Journal of Statistical Physics

, Volume 150, Issue 5, pp 908–939 | Cite as

Stationary Correlations for the 1D KPZ Equation

  • Takashi Imamura
  • Tomohiro Sasamoto


We study exact stationary properties of the one-dimensional Kardar-Parisi-Zhang (KPZ) equation by using the replica approach. The stationary state for the KPZ equation is realized by setting the initial condition the two-sided Brownian motion (BM) with respect to the space variable. Developing techniques for dealing with this initial condition in the replica analysis, we elucidate some exact nature of the height fluctuation for the KPZ equation. In particular, we obtain an explicit representation of the probability distribution of the height in terms of the Fredholm determinants. Furthermore from this expression, we also get the exact expression of the space-time two-point correlation function.


KPZ equation Replica method Exact solution Fredholm determinant 



T.S. thanks A. Borodin, I. Corwin, P.L. Ferrari, S. Prolhac, J. Quastel and H. Spohn for useful discussions. Both authors would like to thank R.Y. Inoue for enjoyable conversations on related issues. The work of T.I. and T.S. is supported by KAKENHI (22740251) and KAKENHI (22740054) respectively.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Research Center for Advanced Science and TechnologyThe University of TokyoTokyoJapan
  2. 2.Department of Mathematics and InformaticsChiba UniversityChibaJapan
  3. 3.Zentrum MathematikTechnische Universität MünchenGarchingGermany

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