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.
Similar content being viewed by others
References
Kardar, M., Parisi, G., Zhang, Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889 (1986)
Krug, J., Meakin, P., Halpin-Healy, T.: Amplitude universality for driven interfaces and directed polymers in random media. Phys. Rev. A 45, 638 (1992)
Spohn, H.: KPZ scaling theory and the semi-discrete directed polymer. arXiv:1201.0645
Krug, J., Meakin, P.: Universal finite-size effects in the rate of growth processes. J. Phys. A 23, L987 (1990)
Huse, D.A., Henley, C.L., Fisher, D.S.: Forced Burgers equation, exact exponent, fluctuation-dissipation theorem. Phys. Rev. Lett. 55, 2924 (1985)
Dhar, D.: An exactly solved model for interface growth. Phase Transit. 9, 51 (1987)
Gwa, L.-H., Spohn, H.: Six-vertex model, roughened surfaces, and an asymmetric spin hamiltonian. Phys. Rev. Lett. 68, 725 (1992)
Gwa, L.-H., Spohn, H.: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844 (1992)
Kardar, M.: Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nucl. Phys. B 290, 582 (1987)
Kardar, M., Nelson, D.R.: Commensurate-incommensurate transitions with quenched disorder. Phys. Rev. Lett. 55, 1157 (1985)
Imbrie, J.Z., Spencer, T.: Diffusion of directed polymers in a random environment. J. Stat. Phys. 52, 609 (1988)
Doty, C.A., Kosterlitz, J.M.: Exact dynamical exponent at the Kardar-Parisi-Zhang roughening transition. Phys. Rev. Lett. 69, 1979 (1992)
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)
Frey, E., Täuber, U.C.: Two-loop renormalization-group analysis of the Burgers-Kardar-Parisi-Zhang equation. Phys. Rev. E 50, 1024 (1994)
van Beijeren, H., Kutner, R., Spohn, H.: Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, 2026 (1985)
Frey, E., Täuber, U., Hwa, T.: Mode-coupling and renormalization group results for the noisy Burgers equation. Phys. Rev. E 53, 4424 (1996)
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)
Kardar, M.: Roughening by impurities at finite temperatures. Phys. Rev. Lett. 55, 2923 (1985)
Kim, J.-M., Kosterlitz, M.: Growth in a restricted solid-on-solid model. Phys. Rev. Lett. 62, 2289 (1989)
Forrest, B., Tang, L.-H.: Surface roughening in a hypercube-stacking model. Phys. Rev. Lett. 64, 1405 (1990)
Amar, J.A., Family, F.: Numerical solution of a continuum equation for interface growth in 2+1 dimensions. Phys. Rev. A 41, 3399 (1990)
Tang, L.-H., Forrest, B., Wolf, D.E.: Kinetic surface roughening. II. Hypercube-stacking models. Phys. Rev. A 45, 7162 (1992)
Moser, K., Wolf, D.: Vectorized and parallel simulations of the KPZ equation in 3+1 dimensions. J. Phys. A 27, 4049 (1994)
Hwa, T., Frey, E.: Exact scaling function of interface growth dynamics. Phys. Rev. E 44, R7873 (1991)
Tang, L.-H.: Steady-state scaling function of the (1 + 1)-dimensional single-step model. J. Stat. Phys. 67, 819 (1992)
Kim, J.M., Moore, M.A., Bray, A.-J.: Zero-temperature directed polymers in a random potential. Phys. Rev. A 44, 2345 (1991)
Halpin-Healy, T.: Directed polymers in random media: probability distributions. Phys. Rev. A 44, R3415 (1991)
Meakin, P.: The growth of rough surfaces and interfaces. Phys. Rep. 235, 189 (1993)
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)
Krug, J.: Origins of scale invariance in growth processes. Adv. Phys. 46, 139 (1997)
Maunuksela, J.: Kinetic roughening in the slow combustion of paper. Phys. Rev. Lett. 79, 1515 (1997)
Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)
Schütz, G.M.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86, 1265 (1997)
Lässig, M., Kinzelbach, H.: Upper critical dimension of the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 78, 903 (1997)
