Skip to main content
SpringerLink
Log in

Fluctuation Properties of the TASEP with Periodic Initial Configuration

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us (Sasamoto in J. Phys. A 38:L549–L556, 2005) and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–542 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barabási, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  3. Borodin, A., Rains, E.M.: Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121, 291–317 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borodin, A., Ferrari, P.L., Prähofer, M.: Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1 process. Int. Math. Res. Pap. rpm002 (2007)

  5. Deift, P.: Universality for mathematical and physical systems. arXiv:math-ph/0603038 (2006)

  6. Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Eynard, B., Mehta, M.L.: Matrices coupled in a chain I. Eigenvalue correlations. J. Phys. A 31, 4449–4456 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Ferrari, P.L.: Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues. Commun. Math. Phys. 252, 77–109 (2004)

    Article  MATH  ADS  Google Scholar 

  9. Ferrari, P.L.: Shape fluctuations of crystal facets and surface growth in one dimension. Ph.D. thesis, Technische Universität München. http://tumb1.ub.tum.de/publ/diss/ma/2004/ferrari.html (2004)

  10. Ferrari, P.L., Prähofer, M.: One-dimensional stochastic growth and Gaussian ensembles of random matrices. Markov Process. Relat. Fields 12, 203–234 (2006)

    Google Scholar 

  11. Ferrari, P.L., Spohn, H.: A determinantal formula for the GOE Tracy-Widom distribution. J. Phys. A 38, L557–L561 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  12. 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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. 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, 601–643 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Hough, J.B., Krishnapur, M., Peres, Y., Virag, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)

    Article  MathSciNet  Google Scholar 

  15. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)

    MATH  ADS  MathSciNet  Google Scholar 

  17. Johansson, K.: Random matrices and determinantal processes. In: Bovier, A., Dunlop, F., van Enter, A., den Hollander, F., Dalibard, J. (eds.) Mathematical Statistical Physics. Lecture Notes of the Les Houches Summer School 2005, vol. LXXXIII, pp. 1–56. Elsevier, Amsterdam (2006)

    Google Scholar 

  18. Kardar, K., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)

    Article  MATH  ADS  Google Scholar 

  19. Karlin, S., McGregor, L.: Coincidence probabilities. Pac. J. 9, 1141–1164 (1959)

    MATH  MathSciNet  Google Scholar 

  20. Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. arXiv:math.CA/9602214 (1996)

  21. Liggett, T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)

    MATH  Google Scholar 

  22. Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Hautes Etudes Sci. 98, 167–212 (2003)

    MATH  MathSciNet  Google Scholar 

  23. Meakin, P.: Fractals, Scaling and Growth Far from Equilibrium. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  24. Nagao, T., Sasamoto, T.: Asymmetric simple exclusion process and modified random matrix ensembles. Nucl. Phys. B 699, 487–502 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. 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)

    Article  MATH  MathSciNet  Google Scholar 

  26. Øksendal, B.K.: Stochastic Differential Equations, 5th ed. Springer, Berlin (1998)

    MATH  Google Scholar 

  27. Prähofer, M.: Stochastic surface growth. Ph.D. thesis, Ludwig-Maximilians-Universität, München. http://edoc.ub.uni-muenchen.de/archive/00001381 (2003)

  28. Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000)

    Article  ADS  Google Scholar 

  29. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)

    Article  MATH  Google Scholar 

  30. Rákos, A., Schütz, G.: Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118, 511–530 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Rákos, A., Schütz, G.: Bethe Ansatz and current distribution for the TASEP with particle-dependent hopping rates. Markov Process. Relat. Fields 12, 323–334 (2006)

    Google Scholar 

  32. Rezakhanlou, F.: Hydrodynamic limit for attractive particle systems on ℤd. Commun. Math. Phys. 140, 417–448 (1991)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Rost, H.: Non-equilibrium behavior of a many particle system: density profile and local equilibrium. Z. Wahrsch. Verw. Gebiete 58, 41–53 (1981)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  34. Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  35. Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88, 427–445 (1997)

    Article  MATH  Google Scholar 

  36. Schütz, G.M.: Exactly solvable models for many-body systems far from equilibrium. In: Domb, C., Lebowitz, J. (eds.) Phase Transitions and Critical Phenomena, vol. 19, pp. 1–251. Academic Press, London (2000)

    Chapter  Google Scholar 

  37. Soshnikov, A.: Determinantal random fields. In: Francoise, J.-P., Naber, G., Tsun, T.S. (eds.) Encyclopedia of Mathematical Physics, pp. 47–53. Elsevier, Oxford (2006)

    Google Scholar 

  38. Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Physica A 369, 71–99 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  39. Stembridge, J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83, 96–131 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  40. Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  42. Viennot, G.: Une forme géométrique de la correspondence de Robinson-Schensted. In: Combinatoire et Représentation du Groupe Symétrique. Lecture Notes in Mathematics, vol. 579, pp. 29–58. Springer, Berlin (1977)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. California Institute of Technology, Pasadena, CA, 91125, USA

    Alexei Borodin

  2. Technische Universität München, 85747, Garching, Germany

    Patrik L. Ferrari & Michael Prähofer

  3. Chiba University, Chiba, 263-8522, Japan

    Tomohiro Sasamoto

Authors
  1. Alexei Borodin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrik L. Ferrari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Prähofer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tomohiro Sasamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrik L. Ferrari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borodin, A., Ferrari, P.L., Prähofer, M. et al. Fluctuation Properties of the TASEP with Periodic Initial Configuration. J Stat Phys 129, 1055–1080 (2007). https://doi.org/10.1007/s10955-007-9383-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-007-9383-0

Keywords

Search

Navigation