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A KPZ Cocktail-Shaken, not Stirred...

Toasting 30 Years of Kinetically Roughened Surfaces

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Abstract

The stochastic partial differential equation proposed nearly three decades ago by Kardar, Parisi and Zhang (KPZ) continues to inspire, intrigue and confound its many admirers. Here, we (i) pay debts to heroic predecessors, (ii) highlight additional, experimentally relevant aspects of the recently solved 1+1 KPZ problem, (iii) use an expanding substrates formalism to gain access to the 3d radial KPZ equation and, lastly, (iv) examining extremal paths on disordered hierarchical lattices, set our gaze upon the fate of \(d=\infty \) KPZ. Clearly, there remains ample unexplored territory within the realm of KPZ and, for the hearty, much work to be done, especially in higher dimensions, where numerical and renormalization group methods are providing a deeper understanding of this iconic equation.

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References

  1. Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889 (1986)

    ADS  MATH  Google Scholar 

  2. Krug, J., Meakin, P., Halpin-Healy, T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45, 638 (1992)

    ADS  Google Scholar 

  3. Spohn, H.: KPZ scaling theory and the semi-discrete directed polymer. arXiv:1201.0645

  4. Krug, J., Meakin, P.: Universal finite-size effects in the rate of growth processes. J. Phys. A 23, L987 (1990)

    ADS  Google Scholar 

  5. Huse, D.A., Henley, C.L., Fisher, D.S.: Forced Burgers equation, exact exponent, fluctuation-dissipation theorem. Phys. Rev. Lett. 55, 2924 (1985)

    ADS  Google Scholar 

  6. Dhar, D.: An exactly solved model for interface growth. Phase Transit. 9, 51 (1987)

    Google Scholar 

  7. Gwa, L.-H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin hamiltonian. Phys. Rev. Lett. 68, 725 (1992)

    MathSciNet  ADS  MATH  Google Scholar 

  8. Gwa, L.-H., Spohn, H.: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844 (1992)

    ADS  Google Scholar 

  9. Kardar, M.: Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nucl. Phys. B 290, 582 (1987)

    MathSciNet  ADS  Google Scholar 

  10. Kardar, M., Nelson, D.R.: Commensurate-incommensurate transitions with quenched disorder. Phys. Rev. Lett. 55, 1157 (1985)

    ADS  Google Scholar 

  11. Imbrie, J.Z., Spencer, T.: Diffusion of directed polymers in a random environment. J. Stat. Phys. 52, 609 (1988)

    MathSciNet  ADS  MATH  Google Scholar 

  12. Doty, C.A., Kosterlitz, J.M.: Exact dynamical exponent at the Kardar-Parisi-Zhang roughening transition. Phys. Rev. Lett. 69, 1979 (1992)

    ADS  Google Scholar 

  13. Medina, E., Hwa, T., Kardar, M., Zhang, Y.-C.: Burgers equation with correlated noise: renormalization-group analysis and applications to directed polymers and interface growth. Phys. Rev. A 39, 3053 (1989)

    MathSciNet  ADS  Google Scholar 

  14. Frey, E., Täuber, U.C.: Two-loop renormalization-group analysis of the Burgers-Kardar-Parisi-Zhang equation. Phys. Rev. E 50, 1024 (1994)

    MathSciNet  ADS  Google Scholar 

  15. van Beijeren, H., Kutner, R., Spohn, H.: Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, 2026 (1985)

    MathSciNet  ADS  Google Scholar 

  16. Frey, E., Täuber, U., Hwa, T.: Mode-coupling and renormalization group results for the noisy Burgers equation. Phys. Rev. E 53, 4424 (1996)

    ADS  Google Scholar 

  17. Huse, D.A., Henley, C.L.: Pinning and roughening of domain walls in Ising systems due to random impurities. Phys. Rev. Lett. 54, 2708 (1985)

    ADS  Google Scholar 

  18. Kardar, M.: Roughening by impurities at finite temperatures. Phys. Rev. Lett. 55, 2923 (1985)

    ADS  Google Scholar 

  19. Kim, J.-M., Kosterlitz, M.: Growth in a restricted solid-on-solid model. Phys. Rev. Lett. 62, 2289 (1989)

    ADS  Google Scholar 

  20. Forrest, B., Tang, L.-H.: Surface roughening in a hypercube-stacking model. Phys. Rev. Lett. 64, 1405 (1990)

    ADS  Google Scholar 

  21. Amar, J.A., Family, F.: Numerical solution of a continuum equation for interface growth in 2+1 dimensions. Phys. Rev. A 41, 3399 (1990)