Lässig, M.: Quantized scaling of growing surfaces. Phys. Rev. Lett. 80, 2366 (1998)
Ala-Nissila, T.: Upper critical dimension of the Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 80, 887 (1998)
Kim, J.M.: Phase transition of the KPZ equation in four substrate dimensions. Phys. Rev. Lett. 80, 888 (1998)
Castellano, C., Marsili, M., Pietronero, L.: Nonperturbative renormalization of the Kardar-Parisi-Zhang growth dynamics. Phys. Rev. Lett. 80, 3527 (1998)
Marinari, E., Pagnani, A., Parisi, G.: Critical exponents of the KPZ equation via multi-surface coding numerical simulations. J. Phys. A 33, 8181 (2000)
Halpin-Healy, T.: Disorder-induced roughening of diverse manifolds. Phys. Rev. A 42, 711 (1990)
Frey, E., Täuber, U., Janssen, H.K.: Scaling regimes and critical dimensions in the Kardar-Parisi-Zhang problem. Europhys. Lett. 47, 14 (1999)
Perlsman, E., Schwartz, M.: UCD of the KPZ equation. Phys. Rev. E 85, 050103 (2012)
Pagnani, A., Parisi, G.: Multisurface coding simulations of the RSOS model in four dimensions. Phys. Rev. E 87, 010102 (2013)
Kim, J.M., Kim, S.-W.: RSOS model with a proper restriction parameter N in 4+1 dimensions. Phys. Rev. E 88, 034102 (2013)
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)
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)
Canét, L., Chaté, H., Delamotte, B., Wschebor, N.: Nonperturbative renormalization group for the KPZ Equation. Phys. Rev. Lett. 104, 150601 (2010)
Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882 (2000)
Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437 (2000)
Tracy, C., Widom, H.: Level-spacing distributions and the Airy kernel. Commun. Math. Phys. 159, 151 (1994)
Tracy, C., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727 (1996)
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)
Aldous, D., Diaconis, P.: Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Am. Math. Soc. 36, 413 (1999)
Okounkov, A.: Random matrices and random permutations. Int. Math. Res. Not. 2000, 1043 (2000)
Odylyzko, A.M., Rains, E.M.: ATT Bell Labs Technical Report (1999)
Baer, R.M., Brock, P.: Natural sorting over permutation spaces. Math. Comput. 22, 385 (1968)
Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523 (2000)
Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071 (2002)
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)
Myllys, M.: Effect of a columnar defect on the shape of slow-combustion fronts. Phys. Rev. E 68, 051103 (2003)
Miettinen, L., Myllys, M., Merikoski, J., Timonen, J.: Experimental determination of KPZ height-fluctuation distributions. Eur. Phys. J. B 46, 55 (2005)
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)
Fogedby, H.: Localized growth modes, dynamic textures, and UCD for the KPZ equation in the weak-noise limit. Phys. Rev. Lett. 94, 195702 (2005)
Fogedby, H.: Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension. Phys. Rev. E 73, 031104 (2006)
Palasantzas, G.: Roughening aspects of room temperature vapor deposited oligomer thin films onto Si substrates. Surf. Sci 507, 357 (2002)
Halpin-Healy, T., Lin, Y.: Universal aspects of curved, flat and stationary-state KPZ statistics. Phys. Rev. E 89, 010103 (2014)
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)
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)
Kriecherbauer, T., Krug, J.: A pedestrian’s view on interacting particle systems, KPZ universality and random matrices. J. Phys. A. 43, 403001 (2010)
Corwin, I.: The KPZ equation and universality class. Random Matrices 1, 1130001 (2012)
Sasamoto, T., Spohn, H.: One-dimensional Kardar-Parisi-Zhang equation: an exact solution and its universality. Phys. Rev. Lett. 104, 230602 (2010)
