Statistics of TASEP with Three Merging Characteristics

  • Patrik L. FerrariEmail author
  • Peter Nejjar


In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing. Consequently, there are three characteristics which meet, i.e. two shocks merge. We study the particle fluctuations at this merging point and show that they are given by a product of three (properly scaled) GOE Tracy–Widom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.


Shock fluctuations Exclusion process KPZ universality class 



This work is supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) by the CRC 1060 (Projektnummer 211504053) and Germany’s Excellence Strategy - GZ 2047/1, Projekt ID 390685813.


  1. 1.
    Andjel, E.D., Vares, M.E.: Hydrodynamic equations for attractive particle systems on \({\mathbb{Z}}\). J. Stat. Phys. 47, 265–288 (1987)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Baik, J., Ferrari, P.L., Péché, S.: Convergence of the two-point function of the stationary TASEP. In: Baik, J., Ferrari, P.L., Péché, S. (eds.) Singular Phenomena and Scaling in Mathematical Models, pp. 91–110. Springer, New York (2014)CrossRefGoogle Scholar
  3. 3.
    Basu, R., Sidoravicius, V., Sly, A.: Last passage percolation with a defect line and the solution of the slow bond problem. arXiv:1408.3464 (2014) (preprint)
  4. 4.
    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)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ models. Ann. Inst. H. Poincaré Probab. Statist. 48, 134–150 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    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)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ferrari, P.A.: The simple exclusion process as seen from a tagged particle. Ann. Probab. 14, 1277–1290 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91, 81–101 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ferrari, P.A., Fontes, L.: Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Relat. Fields 99, 305–319 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ferrari, P.A., Kipnis, C., Saada, E.: Microscopic structure of travelling waves in the asymmetric simple exclusion process. Ann. Probab. 19, 226–244 (1991)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ferrari, P.L.: Slow decorrelations in KPZ growth. J. Stat. Mech. (2008). CrossRefGoogle Scholar
  12. 12.
    Ferrari, P.L.: Finite gue distribution with cut-off at a shock. J. Stat. Phys. 172, 505–521 (2018)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Ferrari, P.L., Nejjar, P.: Anomalous shock fluctuations in TASEP and last passage percolation models. Probab. Theory Relat. Fields 161, 61–109 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ferrari, P.L., Nejjar, P.: Fluctuations of the competition interface in presence of shocks. ALEA Lat. Am. J. Probab. Math. Stat. 14, 299–325 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ferrari, P.L., Occelli, A.: Universality of the GOE Tracy-Widom distribution for TASEP with arbitrary particle density. Eletron. J. Probab. 23(51), 1–24 (2018)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Ferrari, P.L., Ghosal, P., Nejjar, P.: Limit law of a second class particle in TASEP with non-random initial condition. Ann. Inst. Henri Poincaré Probab. Statist. 55, 1203–1225 (2019)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. H. Poincaré (A) 53, 1–14 (1990)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Harris, T.: Additive set-valued markov processes and pharical methods. Ann. Probab. 6, 355–378 (1878)CrossRefGoogle Scholar
  19. 19.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Johansson, K.: Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Relat. Fields 116, 445–456 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kardar, M., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)ADSCrossRefGoogle Scholar
  22. 22.
    Ledoux, M., Rider, B.: Small deviations for beta ensembles. Electron. J. Probab. 15, 1319–1343 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Liggett, T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (1985)CrossRefGoogle Scholar
  25. 25.
    Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)CrossRefGoogle Scholar
  26. 26.
    Nejjar, P.: Transition to shocks in TASEP and decoupling of last passage times. ALEA Lat. Am. J. Probab. Math. Stat. 15, 1311–1334 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nejjar, P.: GUE \(\times \) GUE limit law at hard shocks in ASEP. arXiv:1906.07711 (2019) (preprint)
  28. 28.
    Nejjar, P.: KPZ statistics of second class particles in ASEP via mixing. arXiv:1911.09426 (2019) (preprint)
  29. 28.
    Seppalainen, Timo: Coupling the totally asymmetric simple exclusion process with a moving interface. Markov Process. Relat. Fields 4(4), 593–628 (1998)MathSciNetzbMATHGoogle Scholar
  30. 29.
    Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)ADSMathSciNetCrossRefGoogle Scholar
  31. 30.
    Widom, H.: On convergence of moments for random young tableaux and a random growth model. Int. Math. Res. Notices 9, 455–464 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Applied MathematicsBonn UniversityBonnGermany

Personalised recommendations