    ADS  Google Scholar 

  22. Tang, L.-H., Forrest, B., Wolf, D.E.: Kinetic surface roughening. II. Hypercube-stacking models. Phys. Rev. A 45, 7162 (1992)

    Google Scholar 

  23. Moser, K., Wolf, D.: Vectorized and parallel simulations of the KPZ equation in 3+1 dimensions. J. Phys. A 27, 4049 (1994)

    ADS  Google Scholar 

  24. Hwa, T., Frey, E.: Exact scaling function of interface growth dynamics. Phys. Rev. E 44, R7873 (1991)

    ADS  Google Scholar 

  25. Tang, L.-H.: Steady-state scaling function of the (1 + 1)-dimensional single-step model. J. Stat. Phys. 67, 819 (1992)

    ADS  MATH  Google Scholar 

  26. Kim, J.M., Moore, M.A., Bray, A.-J.: Zero-temperature directed polymers in a random potential. Phys. Rev. A 44, 2345 (1991)

    ADS  Google Scholar 

  27. Halpin-Healy, T.: Directed polymers in random media: probability distributions. Phys. Rev. A 44, R3415 (1991)

    ADS  Google Scholar 

  28. Meakin, P.: The growth of rough surfaces and interfaces. Phys. Rep. 235, 189 (1993)

    ADS  Google Scholar 

  29. 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 (1995)

    ADS  Google Scholar 

  30. Krug, J.: Origins of scale invariance in growth processes. Adv. Phys. 46, 139 (1997)

    ADS  Google Scholar 

  31. Maunuksela, J.: Kinetic roughening in the slow combustion of paper. Phys. Rev. Lett. 79, 1515 (1997)

    ADS  Google Scholar 

  32. Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)

    MathSciNet  ADS  MATH  Google Scholar 

  33. Schütz, G.M.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86, 1265 (1997)

    ADS  MATH  Google Scholar 

  34. Lässig, M., Kinzelbach, H.: Upper critical dimension of the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 78, 903 (1997)

    ADS  Google Scholar 

  35. Lässig, M.: Quantized scaling of growing surfaces. Phys. Rev. Lett. 80, 2366 (1998)

    ADS  Google Scholar 

  36. Ala-Nissila, T.: Upper critical dimension of the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 80, 887 (1998)

    ADS  Google Scholar 

  37. Kim, J.M.: Phase transition of the KPZ equation in four substrate dimensions. Phys. Rev. Lett. 80, 888 (1998)

    ADS  Google Scholar 

  38. Castellano, C., Marsili, M., Pietronero, L.: Nonperturbative renormalization of the Kardar-Parisi-Zhang growth dynamics. Phys. Rev. Lett. 80, 3527 (1998)

    ADS  Google Scholar 

  39. Marinari, E., Pagnani, A., Parisi, G.: Critical exponents of the KPZ equation via multi-surface coding numerical simulations. J. Phys. A 33, 8181 (2000)

    MathSciNet  ADS  MATH  Google Scholar 

  40. Halpin-Healy, T.: Disorder-induced roughening of diverse manifolds. Phys. Rev. A 42, 711 (1990)

    ADS  Google Scholar 

  41. Frey, E., Täuber, U., Janssen, H.K.: Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem. Europhys. Lett. 47, 14 (1999)

    ADS  Google Scholar 

  42. Perlsman, E., Schwartz, M.: UCD of the KPZ equation. Phys. Rev. E 85, 050103 (2012)

    Google Scholar 

  43. Pagnani, A., Parisi, G.: Multisurface coding simulations of the RSOS model in four dimensions. Phys. Rev. E 87, 010102 (2013)

    ADS  Google Scholar 

  44. Kim, J.M., Kim, S.-W.: RSOS model with a proper restriction parameter N in 4+1 dimensions. Phys. Rev. E 88, 034102 (2013)

    ADS  Google Scholar 

  45. Alves, S.G., Oliveira, T.J., Ferreira, S.C.: Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension. Phys. Rev. E 90, 020103 (2014)

