Abstract
The continuum Kardar-Parisi-Zhang equation in one dimension is lattice discretized in such a way that the drift part is divergence free. This allows to determine explicitly the stationary measures. We map the lattice KPZ equation to a bosonic field theory which has a cubic anti-hermitian nonlinearity. Thereby it is established that the stationary two-point function spreads superdiffusively.
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Johansson, K.: Random matrices and determinantal processes. In: Bovier, A., et al. (eds.) Math. Stat. Phys., Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005, pp. 1–56. Elsevier Science, Amsterdam (2006)
Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals. Physica A 369, 71–99 (2006)
Sasamoto, T.: Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques. J. Stat. Mech. P07007 (2007)
Prähofer, M., Spohn, H.: Exact scaling function for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2004)
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)
Kardar, M., Parisi, G., Zhang, Y.Z.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Katzav, E., Schwartz, M.: Numerical evidence for stretched exponential relaxations in the Kardar-Parisi-Zhang equation. Phys. Rev. E 69, 052603 (2004)
Bertini, L., Giacomin, G.: Stochastic Burgers and KPZ equations from particle systems. Commun. Math. Phys. 183, 571–607 (1997)
Balazs, M., Quastel, J., Seppalainen, T.: Scaling exponent for the Cole-Hopf solution of KPZ/stochastic Burgers. arXiv:0909.4816
Bernardin, C.: Superdiffusivity of asymmetric energy model in dimension one and two. J. Math. Phys. 49, 103301 (2008)
Miyao, T.: Private communication (2009)
Gotoh, T., Kraichnan, R.H.: Burgers turbulence with large scale forcing. Phys. Fluids A 10, 2859–2866 (1998)
E, W., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. 151, 877–960 (2000)
Krug, J., Spohn, H.: Kinetic roughening of growing surfaces. In: Godrèche, C. (ed.) Solids Far from Equilibrium, pp. 412–525. Cambridge University Press, Cambridge (1991)
Lam, C.-H., Shin, F.G.: Improved discretization of the Kardar-Parisi-Zhang equation. Phys. Rev. E 58, 5592–5595 (1998)
Schweber, S.: An Introduction to Relativistic Quantum Field Theory. Harper & Row, New York (1966)
Landim, C., Quastel, J., Salmhofer, M., Yau, H.-T.: Superdiffusivity of asymmetric exclusion process in dimensions one and two. Commun. Math. Phys. 244, 455–481 (2004)
Baik, J., Ferrari, P., Péché, S.: Limit process for the TASEP near a characteristic line. arXiv:0907.0226
Ziman, J.M.: Electrons and Phonons: The Theory of Transport Phenomena in Solids. Oxford University Press, New York (1987)
Frachebourg, I., Martin, Ph.A.: Exact statistical properties of the Burgers equation. J. Fluid Mech. 417, 323–349 (2000)
Chabanol, M.-L., Duchon, J.: Markovian solutions of inviscid Burgers equation. J. Stat. Phys. 114, 525–534 (2004)
Burgers, J.M.: The Nonlinear Diffusion Equation. Reidel, Dordrecht (1974)
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On the occasion of the 100th Statistical Mechanics Meeting, December 2008 at Rutgers University, we dedicate this article to Joel Lebowitz as mentor and friend for so many years.
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Sasamoto, T., Spohn, H. Superdiffusivity of the 1D Lattice Kardar-Parisi-Zhang Equation. J Stat Phys 137, 917–935 (2009). https://doi.org/10.1007/s10955-009-9831-0
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DOI: https://doi.org/10.1007/s10955-009-9831-0