Abstract
The two-time distribution gives the limiting joint distribution of the heights at two different times of a local 1D random growth model in the curved geometry. This distribution has been computed in a specific model but is expected to be universal in the KPZ universality class. Its marginals are the GUE Tracy-Widom distribution. In this paper we study two limits of the two-time distribution. The first, is the limit of long time separation when the quotient of the two times goes to infinity, and the second, is the short time limit when the quotient goes to zero.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baik, J., Liu, Z.: Multi-point distribution of periodic TASEP. J. Amer. Math. Soc. 32(3), 609–674 (2019)
Basu, R., Ganguly, S.: Time correlation in last-passage percolation, arXiv:1807.09260
Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1(1), 1130001 (2012)
Corwin, I., Hammond, A.: Brownian Gibbs property for Airy line ensembles. Invent. Math. 195(2), 441–508 (2014)
De Nardis, J., Le Doussal, P.: Tail of the two-time height distribution for KPZ growth in one dimension, J. Stat. Mech. Theory Exp. (5):053212 (2017)
De Nardis, J., Le Doussal, P., Takeuchi, K. A.: Memory and universality in interface growth. Phys. Rev. Lett. 118, 125701 (2017)
De Nardis, J., Le Doussal, P.: Two-time height distribution for 1D KPZ growth: the recent exact result and its tail via replica. J. Stat. Mech, 093203 (2018)
Le Doussal, P.: Maximum of an Airy process plus Brownian motion and memory in KPZ growth. Phys. Rev. E 96, 060101 (2017)
Dauvergne, D., Ortmann, J., Virág, B.: The directed landscape, arXiv:1812.00309
Ferrari, P.L., Spohn, H.: Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265(1), 1–44 (2006)
Ferrari, P.L., Spohn, H.: On time correlations for KPZ growth in one dimension SIGMA Symmetry Integrability Geom. Methods Appl. 12, Paper no. 074, 23 (2016)
Ferrari, P.L., Occelli, A.: Time-time covariance for last passage percolation with generic initial profile. Math. Phys. Anal. Geom. 22(1), Art. 1, 33 (2019)
Gesztesy, F., Latushkin, Y., Zumbrun, K.: Derivatives of (modified) Fredholm determinants and stability of standing and traveling. J. Math. Pures Appl. (9) 90(2), 160–200 (2008)
Hägg, J.: Local Gaussian fluctuations in the Airy and discrete PNG processes. Ann. Probab. 36(3), 1059–1092 (2008)
Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Johansson, K.: Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242, 277–329 (2003)
Johansson, K.: Two time distribution function in Brownian directed percolation. Comm. Math. Phys. 351, 441–492 (2017)
Johansson, K.: The two-time distribution in geometric last-passage percolation. Probab. Th. Rel. Fields. 175, 849–895 (2019)
Johansson, K., Rahman, M.: Multi-time distribution in discrete polynuclear growth, arXiv:1906.01053, to appear in Commun. Pure and Applied Math.
Liu, Z.: Multi-time distribution of TASEP, arXiv:1907.09876
Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point, arXiv:1701.00018
Quastel, J.: Introduction to KPZ. Current Developments in Mathematics, pp 125–194. Int. Press, Somerville (2012)
Pimentel, L.: Ergodicity of the KPZ Fixed Point, arXiv:1708.06006
Takeuchi, K.A.: Statistics of circular interface fluctuations in an off-lattice Eden model. J. Stat. Mech., P05007 (2012)
Takeuchi, K.A., Sano, M.: Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence. J. Stat. Phys. 147, 853–890 (2012)
Acknowledgements
I thank Pierre Le Doussal and Jacopo de Nardis for helpful discussions and correspondence.
Funding
Open access funding provided by Royal Institute of Technology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by the grant KAW 2015.0270 from the Knut and Alice Wallenberg Foundation and grant 2015-04872 from the Swedish Science Research Council (VR)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Johansson, K. Long and Short Time Asymptotics of the Two-Time Distribution in Local Random Growth. Math Phys Anal Geom 23, 43 (2020). https://doi.org/10.1007/s11040-020-09367-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-020-09367-x