Journal of Statistical Physics

, Volume 140, Issue 2, pp 232–267 | Cite as

Limit Processes for TASEP with Shocks and Rarefaction Fans

  • Ivan Corwin
  • Patrik L. Ferrari
  • Sandrine Péché
Article

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density ρ and right density ρ+. We study the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of ρ±. We characterize the large time scaling limit of the multipoint fluctuations as a function of the densities ρ± and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open.

On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.

Keywords

Interacting particle system Asymmetric exclusion process KPZ class Directed percolation Fluctuations Space-time correlations 

References

  1. 1.
    Adler, M., Delépine, J., van Moerbeke, P.: Dyson’s nonintersecting Brownian motions with a few outliers. Commun. Pure Appl. Math., 62, 334–395 (2010). doi:10.1002/cpa.20264 CrossRefGoogle Scholar
  2. 2.
    Adler, M., van Moerbeke, P.: PDE’s for the joint distribution of the Dyson, Airy and Sine processes. Ann. Probab. 33, 1326–1361 (2005) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aldous, D.J., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103, 199–213 (1995) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. (2010, to appear). arXiv:1003.0443
  5. 5.
    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
  6. 6.
    Baik, J., Ferrari, P.L., Péché, S.: Limit process of stationary TASEP near the characteristic line. Commun. Pure Appl. Math. 63, 1017–1070 (2010). doi:10.1002/cpa.20316 Google Scholar
  7. 7.
    Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–542 (2000) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Balázs, M., Seppäläinen, T.: Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. Math. 171, 1237–1265 (2010) MATHCrossRefGoogle Scholar
  9. 9.
    Balázs, M., Seppäläinen, T.: Fluctuation bounds for the asymmetric simple exclusion process. Alea 6, 1–24 (2009) MATHGoogle Scholar
  10. 10.
    Ben Arous, G., Corwin, I.: Current fluctuations for TASEP: a proof of the Prähofer-Spohn conjecture. arXiv:0905.2993. To appear in Ann. Probab. (2010)
  11. 11.
    Borodin, A.: Private communication (2008) Google Scholar
  12. 12.
    Borodin, A., Ferrari, P.L.: Anisotropic growth of random surfaces in 2+1 dimensions (2008). arXiv:0804.3035
  13. 13.
    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
  14. 14.
    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
  15. 15.
    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
  16. 16.
    Borodin, A., Ferrari, P.L., Sasamoto, T.: Two speed TASEP. J. Stat. Phys. 137, 936–977 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    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
  18. 18.
    Burke, P.J.: The output of a queuing system. Oper. Res. 4, 699–704 (1956) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature (2010). arXiv:1002.4560
  20. 20.
    Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ models (2010). Preprint, arXiv:1001.5345
  21. 21.
    Derrida, B., Gerschenfeld, A.: Current fluctuations of the one dimensional symmetric simple exclusion process with a step initial condition. J. Stat. Phys. 136, 1–15 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Dieker, A.B., Warren, J.: On the largest-eigenvalue process for generalized wishart random matrices. Alea 6, 369–376 (2009) MathSciNetGoogle Scholar
  23. 23.
    Dotsenko, V.: Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers (2010). arXiv:1004.4455
  24. 24.
    Durrett, R.: Probability: Theory and Examples. Thompson, Washington (2005) Google Scholar
  25. 25.
    Evans, L.C.: Partial Differential Equations. AMS, Providence (1998) MATHGoogle Scholar
  26. 26.
    Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91, 81–101 (1992) MATHCrossRefGoogle Scholar
  27. 27.
    Ferrari, P.A., Fontes, L.: Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22, 820–832 (1994) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ferrari, P.A., Fontes, L.: Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Relat. Fields 99, 305–319 (1994) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Ferrari, P.A., Kipnis, C.: Second class particles in the rarefaction fan. Ann. Inst. Henri Poincaré 31, 143–154 (1995) MATHMathSciNetGoogle Scholar
  30. 30.
    Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19, 226–244 (1991) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Ferrari, P.L.: Slow decorrelations in KPZ growth. J. Stat. Mech. P07022 (2008) Google Scholar
  32. 32.
    Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Commun. Math. Phys. 265, 1–44 (2006) MATHCrossRefMathSciNetADSGoogle Scholar
  33. 33.
    Imamura, T., Sasamoto, T.: Fluctuations of the one-dimensional polynuclear growth model with external sources. Nucl. Phys. B 699, 503–544 (2004) MATHCrossRefMathSciNetADSGoogle Scholar
  34. 34.
    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
  35. 35.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) MATHCrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003) MATHMathSciNetADSGoogle Scholar
  37. 37.
    Johansson, K.: Random matrices and determinantal processes. In: Bovier, A., Dunlop, F., van Enter, A., den Hollander, F., Dalibard, J. (eds.) Mathematical Statistical Physics, Session LXXXIII. Lecture Notes of the Les Houches Summer School 2005, pp. 1–56. Elsevier, Amsterdam (2006) CrossRefGoogle Scholar
  38. 38.
    Kallabis, H., Krug, J.: Persistence of Kardar-Parisi-Zhang interfaces. Europhys. Lett. 45, 20–25 (1999) CrossRefADSGoogle Scholar
  39. 39.
    Kardar, K., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) MATHCrossRefADSGoogle Scholar
  40. 40.
    Krug, J., Kallabis, H., Majumdar, S.N., Cornell, S.J., Bray, A.J., Sire, C.: Persistence exponents for fluctuating interfaces. Phys. Rev. E 56, 2702 (1997) CrossRefADSGoogle Scholar
  41. 41.
    Krug, J., Spohn, H.: Kinetic roughening of growning surfaces. In: Solids Far from Equilibrium: Growth, Morphology and Defects, pp. 479–582. Cambridge University Press, Cambridge (1992) Google Scholar
  42. 42.
    Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976) MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (1985) MATHGoogle Scholar
  44. 44.
    Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999) MATHGoogle Scholar
  45. 45.
    Mountford, T., Guiol, H.: The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15, 1227–1259 (2005) MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Okounkov, A.: Infinite wedge and random partitions. Sel. Math. 7, 57–81 (2001) MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Am. Math. Soc. 16, 581–603 (2003) MATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    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. Birkhäuser, Basel (2002) Google Scholar
  49. 49.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002) MATHCrossRefGoogle Scholar
  50. 50.
    Quastel, J., Valko, B.: t 1/3 superdiffusivity of finite-range asymmetric exclusion processes on ℤ. Commun. Math. Phys. 273, 379–394 (2007) MATHCrossRefMathSciNetADSGoogle Scholar
  51. 51.
    Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on ℤd. Commun. Math. Phys. 140, 417–448 (1991) MATHCrossRefMathSciNetADSGoogle Scholar
  52. 52.
    Rezakhanlou, F.: A central limit theorem for the asymmetric simple exclusion process. Ann. Inst. Henri Poincaré (B) 38, 437–464 (2002) MATHCrossRefMathSciNetADSGoogle Scholar
  53. 53.
    Sasamoto, T., Spohn, H.: Universality of the one-dimensional KPZ equation (2010). arXiv:1002.1883
  54. 54.
    Seppäläinen, T.: Hydrodynamic scaling, convex duality and asymptotic shapes of growth models. Markov Processes Relat. Fields 4, 1–26 (1998) MATHGoogle Scholar
  55. 55.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) MATHCrossRefMathSciNetADSGoogle Scholar
  56. 56.
    Tracy, C.A., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291–300 (2008) MATHCrossRefMathSciNetADSGoogle Scholar
  57. 57.
    Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008) MATHCrossRefMathSciNetADSGoogle Scholar
  58. 58.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  59. 59.
    Tracy, C.A., Widom, H.: On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137, 825–838 (2009) MATHCrossRefMathSciNetADSGoogle Scholar
  60. 60.
    Tracy, C.A., Widom, H.: Total current fluctuations in ASEP. J. Math. Phys. 50, 095204 (2009) CrossRefMathSciNetADSGoogle Scholar
  61. 61.
    Varadhan, S.R.S.: Large deviations for the asymmetric simple exclusion process. Adv. Stud. Pure Math. 39, 1–27 (2004) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Ivan Corwin
    • 1
  • Patrik L. Ferrari
    • 2
  • Sandrine Péché
    • 3
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany
  3. 3.Institut FourierSaint Martin d’HeresFrance

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