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

Journal of Statistical Physics

Volumn 147, Issue 5, 2012, Pages 853-890

  Takeuchi, Kazumasa A ^a   Sano, Masaki ^a  

^a UNIVERSITY OF TOKYO   (Japan)

Author keywords

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

Indexed keywords

EID: 84862703785     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-012-0503-0     Document Type: Article

References (101)

1. Amar, J. G., Family, F.: Universality in surface growth: scaling functions and amplitude ratios. Phys. Rev. A 45, 5378-5393 (1992).
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).
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).
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).
6. Baik, J., Rains, E. M.: The asymptotics of monotone subsequences of involutions. Duke Math. J. 109, 205-281 (2001).
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).
8. Barabasi, A. L., Stanley, H. E.: Fractal Concepts in Surface Growth. Cambridge University Press, Cambridge (1995).
9. Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79, 871-915 (2010).
10. 1 process is not the limit of the largest eigenvalue in GOE matrix diffusion. J. Stat. Phys. 133, 405-415 (2008).
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).
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).
13. Calabrese, P., Le Doussal, P.: Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions. Phys. Rev. Lett. 106, 250603 (2011).
14. Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Europhys. Lett. 90, 20002 (2010).
15. Canet, L., Chaté, H., Delamotte, B., Wschebor, N.: Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 104, 150601 (2010).
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).
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).
18. Constantin, M., Das Sarma, S., Dasgupta, C.: Spatial persistence and survival probabilities for fluctuating interfaces. Phys. Rev. E 69, 051603 (2004).
19. Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices: Theory Appl. 1, 1130001 (2012).
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).
21. Corwin, I., Quastel, J.: Renormalization fixed point of the KPZ universality class (2011). arXiv: 1103. 3422.
22. 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).
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).
25. Ferrari, P. L.: Slow decorrelations in Kardar-Parisi-Zhang growth. J. Stat. Mech. 2008, P07022 (2008).
26. Ferrari, P. L., Frings, R.: Finite time corrections in KPZ growth models. J. Stat. Phys. 144, 1123-1150 (2011).
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).
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).
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).
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).
31. Frisch, U.: Turbulence: The Legacy of A. N. Kolmogorov. Cambridge Univ. Press, Cambridge (1995).
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).
33. Gillespie, D. T.: Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral. Phys. Rev. E 54, 2084-2091 (1996).
34. Gumbel, E. J.: Statistics of Extremes. Columbia Univ. Press. New York (1958). Republished by Dover, New York (2004).
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).
36. Harris, T. E.: Contact interactions on a lattice. Ann. Probab. 2, 969-988 (1974).
37. Henkel, M.: Conformal Invariance and Critical Phenomena. Springer, Berlin, Heidelberg, New York (1999).
38. Henkel, M., Noh, J. D., Pleimling, M.: Phenomenology of aging in the Kardar-Parisi-Zhang equation. Phys. Rev. E 85, 030102 (2012).
39. Hinrichsen, H.: Non-equilibrium critical phenomena and phase transitions into absorbing states. Adv. Phys. 49, 815-958 (2000).
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).
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).
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).
44. Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277-329 (2003).
45. Kai, S., Zimmermann, W.: Pattern dynamics in the electrohydrodynamics of nematic liquid crystals. Prog. Theor. Phys. Suppl. 99, 458-492 (1989).
46. Kai, S., Zimmermann, W., Andoh, M., Chizumi, N.: Local transition to turbulence in electrohydrodynamic convection. Phys. Rev. Lett. 64, 1111-1114 (1990).
47. Kallabis, H., Krug, J.: Persistence of Kardar-Parisi-Zhang interfaces. Europhys. Lett. 45, 20-25 (1999).
48. Kardar, M., Parisi, G., Zhang, Y. C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889-892 (1986).
49. Kriecherbauer, T., Krug, J.: A pedestrian's view on interacting particle systems, KPZ universality and random matrices. J. Phys. A 43, 403001 (2010).
