Facilitated Exclusion Process

  • Jinho Baik
  • Guillaume Barraquand
  • Ivan Corwin
  • Toufic Suidan
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


We study the Facilitated TASEP, an interacting particle system on the one dimensional integer lattice. We prove that starting from step initial condition, the position of the rightmost particle has Tracy Widom GSE statistics on a cube root time scale, while the statistics in the bulk of the rarefaction fan are GUE. This uses a mapping with last-passage percolation in a half-quadrant which is exactly solvable through Pfaffian Schur processes. Our results further probe the question of how first particles fluctuate for exclusion processes with downward jump discontinuities in their limiting density profiles. Through the Facilitated TASEP and a previously studied MADM exclusion process we deduce that cube-root time fluctuations seem to be a common feature of such systems. However, the statistics which arise are shown to be model dependent (here they are GSE, whereas for the MADM exclusion process they are GUE). We also discuss a two-dimensional crossover between GUE, GOE and GSE distribution by studying the multipoint distribution of the first particles when the rate of the first one varies. In terms of half-space last passage percolation, this corresponds to last passage times close to the boundary when the size of the boundary weights is simultaneously scaled close to the critical point.



G.B and I.C. would like to thank Sidney Redner for drawing their attention to the Facilitated TASEP. Part of this work was done during the stay of J.B, G.B and I.C. at the Kavli Institute of Theoretical Physics and supported by the National Science Foundation under Grant No. NSF PHY11-25915. J.B. was supported in part by NSF grant DMS-1361782, DMS- 1664692 and DMS-1664531, and the Simons Fellows program. G.B. was partially supported by the Laboratoire de Probabilités et Modèles Aléatoires UMR CNRS 7599, Université Paris-Diderot–Paris 7 and the Packard Foundation through I.C.’s Packard Fellowship. I.C. was partially supported by the NSF through DMS-1208998 and DMS-1664650, the Clay Mathematics Institute through a Clay Research Fellowship, the Institute Henri Poincaré through the Poincaré Chair, and the Packard Foundation through a Packard Fellowship for Science and Engineering.


  1. 1.
    Baik, J., Barraquand, G., Corwin, I., Suidan, T.: Pfaffian Schur processes and last passage percolation in a half-quadrant. To appear in Ann. Probab. (2017). arXiv:1606.00525Google Scholar
  2. 2.
    Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barraquand, G., Corwin, I.: The q-Hahn asymmetric exclusion process. Ann. Appl. Probab. 26(4), 2304–2356 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Basu, U., Mohanty, P.K.: Active absorbing-state phase transition beyond directed percolation: a class of exactly solvable models. Phys. Rev. E 79, 041143 (2009). CrossRefGoogle Scholar
  5. 5.
    Ben Arous, G., Corwin, I.: Current fluctuations for TASEP: a proof of the Prähofer–Spohn conjecture. Ann. Probab. 39(1), 104–138 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. AMS, Providence (1998)Google Scholar
  8. 8.
    Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl. Phys. B 553(3), 601–643 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gabel, A., Krapivsky, P.L., Redner, S.: Facilitated asymmetric exclusion. Phys. Rev. Lett. 105(21), 210603 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gabel, A., Redner, S.: Cooperativity-driven singularities in asymmetric exclusion. J. Stat. Mech. 2011(6), P06008 (2011)CrossRefGoogle Scholar
  11. 11.
    Halpin-Healy, T., Takeuchi, K.A.: A KPZ cocktail-shaken, not stirred…. J. Stat. Phys. 160(4), 794–814 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Krug, J., Meakin, P., Halpin-Healy, T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45, 638–653 (1992). CrossRefGoogle Scholar
  13. 13.
    Lee, E., Wang, D.: Distributions of a particle’s position and their asymptotics in the q-deformed totally asymmetric zero range process with site dependent jumping rates. arXiv preprint:1703.08839 (2017)Google Scholar
  14. 14.
    Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (2005)CrossRefGoogle Scholar
  15. 15.
    Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability, vol. 51, pp. 185–204. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  16. 16.
    Rains, E.M.: Correlation functions for symmetrized increasing subsequences. arXiv preprint math/0006097 (2000)Google Scholar
  17. 17.
    Rost, H.: Non-equilibrium behaviour of a many particle process: density profile and local equilibria. Probab. Theory Rel. Fields 58(1), 41–53 (1981)zbMATHGoogle Scholar
  18. 18.
    Spohn, H.: KPZ scaling theory and the semi-discrete directed polymer model. MSRI Proceedings. arXiv:1201.0645 (2012)Google Scholar
  19. 19.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290(1), 129–154 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Jinho Baik
    • 1
  • Guillaume Barraquand
    • 2
  • Ivan Corwin
    • 2
  • Toufic Suidan
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations