGermany
  3. 3.Chiba UniversityChibaJapan
  4. 4.Technische Universität MünchenMunichGermany

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