50. Krug, J.: Classification of some deposition and growth processes. J. Phys. A 22, L769-L773 (1989).
51. Krug, J.: Origins of scale invariance in growth processes. Adv. Phys. 46, 139-282 (1997).
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).
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).
54. Kuennen, E. W., Wang, C. Y.: Off-lattice radial Eden cluster growth in two and three dimensions. J. Stat. Mech. 2008, P05014 (2008).
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).
57. Majumdar, S. N., Bray, A. J.: Spatial persistence of fluctuating interfaces. Phys. Rev. Lett. 86, 3700-3703 (2001).
58. Majumdar, S. N., Dasgupta, C.: Spatial survival probability for one-dimensional fluctuating interfaces in the steady state. Phys. Rev. E 73, 011602 (2006).
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).
60. Meakin, P.: The growth of rough surfaces and interfaces. Phys. Rep. 235, 189-289 (1993).
61. Mehta, M. L.: Random Matrices. Pure and Applied Mathematics, vol. 142, 3rd edn. Elsevier, San Diego (2004).
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).
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).
64. Miettinen, L., Myllys, M., Merikoski, J., Timonen, J.: Experimental determination of KPZ height-fluctuation distributions. Eur. Phys. J. B 46, 55-60 (2005).
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).
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).
68. Newell, G. F., Rosenblatt, M.: Zero crossing probabilities for Gaussian stationary processes. Ann. Math. Stat. 33, 1306-1313 (1962).
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).
70. Paiva, L. R., Ferreira, S. C. Jr: Universality class of isotropic on-lattice eden clusters. J. Phys. A 40, F43-F49 (2007).
71. Prähofer, M., Spohn, H.: Statistical self-similarity of one-dimensional growth processes. Physica A 279, 342-352 (2000).
72. Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882-4885 (2000).
73. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the airy process. J. Stat. Phys. 108, 1071-1106 (2002).
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).
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).
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).
78. Rambeau, J., Schehr, G.: Distribution of the time at which n vicious walkers reach their maximal height. Phys. Rev. E 83, 061146 (2011).
79. Rodríguez-Laguna, J., Santalla, S. N., Cuerno, R.: Intrinsic geometry approach to surface kinetic roughening. J. Stat. Mech. 2011, P05032 (2011).
80. Sasamoto, T.: Spatial correlations of the 1D KPZ surface on a flat substrate. J. Phys. A 38, L549-L556 (2005).
81. Sasamoto, T.: Private communication (2012).
82. Sasamoto, T., Imamura, T.: Fluctuations of the one-dimensional polynuclear growth model in half-space. J. Stat. Phys. 115, 749-803 (2004).
83. Sasamoto, T., Spohn, H.: The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class. J. Stat. Mech. 2010, P11013 (2010).
84. Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nucl. Phys. B 834, 523-542 (2010).
85. Sasamoto, T., Spohn, H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010).
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).
88. Singha, S. B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech. 2005, P08006 (2005).
89. Stanley, H. E.: Introduction to Phase Transitions and Critical Phenomena. International Series of Monographs on Physics, vol. 46. Oxford University Press, Oxford (1987).
90. Takeuchi, K. A.: Scaling of hysteresis loops at phase transitions into a quasiabsorbing state. Phys. Rev. E 77, 030103(R) (2008).
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).
93. Takeuchi, K. A., Kuroda, M., Chaté, H., Sano, M.: Directed percolation criticality in turbulent liquid crystals. Phys. Rev. Lett. 99, 234503 (2007).
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).
95. Takeuchi, K. A., Sano, M.: Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals. Phys. Rev. Lett. 104, 230601 (2010).
96. Takeuchi, K. A., Sano, M., Sasamoto, T., Spohn, H.: Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. 1, 34 (2011).
97. Tracy, C., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129-154 (2009).
98. Tracy, C. A., Widom, H.: Level-spacing distributions and the airy kernel. Commun. Math. Phys. 159, 151-174 (1994).
99. Tracy, C. A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727-754 (1996).
100. Vicsek, T., Cserzo{double acute}, M., Horváth, V. K.: Self-affine growth of bacterial colonies. Physica A 167, 315-321 (1990).
101. Wakita, J., Itoh, H., Matsuyama, T., Matsushita, M.: Self-affinity for the growing interface of bacterial colonies. J. Phys. Soc. Jpn. 66, 67-72 (1997).