    ADS  Google Scholar 

  46. Moore, M.A., Blum, T., Doherty, J.P., Marsili, M., Bouchaud, J.-P., Claudin, P.: Glassy solutions of the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 74, 4257 (1995)

    ADS  Google Scholar 

  47. Canét, L., Chaté, H., Delamotte, B., Wschebor, N.: Nonperturbative renormalization group for the KPZ Equation. Phys. Rev. Lett. 104, 150601 (2010)

    ADS  Google Scholar 

  48. Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882 (2000)

    ADS  Google Scholar 

  49. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437 (2000)

    ADS  MATH  Google Scholar 

  50. Tracy, C., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151 (1994)

    MathSciNet  ADS  MATH  Google Scholar 

  51. Tracy, C., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727 (1996)

    MathSciNet  ADS  MATH  Google Scholar 

  52. 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 (1999)

    MathSciNet  MATH  Google Scholar 

  53. Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Am. Math. Soc. 36, 413 (1999)

    MathSciNet  MATH  Google Scholar 

  54. Okounkov, A.: Random matrices and random permutations. Int. Math. Res. Not. 2000, 1043 (2000)

    MathSciNet  MATH  Google Scholar 

  55. Odylyzko, A.M., Rains, E.M.: ATT Bell Labs Technical Report (1999)

  56. Baer, R.M., Brock, P.: Natural sorting over permutation spaces. Math. Comput. 22, 385 (1968)

    MathSciNet  Google Scholar 

  57. Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523 (2000)

    MathSciNet  MATH  Google Scholar 

  58. Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071 (2002)

    MATH  Google Scholar 

  59. 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)

    ADS  Google Scholar 

  60. Myllys, M.: Effect of a columnar defect on the shape of slow-combustion fronts. Phys. Rev. E 68, 051103 (2003)

    ADS  Google Scholar 

  61. Miettinen, L., Myllys, M., Merikoski, J., Timonen, J.: Experimental determination of KPZ height-fluctuation distributions. Eur. Phys. J. B 46, 55 (2005)

    ADS  Google Scholar 

  62. Colaiori, F., Moore, M.A.: Upper critical dimension, dynamic exponent, and scaling functions in the mode-coupling theory for the KPZ equation. Phys. Rev. Lett. 86, 3946 (2001)

    ADS  Google Scholar 

  63. Fogedby, H.: Localized growth modes, dynamic textures, and UCD for the KPZ equation in the weak-noise limit. Phys. Rev. Lett. 94, 195702 (2005)

    ADS  Google Scholar 

  64. Fogedby, H.: Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension. Phys. Rev. E 73, 031104 (2006)

    MathSciNet  ADS  Google Scholar 

  65. Palasantzas, G.: Roughening aspects of room temperature vapor deposited oligomer thin films onto Si substrates. Surf. Sci 507, 357 (2002)

    ADS  Google Scholar 

  66. Halpin-Healy, T., Lin, Y.: Universal aspects of curved, flat and stationary-state KPZ statistics. Phys. Rev. E 89, 010103 (2014)

    ADS  Google Scholar 

  67. Majumdar, S.N., Nechaev, S.: Anisotropic ballistic deposition model with links to the Ulam problem and the Tracy-Widom distribution. Phys. Rev. E 69, 011103 (2004)

    MathSciNet  ADS  Google Scholar 

  68. Barkema, G.T., Ferrari, P.L., Lebowitz, J.L., Spohn, H.: Kardar-Parisi-Zhang universality class and the anchored Toom interface. Phys. Rev. E 90, 042116 (2014)

    ADS  Google Scholar 

  69. Kriecherbauer, T., Krug, J.: A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. J. Phys. A. 43, 403001 (2010)

    MathSciNet  Google Scholar 

  70. Corwin, I.: The KPZ equation and universality class. Random Matrices 1, 1130001 (2012)

    MathSciNet  Google Scholar 

  71. Sasamoto, T., Spohn, H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010)

    ADS  Google Scholar 

  72. 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 (2011)

    MathSciNet  MATH  Google Scholar 

  73. Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Europhys. Lett. 90, 20002 (2010)

