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Stationary Correlations for the 1D KPZ Equation

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Research Center for Advanced Science and Technology, The University of Tokyo, Tokyo, Japan

    Takashi Imamura

  2. Department of Mathematics and Informatics, Chiba University, Chiba, Japan

    Tomohiro Sasamoto

  3. Zentrum Mathematik, Technische Universität München, Garching, Germany

    Tomohiro Sasamoto

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  1. Takashi Imamura
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  2. Tomohiro Sasamoto
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Correspondence to Takashi Imamura.

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Imamura, T., Sasamoto, T. Stationary Correlations for the 1D KPZ Equation. J Stat Phys 150, 908–939 (2013). https://doi.org/10.1007/s10955-013-0710-3

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