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

## Abstract

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.

## Keywords

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

### Acknowledgements

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 Airy_{1} and Airy_{2} 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.”

## References

- 1.Amar, J.G., Family, F.: Universality in surface growth: scaling functions and amplitude ratios. Phys. Rev. A
**45**, 5378–5393 (1992) ADSCrossRefGoogle Scholar - 2.Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Commun. Pure Appl. Math.
**64**, 466–537 (2011) MathSciNetMATHCrossRefGoogle Scholar - 3.Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc.
**12**, 1119–1178 (1999) MathSciNetMATHCrossRefGoogle Scholar - 4.Baik, J., Jenkins, R.: Limiting distribution of maximal crossing and nesting of Poissonized random matchings (2011). arXiv:1111.0269
- 5.Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys.
**100**, 523–541 (2000) MathSciNetMATHCrossRefGoogle Scholar - 6.Baik, J., Rains, E.M.: The asymptotics of monotone subsequences of involutions. Duke Math. J.
**109**, 205–281 (2001) MathSciNetMATHCrossRefGoogle Scholar - 7.Baik, J., Rains, E.M.: Symmetrized random permutations. In: Bleher, P., Its, A. (eds.) Random Matrix Models and Their Applications. MSRI Publications, vol. 40, pp. 1–19. Cambridge University Press, Cambridge (2001) Google Scholar
- 8.Barabasi, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995) MATHCrossRefGoogle Scholar
- 9.Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput.
**79**, 871–915 (2010) MathSciNetADSMATHGoogle Scholar - 10.Bornemann, F., Ferrari, P., Prähofer, M.: The Airy
_{1}process is not the limit of the largest eigenvalue in GOE matrix diffusion. J. Stat. Phys.**133**, 405–415 (2008) MathSciNetADSMATHCrossRefGoogle Scholar - 11.Borodin, A., Ferrari, P., Prähofer, M., Sasamoto, T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys.
**129**, 1055–1080 (2007) MathSciNetADSMATHCrossRefGoogle Scholar - 12.Borodin, A., Ferrari, P., Sasamoto, T.: Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP. Commun. Math. Phys.
**283**, 417–449 (2008) MathSciNetADSMATHCrossRefGoogle Scholar - 13.Calabrese, P., Le Doussal, P.: Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions. Phys. Rev. Lett.
**106**, 250603 (2011) ADSCrossRefGoogle Scholar - 14.Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Europhys. Lett.
**90**, 20002 (2010) ADSCrossRefGoogle Scholar - 15.Canet, L., Chaté, H., Delamotte, B., Wschebor, N.: Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation. Phys. Rev. Lett.
**104**, 150601 (2010) ADSCrossRefGoogle Scholar - 16.Canet, L., Chaté, H., Delamotte, B., Wschebor, N.: Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation: general framework and first applications. Phys. Rev. E
**84**, 061128 (2011) ADSCrossRefGoogle Scholar - 17.Clusel, M., Bertin, E.: Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics. Int. J. Mod. Phys. B
**22**, 3311–3368 (2008) MathSciNetADSMATHCrossRefGoogle Scholar - 18.Constantin, M., Das Sarma, S., Dasgupta, C.: Spatial persistence and survival probabilities for fluctuating interfaces. Phys. Rev. E
**69**, 051603 (2004) ADSCrossRefGoogle Scholar - 19.Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices: Theory Appl.
**1**, 1130001 (2012) Google Scholar - 20.Corwin, I., Ferrari, P.L., Péché, S.: Universality of slow decorrelation in KPZ growth. Ann. Inst. Henri Poincaré B, Probab. Stat.
**48**, 134–150 (2012) ADSMATHCrossRefGoogle Scholar - 21.Corwin, I., Quastel, J.: Renormalization fixed point of the KPZ universality class (2011). arXiv:1103.3422
- 22.Corwin, I., Quastel, J., Remenik, D.: Continuum statistics of the Airy
_{2}process (2011). arXiv:1106.2717 - 23.Dotsenko, V.: Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. Europhys. Lett.
