Mathematical Physics, Analysis and Geometry

, Volume 14, Issue 3, pp 211–235 | Cite as

Formulas and Asymptotics for the Asymmetric Simple Exclusion Process

  • Craig A. Tracy
  • Harold Widom


This is an expanded version of a series of lectures delivered by the second author in June, 2009. It describes the results of three of the authors’ papers on the asymmetric simple exclusion process, from the derivation of exact formulas for configuration probabilities, through Fredholm determinant representation, to asymptotics with step initial condition establishing KPZ universality. Although complete proofs are in general not given, at least the main elements of them are.


Asymmetric simple exclusion process Fredholm determinant Asymptotics KPZ universality 

Mathematics Subject Classification (2010)



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  1. 1.
    Bethe, H.A.: On the theory of metals, I. Eigenvalues and eigenfunctions of a linear chain of atoms (German). Zeits. Phys. 74, 205–226 (1931)ADSCrossRefGoogle Scholar
  2. 2.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)ADSCrossRefMATHGoogle Scholar
  4. 4.
    Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (2005). Reprint of the 1985 originalMATHGoogle Scholar
  5. 5.
    Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis. Springer, Berlin (1964)MATHGoogle Scholar
  6. 6.
    Rákos, A., Schütz, G.M.: Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118, 511–530 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Schütz, G.M.: Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88, 427–445 (1997)ADSCrossRefMATHGoogle Scholar
  8. 8.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246–290 (1970)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Tracy, C.A., Widom, H.: Integral formulas for the asymmetric simple exclusion process. Commun. Math. Phys. 279, 815–844 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  10. 10.
    Tracy, C.A., Widom, H.: A Fredholm determinant representation in ASEP. J. Stat. Phys. 132, 291–300 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  11. 11.
    Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009)MathSciNetADSCrossRefMATHGoogle Scholar
  12. 12.
    Tracy, C.A., Widom, H.: Formulas for joint probabilities for the asymmetric simple exclusion process. J. Math. Phys. 51, 063302 (2010)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Tracy, C.A., Widom, H.: Erratum to “Integral formulas for the asymmetric simple exclusion process”. Commun. Math. Phys. 304, 875–878 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

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