Journal of Statistical Physics

, Volume 160, Issue 4, pp 985–1004

Shock Fluctuations in Flat TASEP Under Critical Scaling

Article

Abstract

We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate \(1\), while particles to the right have jump rate \(\alpha \). When \(\alpha <1\) there is a formation of a shock where the density jumps to \((1-\alpha )/2\). For \(\alpha <1\) fixed, the statistics of the associated height functions around the shock is asymptotically (as time \(t\rightarrow \infty \)) a maximum of two independent random variables as shown in Ferrari and Nejjar (Probab Theory Rel Fields 161:61–109, 2015). In this paper we consider the critical scaling when \(1-\alpha =a t^{-1/3}\), where \(t\gg 1\) is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of \(a\). We see that the convergence to \(F_\mathrm{GOE}^2\) occurs quite rapidly as \(a\) increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes (Baik et al. in Ann Probab 33:1643–1697, 2006).

Keywords

KPZ universality class Exclusion process Random matrices Shock fluctuations 

References

  1. 1.
    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
  2. 2.
    Baik, J., Ferrari, P.L., Péché, S.: Limit process of stationary TASEP near the characteristic line. Commun. Pure Appl. Math. 63, 1017–1070 (2010)MATHGoogle Scholar
  3. 3.
    Battles, Z., Trefethen, L.: An extension of Matlab to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: a review. Markov Process. Relat. Fields 16, 803–866 (2010)MathSciNetMATHGoogle Scholar
  5. 5.
    Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79, 871–915 (2009)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Bornemann, F., Ferrari, P.L., Prähofer, M.: The Airy\(_1\) process is not the limit of the largest eigenvalue in GOE matrix diffusion. J. Stat. Phys. 133, 405–415 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  7. 7.
    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), rpm002Google Scholar
  8. 8.
    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)MathSciNetADSCrossRefMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Borodin, A., Ferrari, P.L., Sasamoto, T.: Two speed TASEP. J. Stat. Phys. 137, 936–977 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  11. 11.
    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)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Corwin, I., Ferrari, P.L., Péché, S.: Limit processes of non-equilibrium TASEP. J. Stat. Phys. 140, 232–267 (2010)MathSciNetADSCrossRefMATHGoogle Scholar
  13. 13.
    Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ models. Ann. Inst. H. Poincaré Probab. Statist. 48, 134–150 (2012)ADSCrossRefMATHGoogle Scholar
  14. 14.
    Ferrari, P.A.: Shock fluctuations in asymmetric simple exclusion. Probab. Theory Relat. Fields 91, 81–101 (1992)CrossRefMATHGoogle Scholar
  15. 15.
    Ferrari, P.A., Fontes, L.: Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22, 820–832 (1994)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ferrari, P.L.: Slow decorrelations in KPZ growth. J. Stat. Mech. P07022 (2008)Google Scholar
  17. 17.
    Ferrari, P.L., Nejjar, P.: Anomalous shock fluctuations in TASEP and last-passage percolation models. Probab. Theory Rel. Fields 161, 61–109 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ferrari, P.L., Spohn, H.: Last branching in directed last passage percolation. Markov Process. Relat. Fields 9, 323–339 (2003)MathSciNetMATHGoogle Scholar
  19. 19.
    Gärtner, J., Presutti, E.: Shock fluctuations in a particle system. Ann. Inst. H. Poincaré A 53, 1–14 (1990)MATHGoogle Scholar
  20. 20.
    Gravner, J., Tracy, C.A., Widom, H.: Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102, 1085–1132 (2001)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Harris, T.: Additive set-valued markov processes and pharical methods. Ann. Probab. 6, 355–378 (1878)CrossRefGoogle Scholar
  22. 22.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)ADSCrossRefMATHGoogle Scholar
  23. 23.
    Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (1985)CrossRefGoogle Scholar
  24. 24.
    Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
  25. 25.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)CrossRefMATHGoogle Scholar
  26. 26.
    Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)MathSciNetADSCrossRefMATHGoogle Scholar
  28. 28.
    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)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institute for Applied MathematicsBonn UniversityBonnGermany

Personalised recommendations