**90**, 20003 (2010) ADSCrossRefGoogle Scholar - 24.Family, F., Vicsek, T.: Scaling of the active zone in the eden process on percolation networks and the ballistic deposition model. J. Phys. A
**18**, L75–L81 (1985) ADSCrossRefGoogle Scholar - 25.Ferrari, P.L.: Slow decorrelations in Kardar-Parisi-Zhang growth. J. Stat. Mech.
**2008**, P07022 (2008) CrossRefGoogle Scholar - 26.Ferrari, P.L., Frings, R.: Finite time corrections in KPZ growth models. J. Stat. Phys.
**144**, 1123–1150 (2011) MathSciNetADSMATHCrossRefGoogle Scholar - 27.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) MathSciNetADSMATHCrossRefGoogle Scholar - 28.Ferreira, S.C. Jr, Alves, S.G.: Pitfalls in the determination of the universality class of radial clusters. J. Stat. Mech.
**2006**, P11007 (2006) CrossRefGoogle Scholar - 29.Forrester, P.J., Majumdar, S.N., Schehr, G.: Non-intersecting Brownian walkers and Yang-Mills theory on the sphere. Nucl. Phys. B
**844**, 500–526 (2011) MathSciNetADSMATHCrossRefGoogle Scholar - 30.Forster, D., Nelson, D.R., Stephen, M.J.: Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A
**16**, 732–749 (1977) MathSciNetADSCrossRefGoogle Scholar - 31.Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge Univ. Press, Cambridge (1995) MATHGoogle Scholar
- 32.de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals. International Series of Monographs on Physics, vol. 83, 2nd edn. Oxford Univ. Press, New York (1995) Google Scholar
- 33.Gillespie, D.T.: Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral. Phys. Rev. E
**54**, 2084–2091 (1996) MathSciNetADSCrossRefGoogle Scholar - 34.Gumbel, E.J.: Statistics of Extremes. Columbia Univ. Press. New York (1958). Republished by Dover, New York (2004) MATHGoogle Scholar
- 35.Halpin-Healy, T., Zhang, Y.C.: Kinetic roughening phenomena, stochastic growth, directed polymers and all that. Aspects of multidisciplinary statistical mechanics. Phys. Rep.
**254**, 215–414 (1995) ADSCrossRefGoogle Scholar - 36.Harris, T.E.: Contact interactions on a lattice. Ann. Probab.
**2**, 969–988 (1974) MATHCrossRefGoogle Scholar - 37.Henkel, M.: Conformal Invariance and Critical Phenomena. Springer, Berlin, Heidelberg, New York (1999) MATHGoogle Scholar
- 38.Henkel, M., Noh, J.D., Pleimling, M.: Phenomenology of aging in the Kardar-Parisi-Zhang equation. Phys. Rev. E
**85**, 030102 (2012) ADSCrossRefGoogle Scholar - 39.Hinrichsen, H.: Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys.
**49**, 815–958 (2000) ADSCrossRefGoogle Scholar - 40.Huergo, M.A.C., Pasquale, M.A., Bolzán, A.E., Arvia, A.J., González, P.H.: Morphology and dynamic scaling analysis of cell colonies with linear growth fronts. Phys. Rev. E
**82**, 031903 (2010) ADSCrossRefGoogle Scholar - 41.Huergo, M.A.C., Pasquale, M.A., González, P.H., Bolzán, A.E., Arvia, A.J.: Dynamics and morphology characteristics of cell colonies with radially spreading growth fronts. Phys. Rev. E
**84**, 021917 (2011) ADSCrossRefGoogle Scholar - 42.Imamura, T., Sasamoto, T.: Exact solution for the stationary KPZ equation (2011). arXiv:1111.4634
- 43.Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys.
**209**, 437–476 (2000) MathSciNetADSMATHCrossRefGoogle Scholar - 44.Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys.
**242**, 277–329 (2003) MathSciNetADSMATHGoogle Scholar - 45.Kai, S., Zimmermann, W.: Pattern dynamics in the electrohydrodynamics of nematic liquid crystals. Prog. Theor. Phys. Suppl.
**99**, 458–492 (1989) ADSCrossRefGoogle Scholar - 46.Kai, S., Zimmermann, W., Andoh, M., Chizumi, N.: Local transition to turbulence in electrohydrodynamic convection. Phys. Rev. Lett.
**64**, 1111–1114 (1990) ADSCrossRefGoogle Scholar - 47.Kallabis, H., Krug, J.: Persistence of Kardar-Parisi-Zhang interfaces. Europhys. Lett.
**45**, 20–25 (1999) ADSCrossRefGoogle Scholar - 48.Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett.
**56**, 889–892 (1986) ADSMATHCrossRefGoogle Scholar - 49.Kriecherbauer, T., Krug, J.: A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. J. Phys. A
**43**, 403001 (2010) MathSciNetCrossRefGoogle Scholar - 50.Krug, J.: Classification of some deposition and growth processes. J. Phys. A
**22**, L769–L773 (1989) ADSCrossRefGoogle Scholar - 51.Krug, J.: Origins of scale invariance in growth processes. Adv. Phys.
