Journal of Statistical Physics

, Volume 147, Issue 5, pp 853–890 | Cite as

Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence

  • Kazumasa A. Takeuchi
  • Masaki Sano


We provide a comprehensive report on scale-invariant fluctuations of growing interfaces in liquid-crystal turbulence, for which we recently found evidence that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1 dimensions [Takeuchi and Sano in Phys. Rev. Lett. 104:230601, 2010; Takeuchi et al. in Sci. Rep. 1:34, 2011]. Here we investigate both circular and flat interfaces and report their statistics in detail. First we demonstrate that their fluctuations show not only the KPZ scaling exponents but beyond: they asymptotically share even the precise forms of the distribution function and the spatial correlation function in common with solvable models of the KPZ class, demonstrating also an intimate relation to random matrix theory. We then determine other statistical properties for which no exact theoretical predictions were made, in particular the temporal correlation function and the persistence probabilities. Experimental results on finite-time effects and extreme-value statistics are also presented. Throughout the paper, emphasis is put on how the universal statistical properties depend on the global geometry of the interfaces, i.e., whether the interfaces are circular or flat. We thereby corroborate the powerful yet geometry-dependent universality of the KPZ class, which governs growing interfaces driven out of equilibrium.


Growth phenomenon Scaling laws KPZ universality class Electroconvection Liquid crystal Random matrix 



The authors acknowledge enlightening discussions with many theoreticians: T. Sasamoto, H. Spohn, M. Prähofer, G. Schehr, J. Rambeau, H. Chaté, P. Ferrari, to name but a few. We are grateful to T. Sasamoto for his continuing and scrupulous support on the theoretical side of the subject, to G. Schehr for drawing our attention to extreme-value statistics and to P. Ferrari with respect to the finite-time corrections in the second- and higher-order cumulants of the local height. We also wish to thank our colleagues who kindly sent us theoretical curves and numerical data used in this paper: M. Prähofer for the theoretical curves of the TW distributions, F. Bornemann for those of the Airy1 and Airy2 covariance obtained by his accurate algorithm [9], J. Rambeau and G. Schehr for their numerical data on the PNG model partly presented in their work [77, 78], and J. Quastel and D. Remenik for the theoretical curve of the asymptotic distribution of X max, numerically evaluated very recently by them [76]. Critical reading of the manuscript and useful comments by J. Krug, J. Rambeau, T. Sasamoto, G. Schehr and H. Spohn are also much appreciated. This work is supported in part by Grant for Basic Science Research Projects from The Sumitomo Foundation and by the JSPS Core-to-Core Program “International research network for non-equilibrium dynamics of soft matter.”


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Authors and Affiliations

  1. 1.Department of PhysicsThe University of TokyoTokyoJapan

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