    ADS  Google Scholar 

  74. Dotsenko, V.: Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. Europhys. Lett. 90, 20003 (2010)

    ADS  Google Scholar 

  75. Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129 (2009)

    MathSciNet  ADS  MATH  Google Scholar 

  76. Calabrese, P., Le Doussal, P.: Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions. Phys. Rev. Lett. 106, 250603 (2011)

    ADS  Google Scholar 

  77. Le Doussal, P., Calabrese, P.: The KPZ equation with flat initial condition and the directed polymer with one free end. J. Stat. Mech. P06001 (2012)

  78. Gueudré, T., Le Doussal, P.: Directed polymer near a hard wall and KPZ equation in the half-space. EPL 100, 26006 (2012)

    ADS  Google Scholar 

  79. Imamura, T., Sasamoto, T.: Exact solution for the stationary Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 108, 190603 (2012)

    ADS  Google Scholar 

  80. Borodin, A., Corwin, I., Ferrari, P.L., Vető, B.: Height fluctuations for the stationary KPZ equation. arXiv:1407.6977

  81. Takeuchi, K.A., Sano, M.: Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals. Phys. Rev. Lett. 104, 230601 (2010)

    ADS  Google Scholar 

  82. Takeuchi, K.A., Sano, M., Sasamoto, T., Spohn, H.: Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. (Nature) 1, 34 (2011)

    ADS  Google Scholar 

  83. Takeuchi, K.A., Sano, M.: Evidence for geometry-dependent universal fluctuations of KPZ interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012)

    ADS  MATH  Google Scholar 

  84. Takeuchi, K.A.: Crossover from growing to stationary interfaces in the Kardar-Parisi-Zhang class. Phys. Rev. Lett. 110, 210604 (2013)

    ADS  Google Scholar 

  85. Kloss, T., Canet, L., Wschebor, N.: Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1, and 3+1 dimensions. Phys. Rev. E 86, 051124 (2012). see, esp., section IV-E

    ADS  Google Scholar 

  86. Halpin-Healy, T.: (2+1)-Dimensional directed polymer in a random medium: scaling phenomena and universal distributions. Phys. Rev. Lett. 109, 170602 (2012)

    ADS  Google Scholar 

  87. Halpin-Healy, T.: Extremal paths, the stochastic heat equation, and the 3d KPZ universality class. Phys. Rev. E 88, 042118 (2013); Phys. Rev. E 88, 069903 (2013)

  88. Oliveira, T.J., Alves, S.G., Ferreira, S.C.: Kardar-Parisi-Zhang universality class in (2+1) dimensions: universal geometry-dependent distributions and finite-time corrections. Phys. Rev. E 87, 040102 (2013)

    ADS  Google Scholar 

  89. Prähofer, M.: Stochastic Surface Growth. Ludwig-Maximilians-Universitait, München (2003)

    Google Scholar 

  90. Halpin-Healy, T., Palasantzas, G.: Universal correlators & distributions as experimental signatures of (2 + 1)-dimensional Kardar-Parisi-Zhang growth. EPL 105, 50001 (2014)

    ADS  Google Scholar 

  91. Carrasco, I.S.S., Takeuchi, K.A., Ferreira, S.C., Oliveira, T.J.: Interface fluctuations for deposition on enlarging flat substrates. New J. Phys. 16, 123057 (2014)

    ADS  Google Scholar 

  92. Almeida, R.A.L., Ferreira, S.O., Oliveira, T.J., Aarao Reis, F.D.A.: Universal fluctuations in the growth of semiconductor thin films. Phys. Rev. B 89, 045309 (2014)

  93. Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559 (2013)

    MathSciNet  MATH  Google Scholar 

  94. Seppäläinen, T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40, 19 (2012)

    MathSciNet  MATH  Google Scholar 

  95. O‘Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40, 437 (2012)

    MathSciNet  MATH  Google Scholar 

  96. Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Rel. Fields 158, 225 (2014)

    MathSciNet  MATH  Google Scholar 

  97. Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42, 2314 (2014)

    MathSciNet  Google Scholar 

  98. Calabrese, P., Kormos, M., Le Doussal, P.: From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation. EPL 107, 10011 (2014)