**46**, 139–282 (1997) ADSCrossRefGoogle Scholar - 52.Krug, J., Kallabis, H., Majumdar, S.N., Cornell, S.J., Bray, A.J., Sire, C.: Persistence exponents for fluctuating interfaces. Phys. Rev. E
**56**, 2702–2712 (1997) ADSCrossRefGoogle Scholar - 53.Krug, J., Meakin, P., Halpin-Healy, T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A
**45**, 638–653 (1992) ADSCrossRefGoogle Scholar - 54.Kuennen, E.W., Wang, C.Y.: Off-lattice radial Eden cluster growth in two and three dimensions. J. Stat. Mech.
**2008**, P05014 (2008) CrossRefGoogle Scholar - 55.Liechty, K.: The limiting distribution of the maximal height of the outermost path of nonintersecting Brownian excursions and discrete Gaussian orthogonal polynomials (2011). arXiv:1111.4239
- 56.Majumdar, S.N.: Persistence in nonequilibrium systems. Curr. Sci.
**77**, 370–375 (1999) Google Scholar - 57.Majumdar, S.N., Bray, A.J.: Spatial persistence of fluctuating interfaces. Phys. Rev. Lett.
**86**, 3700–3703 (2001) ADSCrossRefGoogle Scholar - 58.Majumdar, S.N., Dasgupta, C.: Spatial survival probability for one-dimensional fluctuating interfaces in the steady state. Phys. Rev. E
**73**, 011602 (2006) ADSCrossRefGoogle Scholar - 59.Maunuksela, J., Myllys, M., Kähkönen, O.P., Timonen, J., Provatas, N., Alava, M.J., Ala-Nissila, T.: Kinetic roughening in slow combustion of paper. Phys. Rev. Lett.
**79**, 1515–1518 (1997) ADSCrossRefGoogle Scholar - 60.Meakin, P.: The growth of rough surfaces and interfaces. Phys. Rep.
**235**, 189–289 (1993) ADSCrossRefGoogle Scholar - 61.Mehta, M.L.: Random Matrices. Pure and Applied Mathematics, vol. 142, 3rd edn. Elsevier, San Diego (2004) MATHGoogle Scholar
- 62.Merikoski, J., Maunuksela, J., Myllys, M., Timonen, J., Alava, M.J.: Temporal and spatial persistence of combustion fronts in paper. Phys. Rev. Lett.
**90**, 024501 (2003) ADSCrossRefGoogle Scholar - 63.Mézard, M., Parisi, G., Virasoro, M.: Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications. Lecture Notes in Physics, vol. 9. World Scientific, Singapore (1987) Google Scholar
- 64.Miettinen, L., Myllys, M., Merikoski, J., Timonen, J.: Experimental determination of KPZ height-fluctuation distributions. Eur. Phys. J. B
**46**, 55–60 (2005) ADSCrossRefGoogle Scholar - 65.Moreno Flores, G., Quastel, J., Remenik, D.: Endpoint distribution of directed polymers in 1+1 dimensions (2011). arXiv:1106.2716
- 66.Myllys, M., Maunuksela, J., Alava, M., Ala-Nissila, T., Merikoski, J., Timonen, J.: Kinetic roughening in slow combustion of paper. Phys. Rev. E
**64**, 036101 (2001) ADSCrossRefGoogle Scholar - 67.Myllys, M., Maunuksela, J., Alava, M.J., Ala-Nissila, T., Timonen, J.: Scaling and noise in slow combustion of paper. Phys. Rev. Lett.
**84**, 1946–1949 (2000) ADSCrossRefGoogle Scholar - 68.Newell, G.F., Rosenblatt, M.: Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Stat.
**33**, 1306–1313 (1962) MathSciNetMATHCrossRefGoogle Scholar - 69.Oliveira, T.J., Ferreira, S.C., Alves, S.G.: Universal fluctuations in Kardar-Parisi-Zhang growth on one-dimensional flat substrates. Phys. Rev. E
**85**, 010601 (2012) ADSCrossRefGoogle Scholar - 70.Paiva, L.R., Ferreira, S.C. Jr: Universality class of isotropic on-lattice eden clusters. J. Phys. A
**40**, F43–F49 (2007) MathSciNetMATHCrossRefGoogle Scholar - 71.Prähofer, M., Spohn, H.: Statistical self-similarity of one-dimensional growth processes. Physica A
**279**, 342–352 (2000) MathSciNetADSMATHCrossRefGoogle Scholar - 72.Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett.
**84**, 4882–4885 (2000) ADSCrossRefGoogle Scholar - 73.Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the airy process. J. Stat. Phys.
**108**, 1071–1106 (2002) MATHCrossRefGoogle Scholar - 74.Prolhac, S., Spohn, H.: Height distribution of the Kardar-Parisi-Zhang equation with sharp-wedge initial condition: numerical evaluations. Phys. Rev. E
**84**, 011119 (2011) ADSCrossRefGoogle Scholar - 75.Prolhac, S., Spohn, H.: Two-point generating function of the free energy for a directed polymer in a random medium. J. Stat. Mech.
**2011**, P01031 (2011) MathSciNetCrossRefGoogle Scholar - 76.Quastel, J., Remenik, D.: Tails of the endpoint distribution of directed polymers (2012). arXiv:1203.2907
- 77.Rambeau, J., Schehr, G.: Extremal statistics of curved growing interfaces in 1+1 dimensions. Europhys. Lett.