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)
Calabrese, P., Le Doussal, P., Rosso, A.: Free-energy distribution of the directed polymer at high temperature. Europhys. Lett. 90, 20002 (2010)
Dotsenko, V.: Bethe ansatz derivation of the Tracy-Widom distribution for one-dimensional directed polymers. Europhys. Lett. 90, 20003 (2010)
Tracy, C.A., Widom, H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129 (2009)
Calabrese, P., Le Doussal, P.: Exact solution for the Kardar-Parisi-Zhang equation with flat initial conditions. Phys. Rev. Lett. 106, 250603 (2011)
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)
Gueudré, T., Le Doussal, P.: Directed polymer near a hard wall and KPZ equation in the half-space. EPL 100, 26006 (2012)
Imamura, T., Sasamoto, T.: Exact solution for the stationary Kardar-Parisi-Zhang equation. Phys. Rev. Lett. 108, 190603 (2012)
Borodin, A., Corwin, I., Ferrari, P.L., Vető, B.: Height fluctuations for the stationary KPZ equation. arXiv:1407.6977
Takeuchi, K.A., Sano, M.: Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals. Phys. Rev. Lett. 104, 230601 (2010)
Takeuchi, K.A., Sano, M., Sasamoto, T., Spohn, H.: Growing interfaces uncover universal fluctuations behind scale invariance. Sci. Rep. (Nature) 1, 34 (2011)
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)
Takeuchi, K.A.: Crossover from growing to stationary interfaces in the Kardar-Parisi-Zhang class. Phys. Rev. Lett. 110, 210604 (2013)
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
Halpin-Healy, T.: (2+1)-Dimensional directed polymer in a random medium: scaling phenomena and universal distributions. Phys. Rev. Lett. 109, 170602 (2012)
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)
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)
Prähofer, M.: Stochastic Surface Growth. Ludwig-Maximilians-Universitait, München (2003)
Halpin-Healy, T., Palasantzas, G.: Universal correlators & distributions as experimental signatures of (2 + 1)-dimensional Kardar-Parisi-Zhang growth. EPL 105, 50001 (2014)
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)
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)
Hairer, M.: Solving the KPZ equation. Ann. Math. 178, 559 (2013)
Seppäläinen, T.: Scaling for a one-dimensional directed polymer with boundary conditions. Ann. Probab. 40, 19 (2012)
O‘Connell, N.: Directed polymers and the quantum Toda lattice. Ann. Probab. 40, 437 (2012)
Borodin, A., Corwin, I.: Macdonald processes. Probab. Theory Rel. Fields 158, 225 (2014)
Borodin, A., Corwin, I., Sasamoto, T.: From duality to determinants for q-TASEP and ASEP. Ann. Probab. 42, 2314 (2014)
Calabrese, P., Kormos, M., Le Doussal, P.: From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation. EPL 107, 10011 (2014)
Barraquand, G., Corwin, I.: The q-Hahn asymmetric exclusion process. arXiv:1501.03445
Corwin, I., Petrov, L.: Stochastic higher spin vertex models on the line. arXiv:1502.07374
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)
Johansson, K.: Two-time distribution in Brownian directed percolation. arXiv:1502.00941
Maritan, A., Toigo, F., Koplik, J., Banavar, J.R.: Dynamics of growing interfaces. Phys. Rev. Lett. 69, 3193 (1992)
Batchelor, M.T., Henry, B.I., Watt, S.D.: Continuum model for radial interface growth. Physica A 260, 11 (1998)
Singha, S.B.: Persistence of surface fluctuations in radially growing surfaces. J. Stat. Mech. 2005, P08006 (2005)
Masoudi, A.A.: Statistical analysis of radial interface growth. JSTAT 2012, L02001 (2012)
Rodriguez-Laguna, J., Santalla, S. N., Cuerno, R.: Intrinsic geometry approach to surface kinetic roughening. J. Stat. Mech. P05032 (2011)