    ADS  Google Scholar 

  99. Barraquand, G., Corwin, I.: The q-Hahn asymmetric exclusion process. arXiv:1501.03445

  100. Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. arXiv:1502.07374

  101. Dean, D.S., Le Doussal, P., Majumdar, S.N., Schehr, G.: Finite-temperature free fermions and the KPZ equation at finite time. Phys. Rev. Lett. 114, 110402 (2015)

    ADS  Google Scholar 

  102. Johansson, K.: Two-time distribution in Brownian directed percolation. arXiv:1502.00941

  103. Maritan, A., Toigo, F., Koplik, J., Banavar, J.R.: Dynamics of growing interfaces. Phys. Rev. Lett. 69, 3193 (1992)

    ADS  Google Scholar 

  104. Batchelor, M.T., Henry, B.I., Watt, S.D.: Continuum model for radial interface growth. Physica A 260, 11 (1998)

    MathSciNet  ADS  Google Scholar 

  105. Singha, S.B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech. 2005, P08006 (2005)

    Google Scholar 

  106. Masoudi, A.A.: Statistical analysis of radial interface growth. JSTAT 2012, L02001 (2012)

    Google Scholar 

  107. Rodriguez-Laguna, J., Santalla, S. N., Cuerno, R.: Intrinsic geometry approach to surface kinetic roughening. J. Stat. Mech. P05032 (2011)

  108. Santalla, S.N., Rodriguez-Laguna, J., Cuerno, R.: The circular Kardar-Parisi-Zhang equation as an inflating, self-avoiding ring polymer. Phys. Rev. E 89, 010401 (2014)

    ADS  Google Scholar 

  109. Santalla, S.N., Rodriguez-Laguna, J., LaGatta, T., Cuerno, R.: Random geometry and the Kardar-Parisi-Zhang universality class. New J. Phys. 17, 033018 (2015)

    MathSciNet  ADS  Google Scholar 

  110. Alves, S.G., Oliveira, T.J., Ferreira, S.C.: Universal fluctuations in radial growth models belonging to the KPZ universality class. Europhys. Lett. 96, 48003 (2011)

    ADS  Google Scholar 

  111. Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice Eden model. J. Stat. Mech. 2012, P05007 (2012)

    Google Scholar 

  112. Alves, S.G., Oliveira, T.J., Ferreira, S.C.: Non-universal parameters, corrections and universality in Kardar-Parisi-Zhang growth. JSTAT 2013, P05007 (2013)

    Google Scholar 

  113. Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79, 871915 (2010)

    MathSciNet  Google Scholar 

  114. Rácz, Z., Plischke, M.: Width distribution for (2+1)-dimensional growth and deposition processes. Phys. Rev. E 50, 3530 (1994)

    ADS  Google Scholar 

  115. Foltin, G., Oerding, K., Rácz, Z., Workman, R.L., Zia, R.K.P.: Width-distribution for random-walk interfaces. Phys. Rev. E 50, R639 (1994)

    ADS  Google Scholar 

  116. Edwards, S.F., Wilkinson, D.R.: The surface statistics of a granular aggregate. Proc. R. Soc. London Ser. A 381, 17 (1982)

    MathSciNet  ADS  Google Scholar 

  117. Antal, T., Droz, M., Győrgyi, G., Rácz, Z.: Roughness distributions for 1/f\(^\alpha \) signals. Phys. Rev. E 65, 046140 (2002)

    ADS  Google Scholar 

  118. Santachiara, R., Rosso, A., Krauth, W.: Universal width distribution in non-Markovian gaussian processes. JSTAT 2007, P02009 (2007)

    Google Scholar 

  119. Raychaudhauri, S., Cranston, M., Przybyla, C., Shapir, Y.: Maximal height scaling of kinetically growing surfaces. Phys. Rev. Lett. 87, 136101 (2001)

    ADS  Google Scholar 

  120. Majumdar, S.N., Comtet, A.: Exact maximal height distribution of fluctuating interfaces. Phys. Rev. Lett. 92, 225501 (2004)

    ADS  Google Scholar 

  121. Majumdar, S.N., Comtet, A.: Airy distribution function: From the area under a Brownian excursion to the maximal height of fluctuating interfaces. J. Stat. Phys. 119, 777 (2005)

    MathSciNet  ADS  MATH  Google Scholar 

  122. Schehr, G., Majumdar, S.: Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models. Phys. Rev. E 73, 056103 (2006)