**91**, 60006 (2010) ADSCrossRefGoogle Scholar - 78.Rambeau, J., Schehr, G.: Distribution of the time at which
*n*vicious walkers reach their maximal height. Phys. Rev. E**83**, 061146 (2011) ADSCrossRefGoogle Scholar - 79.Rodríguez-Laguna, J., Santalla, S.N., Cuerno, R.: Intrinsic geometry approach to surface kinetic roughening. J. Stat. Mech.
**2011**, P05032 (2011) CrossRefGoogle Scholar - 80.Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A
**38**, L549–L556 (2005) MathSciNetADSCrossRefGoogle Scholar - 81.Sasamoto, T.: Private communication (2012) Google Scholar
- 82.Sasamoto, T., Imamura, T.: Fluctuations of the one-dimensional polynuclear growth model in half-space. J. Stat. Phys.
**115**, 749–803 (2004) MathSciNetADSMATHCrossRefGoogle Scholar - 83.Sasamoto, T., Spohn, H.: The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class. J. Stat. Mech.
**2010**, P11013 (2010) CrossRefGoogle Scholar - 84.Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B
**834**, 523–542 (2010) MathSciNetADSMATHCrossRefGoogle Scholar - 85.Sasamoto, T., Spohn, H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett.
**104**, 230602 (2010) ADSCrossRefGoogle Scholar - 86.Schehr, G.: Extremes of
*N*vicious walkers for large*N*: application to the directed polymer and KPZ interfaces (2012). arXiv:1203.1658 - 87.Schehr, G.: Private communication (2012) Google Scholar
- 88.Singha, S.B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech.
**2005**, P08006 (2005) CrossRefGoogle Scholar - 89.Stanley, H.E.: Introduction to Phase Transitions and Critical Phenomena. International Series of Monographs on Physics, vol. 46. Oxford University Press, Oxford (1987) Google Scholar
- 90.Takeuchi, K.A.: Scaling of hysteresis loops at phase transitions into a quasiabsorbing state. Phys. Rev. E
**77**, 030103(R) (2008) ADSCrossRefGoogle Scholar - 91.Takeuchi, K.A.: Comment on “Experimental determination of KPZ height-fluctuation distributions” by L. Miettinen et al. (2012). http://publ.kaztake.org/miet-com.pdf
- 92.Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice eden model. J. Stat. Mech.
**2012**, P05007 (2012) CrossRefGoogle Scholar - 93.Takeuchi, K.A., Kuroda, M., Chaté, H., Sano, M.: Directed percolation criticality in turbulent liquid crystals. Phys. Rev. Lett.
**99**, 234503 (2007) ADSCrossRefGoogle Scholar - 94.Takeuchi, K.A., Kuroda, M., Chaté, H., Sano, M.: Experimental realization of directed percolation criticality in turbulent liquid crystals. Phys. Rev. E
**80**, 051116 (2009) ADSCrossRefGoogle Scholar - 95.Takeuchi, K.A., Sano, M.: Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett.
**104**, 230601 (2010) ADSCrossRefGoogle Scholar - 96.Takeuchi, K.A., Sano, M., Sasamoto, T., Spohn, H.: Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep.
**1**, 34 (2011) CrossRefGoogle Scholar - 97.Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys.
**290**, 129–154 (2009) MathSciNetADSMATHCrossRefGoogle Scholar - 98.Tracy, C.A., Widom, H.: Level-spacing distributions and the airy kernel. Commun. Math. Phys.
**159**, 151–174 (1994) MathSciNetADSMATHCrossRefGoogle Scholar - 99.Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys.
**177**, 727–754 (1996) MathSciNetADSMATHCrossRefGoogle Scholar - 100.Vicsek, T., Cserző, M., Horváth, V.K.: Self-affine growth of bacterial colonies. Physica A
**167**, 315–321 (1990) ADSCrossRefGoogle Scholar - 101.Wakita, J.i., Itoh, H., Matsuyama, T., Matsushita, M.: Self-affinity for the growing interface of bacterial colonies. J. Phys. Soc. Jpn.
**66**, 67–72 (1997) ADSCrossRefGoogle Scholar