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)
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)
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)
Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice Eden model. J. Stat. Mech. 2012, P05007 (2012)
Alves, S.G., Oliveira, T.J., Ferreira, S.C.: Non-universal parameters, corrections and universality in Kardar-Parisi-Zhang growth. JSTAT 2013, P05007 (2013)
Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79, 871915 (2010)
Rácz, Z., Plischke, M.: Width distribution for (2+1)-dimensional growth and deposition processes. Phys. Rev. E 50, 3530 (1994)
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)
Edwards, S.F., Wilkinson, D.R.: The surface statistics of a granular aggregate. Proc. R. Soc. London Ser. A 381, 17 (1982)
Antal, T., Droz, M., Győrgyi, G., Rácz, Z.: Roughness distributions for 1/f\(^\alpha \) signals. Phys. Rev. E 65, 046140 (2002)
Santachiara, R., Rosso, A., Krauth, W.: Universal width distribution in non-Markovian gaussian processes. JSTAT 2007, P02009 (2007)
Raychaudhauri, S., Cranston, M., Przybyla, C., Shapir, Y.: Maximal height scaling of kinetically growing surfaces. Phys. Rev. Lett. 87, 136101 (2001)
Majumdar, S.N., Comtet, A.: Exact maximal height distribution of fluctuating interfaces. Phys. Rev. Lett. 92, 225501 (2004)
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)
Schehr, G., Majumdar, S.: Universal asymptotic statistics of maximal relative height in one-dimensional solid-on-solid models. Phys. Rev. E 73, 056103 (2006)
Győrgyi, G., Moloney, N.R., Ozogány, K., Rácz, Z.: Maximal height statistics for 1/\(f^a\) signals. Phys. Rev. E 75, 021123 (2007)
Rambeau, J., Bustingorry, S., Kolton, A.B., Schehr, G.: MRH of elastic interfaces in random media. Phys. Rev. E 84, 041131 (2011)
Lee, D.-S.: Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces. Phys. Rev. Lett. 95, 150601 (2005)
Kelling, J., Ódor, G.: Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards. Phys. Rev. E 84, 061150 (2011)
Derrida, B., Griffiths, R.: Directed polymers on disordered hierarchical lattices. Europhys. Lett. 8, 111 (1989)
Cook, J., Derrida, B.: Polymers on disordered hierarchical lattices: a nonlinear combination of random variables. J. Stat. Phys. 57, 89 (1989)
Halpin-Healy, T.: Comment–growth in a restricted solid-on-solid model. Phys. Rev. Lett. 63, 917 (1989)
Derrida, B.: Directed polymers in a random medium. Physica A 163, 71 (1990)
Roux, S., Hansen, A., da Silva, L., Lucena, L., Pandey, R.: Minimal path on the hierarchical diamond lattice. J. Stat. Phys. 65, 183 (1991)
Monthus, C., Garel, T.: Disorder-dominated phases of random systems: relations between the tail exponents and scaling exponents. J. Stat. Mech. 2008, P01008 (2008)
Gumbel, E.J.: Statistics of Extremes. Columbia University Press. New York (1958). Republished by Dover, New York (2004)
Ferrari, P.L., Frings, R.: Finite-time corrections in KPZ growth models. J. Stat. Phys. 144, 1123 (2011)
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)
Derrida, B., Golinelli, O.: Thermal properties of directed polymers in random media. Phys. Rev. A 41, 4160 (1990)
Monthus, C., Garel, T.: Numerical study of the directed polymer in a 3+1 dimensional random medium. Eur. Phys. J. B 53, 39 (2006)
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)
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)
Derrida, B., Spohn, H.: Polymers on disordered trees, spin glasses, and traveling waves. J. Stat. Phys. 51, 817 (1988)
Derrida, B., Appert, C.: Universal large deviation function of the KPZ equation in one dimension. J. Stat. Phys. 94, 1 (1999)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-015-1282-1