    ADS  Google Scholar 

  123. Győrgyi, G., Moloney, N.R., Ozogány, K., Rácz, Z.: Maximal height statistics for 1/\(f^a\) signals. Phys. Rev. E 75, 021123 (2007)

    MathSciNet  ADS  Google Scholar 

  124. Rambeau, J., Bustingorry, S., Kolton, A.B., Schehr, G.: MRH of elastic interfaces in random media. Phys. Rev. E 84, 041131 (2011)

    ADS  Google Scholar 

  125. Lee, D.-S.: Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces. Phys. Rev. Lett. 95, 150601 (2005)

    ADS  Google Scholar 

  126. Kelling, J., Ódor, G.: Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards. Phys. Rev. E 84, 061150 (2011)

    ADS  Google Scholar 

  127. Derrida, B., Griffiths, R.: Directed polymers on disordered hierarchical lattices. Europhys. Lett. 8, 111 (1989)

    ADS  Google Scholar 

  128. Cook, J., Derrida, B.: Polymers on disordered hierarchical lattices: a nonlinear combination of random variables. J. Stat. Phys. 57, 89 (1989)

    MathSciNet  ADS  Google Scholar 

  129. Halpin-Healy, T.: Comment–growth in a restricted solid-on-solid model. Phys. Rev. Lett. 63, 917 (1989)

    ADS  Google Scholar 

  130. Derrida, B.: Directed polymers in a random medium. Physica A 163, 71 (1990)

    MathSciNet  ADS  Google Scholar 

  131. Roux, S., Hansen, A., da Silva, L., Lucena, L., Pandey, R.: Minimal path on the hierarchical diamond lattice. J. Stat. Phys. 65, 183 (1991)

    ADS  MATH  Google Scholar 

  132. Monthus, C., Garel, T.: Disorder-dominated phases of random systems: relations between the tail exponents and scaling exponents. J. Stat. Mech. 2008, P01008 (2008)

    Google Scholar 

  133. Gumbel, E.J.: Statistics of Extremes. Columbia University Press. New York (1958). Republished by Dover, New York (2004)

  134. Ferrari, P.L., Frings, R.: Finite-time corrections in KPZ growth models. J. Stat. Phys. 144, 1123 (2011)

    MathSciNet  ADS  MATH  Google Scholar 

  135. Oliveira, T.J., Ferreira, S.C., Alves, S.G.: Universal fluctuations in KPZ growth on one-dimensional flat substrates. Phys. Rev. E 85, 010601 (2012)

    ADS  Google Scholar 

  136. Derrida, B., Golinelli, O.: Thermal properties of directed polymers in random media. Phys. Rev. A 41, 4160 (1990)

    ADS  Google Scholar 

  137. Monthus, C., Garel, T.: Numerical study of the directed polymer in a 3+1 dimensional random medium. Eur. Phys. J. B 53, 39 (2006)

    ADS  Google Scholar 

  138. Monthus, C., Garel, T.: Probing the tails of the ground-state energy distribution for the directed polymer in a random medium of dimension d=1,2,3 via a Monte Carlo procedure in the disorder. Phys. Rev. E 74, 051109 (2006)

    ADS  Google Scholar 

  139. Dean, D.S., Majumdar, S.N.: Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence. Phys. Rev. E 64, 046121 (2001)

    ADS  Google Scholar 

  140. Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51, 817 (1988)

    MathSciNet  ADS  MATH  Google Scholar 

  141. Derrida, B., Appert, C.: Universal large deviation function of the KPZ equation in one dimension. J. Stat. Phys. 94, 1 (1999)

    MathSciNet  ADS  MATH  Google Scholar 

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Acknowledgments

The authors would like to express their gratitude to Herbert Spohn for his many years of inspired work, wisdom, and stamina on behalf of the KPZ cause. Thanks, too, to Joel Lebowitz for keeping the statistical mechanical fire well-lit through the generations. This work is supported in part by KAKENHI (No. 25707033 from JSPS and No. 25103004 “Fluctuation & Structure” from MEXT in Japan).

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Halpin-Healy, T., Takeuchi, K.A. A KPZ Cocktail-Shaken, not Stirred.... J Stat Phys 160, 794–814 (2015). https://doi.org/10.1007/s10955-015-1282-1

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