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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
Article

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 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.”

References

  1. 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. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baik, J., Jenkins, R.: Limiting distribution of maximal crossing and nesting of Poissonized random matchings (2011). arXiv:1111.0269
  5. 5.
    Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–541 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Baik, J., Rains, E.M.: The asymptotics of monotone subsequences of involutions. Duke Math. J. 109, 205–281 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 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. 8.
    Barabasi, A.L., Stanley, H.E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995) zbMATHCrossRefGoogle Scholar
  9. 9.
    Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79, 871–915 (2010) MathSciNetADSzbMATHGoogle Scholar
  10. 10.
    Bornemann, F., Ferrari, P., Prähofer, M.: The Airy1 process is not the limit of the largest eigenvalue in GOE matrix diffusion. J. Stat. Phys. 133, 405–415 (2008) MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 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) MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 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) MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 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. 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. 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. 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. 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) MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 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. 19.
    Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices: Theory Appl. 1, 1130001 (2012) Google Scholar
  20. 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) ADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Corwin, I., Quastel, J.: Renormalization fixed point of the KPZ universality class (2011). arXiv:1103.3422
  22. 22.
    Corwin, I., Quastel, J., Remenik, D.: Continuum statistics of the Airy2 process (2011). arXiv:1106.2717
  23. 23.
    Dotsenko, V.: Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. Europhys. Lett. 90, 20003 (2010) ADSCrossRefGoogle Scholar
  24. 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. 25.
    Ferrari, P.L.: Slow decorrelations in Kardar-Parisi-Zhang growth. J. Stat. Mech. 2008, P07022 (2008) CrossRefGoogle Scholar
  26. 26.
    Ferrari, P.L., Frings, R.: Finite time corrections in KPZ growth models. J. Stat. Phys. 144, 1123–1150 (2011) MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. 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) MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 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. 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) MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 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. 31.
    Frisch, U.: Turbulence: The Legacy of A.N. Kolmogorov. Cambridge Univ. Press, Cambridge (1995) zbMATHGoogle Scholar
  32. 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. 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. 34.
    Gumbel, E.J.: Statistics of Extremes. Columbia Univ. Press. New York (1958). Republished by Dover, New York (2004) zbMATHGoogle Scholar
  35. 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. 36.
    Harris, T.E.: Contact interactions on a lattice. Ann. Probab. 2, 969–988 (1974) zbMATHCrossRefGoogle Scholar
  37. 37.
    Henkel, M.: Conformal Invariance and Critical Phenomena. Springer, Berlin, Heidelberg, New York (1999) zbMATHGoogle Scholar
  38. 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. 39.
    Hinrichsen, H.: Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815–958 (2000) ADSCrossRefGoogle Scholar
  40. 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. 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. 42.
    Imamura, T., Sasamoto, T.: Exact solution for the stationary KPZ equation (2011). arXiv:1111.4634
  43. 43.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000) MathSciNetADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003) MathSciNetADSzbMATHGoogle Scholar
  45. 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. 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. 47.
    Kallabis, H., Krug, J.: Persistence of Kardar-Parisi-Zhang interfaces. Europhys. Lett. 45, 20–25 (1999) ADSCrossRefGoogle Scholar
  48. 48.
    Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986) ADSzbMATHCrossRefGoogle Scholar
  49. 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. 50.
    Krug, J.: Classification of some deposition and growth processes. J. Phys. A 22, L769–L773 (1989) ADSCrossRefGoogle Scholar
  51. 51.
    Krug, J.: Origins of scale invariance in growth processes. Adv. Phys. 46, 139–282 (1997) ADSCrossRefGoogle Scholar
  52. 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. 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. 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. 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. 56.
    Majumdar, S.N.: Persistence in nonequilibrium systems. Curr. Sci. 77, 370–375 (1999) Google Scholar
  57. 57.
    Majumdar, S.N., Bray, A.J.: Spatial persistence of fluctuating interfaces. Phys. Rev. Lett. 86, 3700–3703 (2001) ADSCrossRefGoogle Scholar
  58. 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. 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. 60.
    Meakin, P.: The growth of rough surfaces and interfaces. Phys. Rep. 235, 189–289 (1993) ADSCrossRefGoogle Scholar
  61. 61.
    Mehta, M.L.: Random Matrices. Pure and Applied Mathematics, vol. 142, 3rd edn. Elsevier, San Diego (2004) zbMATHGoogle Scholar
  62. 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. 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. 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. 65.
    Moreno Flores, G., Quastel, J., Remenik, D.: Endpoint distribution of directed polymers in 1+1 dimensions (2011). arXiv:1106.2716
  66. 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. 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. 68.
    Newell, G.F., Rosenblatt, M.: Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Stat. 33, 1306–1313 (1962) MathSciNetzbMATHCrossRefGoogle Scholar
  69. 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. 70.
    Paiva, L.R., Ferreira, S.C. Jr: Universality class of isotropic on-lattice eden clusters. J. Phys. A 40, F43–F49 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Prähofer, M., Spohn, H.: Statistical self-similarity of one-dimensional growth processes. Physica A 279, 342–352 (2000) MathSciNetADSzbMATHCrossRefGoogle Scholar
  72. 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. 73.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the airy process. J. Stat. Phys. 108, 1071–1106 (2002) zbMATHCrossRefGoogle Scholar
  74. 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. 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. 76.
    Quastel, J., Remenik, D.: Tails of the endpoint distribution of directed polymers (2012). arXiv:1203.2907
  77. 77.
    Rambeau, J., Schehr, G.: Extremal statistics of curved growing interfaces in 1+1 dimensions. Europhys. Lett. 91, 60006 (2010) ADSCrossRefGoogle Scholar
  78. 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. 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. 80.
    Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549–L556 (2005) MathSciNetADSCrossRefGoogle Scholar
  81. 81.
    Sasamoto, T.: Private communication (2012) Google Scholar
  82. 82.
    Sasamoto, T., Imamura, T.: Fluctuations of the one-dimensional polynuclear growth model in half-space. J. Stat. Phys. 115, 749–803 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar
  83. 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. 84.
    Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834, 523–542 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar
  85. 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. 86.
    Schehr, G.: Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces (2012). arXiv:1203.1658
  87. 87.
    Schehr, G.: Private communication (2012) Google Scholar
  88. 88.
    Singha, S.B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech. 2005, P08006 (2005) CrossRefGoogle Scholar
  89. 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. 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. 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. 92.
    Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice eden model. J. Stat. Mech. 2012, P05007 (2012) CrossRefGoogle Scholar
  93. 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. 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. 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. 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. 97.
    Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  98. 98.
    Tracy, C.A., Widom, H.: Level-spacing distributions and the airy kernel. Commun. Math. Phys. 159, 151–174 (1994) MathSciNetADSzbMATHCrossRefGoogle Scholar
  99. 99.
    Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996) MathSciNetADSzbMATHCrossRefGoogle Scholar
  100. 100.
    Vicsek, T., Cserző, M., Horváth, V.K.: Self-affine growth of bacterial colonies. Physica A 167, 315–321 (1990) ADSCrossRefGoogle Scholar
  101. 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

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Authors and Affiliations

  1. 1.Department of PhysicsThe University of TokyoTokyoJapan

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