Probability Theory and Related Fields

, Volume 161, Issue 1–2, pp 61–109

Anomalous shock fluctuations in TASEP and last passage percolation models

Article

Abstract

We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time \(t\) will have a width of order \(t^{1/3}\). We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy\(_1\) process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models.

Mathematics Subject Classification

60K35 82C22 

References

  1. 1.
    Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \(\mathbb{Z}\). J. Stat. Phys. 47, 265–288 (1987)CrossRefMATHMathSciNetGoogle Scholar
  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.
    Baik, J., Ferrari, P.L., Péché, S.: Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math. 63, 1017–1070 (2010)MATHMathSciNetGoogle Scholar
  4. 4.
    Baik, J., Ferrari, P.L., Péché, S.: Convergence of the two-point function of the stationary TASEP, arXiv:1209.0116 (2012)Google Scholar
  5. 5.
    Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–542 (2000)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Baik, J., Rains, E.M.: Symmetrized Random Permutations, Random Matrix Models and Their Applications, pp. 1–19. Cambridge University Press, Cambridge (2001)Google Scholar
  7. 7.
    van Beijeren, H.: Fluctuations in the motions of mass and of patterns in one-dimensional driven diffusive systems. J. Stat. Phys. 63, 47–58 (1991)CrossRefGoogle Scholar
  8. 8.
    Belitsky, V., Schütz, G.M.: Microscopic structure of shocks and antishocks in the ASEP conditioned on low current. J. Stat. Phys. 152, 93–111 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ben Arous, G., Corwin, I.: Current fluctuations for TASEP: a proof of the Prähofer-Spohn conjecture. Ann. Probab. 39, 104–138 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Borodin, A., Ferrari, P.L.: Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13, 1380–1418 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy\(_1\) process. Int. Math. Res. Papers 2007, rpm002 (2007)Google Scholar
  12. 12.
    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)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Borodin, A., Ferrari, P.L., Sasamoto, T.: Transition between Airy\(_1\) and Airy\(_2\) processes and TASEP fluctuations. Comm. Pure Appl. Math. 61, 1603–1629 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Borodin, A., Ferrari, P.L., Sasamoto, T.: Two speed TASEP. J. Stat. Phys. 137, 936–977 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1 (2012). doi:10.1142/S2010326311300014
  16. 16.
    Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ models. Ann. Inst. H. Poincaré Probab. Statist. 48, 134–150 (2012)CrossRefMATHGoogle Scholar
  17. 17.
    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)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Ferrari, P.A.: The simple exclusion process as seen from a tagged particle. Ann. Probab. 14, 1277–1290 (1986)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91, 81–101 (1992)CrossRefMATHGoogle Scholar
  20. 20.
    Ferrari, P.A., Fontes, L.: Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Relat. Fields 99, 305–319 (1994)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19, 226–244 (1991)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Ferrari, P.L.: Slow decorrelations in KPZ growth. J. Stat. Mech. P07022 (2008)Google Scholar
  23. 23.
    Ferrari, P.L.: The universal Airy\(_1\) and Airy\(_2\) processes in the Totally Asymmetric Simple Exclusion Process. In: Baik, J., Kriecherbauer, T., Li, L.-C., McLaughlin K., Tomei, C. (eds.) Integrable Systems and Random Matrices: In Honor of Percy Deifts, Contemporary Mathematics, vol. 458, pp. 321–332. American Mathematical Society (2008).Google Scholar
  24. 24.
    Ferrari, P.L., Spohn, H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557–L561 (2005)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265, 1–44 (2006)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. H. Poincaré (A) 53, 1–14 (1990)MATHGoogle Scholar
  27. 27.
    Johansson, K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209, 437–476 (2000)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Johansson, K.: Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116, 445–456 (2000)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Johansson, K.: Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242, 277–329 (2003)MATHMathSciNetGoogle Scholar
  30. 30.
    Kardar, M., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)CrossRefMATHGoogle Scholar
  31. 31.
    Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Liggett, T.M.: Stochastic interacting systems: contact, voter and exclusion processes. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
  33. 33.
    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, Basel (2002)Google Scholar
  34. 34.
    Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Spohn, H.: Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  36. 36.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159, 151–174 (1994)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177, 727–754 (1996)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institute for Applied MathematicsBonn UniversityBonnGermany

Personalised recommendations