Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 074, 23 pages      arXiv:1602.00486      https://doi.org/10.3842/SIGMA.2016.074
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

On Time Correlations for KPZ Growth in One Dimension

Patrik L. Ferrari a and Herbert Spohn b
a) Institute for Applied Mathematics, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany
b) Zentrum Mathematik, TU München, Boltzmannstrasse 3, D-85747 Garching, Germany

Received March 17, 2016, in final form July 21, 2016; Published online July 26, 2016

Abstract
Time correlations for KPZ growth in $1+1$ dimensions are reconsidered. We discuss flat, curved, and stationary initial conditions and are interested in the covariance of the height as a function of time at a fixed point on the substrate. In each case the power laws of the covariance for short and long times are obtained. They are derived from a variational problem involving two independent Airy processes. For stationary initial conditions we derive an exact formula for the stationary covariance with two approaches: (1) the variational problem and (2) deriving the covariance of the time-integrated current at the origin for the corresponding driven lattice gas. In the stationary case we also derive the large time behavior for the covariance of the height gradients.

Key words: KPZ universality, space-time correlations, interacting particles, last passage percolation.

pdf (624 kb)   tex (180 kb)

References

  1. Baik J., Ferrari P.L., Péché S., Limit process of stationary TASEP near the characteristic line, Comm. Pure Appl. Math. 63 (2010), 1017-1070, arXiv:0907.0226.
  2. Baik J., Liechty K., Schehr G., On the joint distribution of the maximum and its position of the ${\rm Airy}_2$ process minus a parabola, J. Math. Phys. 53 (2012), 083303, 13 pages, arXiv:1205.3665.
  3. Baik J., Rains E.M., Limiting distributions for a polynuclear growth model with external sources, J. Stat. Phys. 100 (2000), 523-541, math.PR/0003130.
  4. Baik J., Rains E.M., The asymptotics of monotone subsequences of involutions, Duke Math. J. 109 (2001), 205-281, math.CO/9905084.
  5. Basu R., Sidoravicius V., Sly A., Last passage percolation with a defect line and the solution of the slow bond problem, arXiv:1408.3464.
  6. Bornemann F., Ferrari P.L., Prähofer M., The ${\rm Airy}_1$ process is not the limit of the largest eigenvalue in GOE matrix diffusion, J. Stat. Phys. 133 (2008), 405-415, arXiv:0806.3410.
  7. Borodin A., Corwin I., Ferrari P., Vető B., Height fluctuations for the stationary KPZ equation, Math. Phys. Anal. Geom. 18 (2015), Art. 20, 95 pages, arXiv:1407.6977.
  8. Borodin A., Ferrari P.L., Large time asymptotics of growth models on space-like paths. I. PushASEP, Electron. J. Probab. 13 (2008), no. 50, 1380-1418, arXiv:0707.2813.
  9. Borodin A., Ferrari P.L., Prähofer M., Sasamoto T., Fluctuation properties of the TASEP with periodic initial configuration, J. Stat. Phys. 129 (2007), 1055-1080, math-ph/0608056.
  10. Borodin A., Gorin V., Lectures on integrable probability, arXiv:1212.3351.
  11. Burke P.J., The output of a queuing system, Operations Res. 4 (1956), 699-704.
  12. Chang C.-C., Equilibrium fluctuations of gradient reversible particle systems, Probab. Theory Related Fields 100 (1994), 269-283.
  13. Chang C.-C., Equilibrium fluctuations of nongradient reversible particle systems, in Nonlinear Stochastic PDEs (Minneapolis, MN, 1994), IMA Vol. Math. Appl., Vol. 77, Springer, New York, 1996, 41-51.
  14. Corwin I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76 pages, arXiv:1106.1596.
  15. Corwin I., Ferrari P.L., Péché S., Limit processes for TASEP with shocks and rarefaction fans, J. Stat. Phys. 140 (2010), 232-267, arXiv:1002.3476.
  16. Corwin I., Ferrari P.L., Péché S., Universality of slow decorrelation in KPZ growth, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), 134-150, arXiv:1001.5345.
  17. Corwin I., Hammond A., Brownian Gibbs property for Airy line ensembles, Invent. Math. 195 (2014), 441-508, arXiv:1108.2291.
  18. Corwin I., Hammond A., Private communication, 2016.
  19. De Masi A., Ferrari P.A., Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process, J. Stat. Phys. 107 (2002), 677-683, math.PR/0103233.
  20. Dotsenko V., Two-time free energy distribution function in $(1+1)$ directed polymers, J. Stat. Mech. Theory Exp. 2013 (2013), P06017, 23 pages, arXiv:1304.0626.
  21. Dotsenko V., On two-time distribution functions in $(1+1)$ random directed polymers, J. Phys. A: Math. Theor. 49 (2016), 27LT01, 8 pages, arXiv:1603.08945.
  22. Ferrari P.L., Slow decorrelations in Kardar-Parisi-Zhang growth, J. Stat. Mech. Theory Exp. 2008 (2008), P07022, 18 pages, arXiv:0806.1350.
  23. Ferrari P.L., The universal ${\rm Airy}_1$ and ${\rm Airy}_2$ processes in the totally asymmetric simple exclusion process, in Integrable Systems and Random Matrices: in Honor of Percy Deift, Contemp. Math., Vol. 458, Editors J. Baik, T. Kriecherbauer, L.-C. Li, K.D.T.-R. McLaughlin, C. Tomei, Amer. Math. Soc., Providence, RI, 2008, 321-332, math-ph/0701021.
  24. Ferrari P.L., Frings R., Finite time corrections in KPZ growth models, J. Stat. Phys. 144 (2011), 1123-1150, arXiv:1104.2129.
  25. Ferrari P.L., Spohn H., Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process, Comm. Math. Phys. 265 (2006), 1-44, math-ph/0504041.
  26. Ferrari P.L., Spohn H., Random growth models, in The Oxford Handbook of Random Matrix Theory, Editors G. Akemann, J. Baik, P. Di Francesco, Oxford University Press, Oxford, 2011, 782-801, arXiv:1003.0881.
  27. Ferrari P.L., Spohn H., Weiss T., Brownian motions with one-sided collisions: the stationary case, Electron. J. Probab. 20 (2015), no. 69, 41 pages, arXiv:1502.01468.
  28. Hägg J., Local Gaussian fluctuations in the Airy and discrete PNG processes, Ann. Probab. 36 (2008), 1059-1092, math.PR/0701880.
  29. Imamura T., Sasamoto T., Polynuclear growth model with external source and random matrix model with deterministic source, Phys. Rev. E 71 (2005), 041606, 12 pages, math-ph/0411057.
  30. Johansson K., Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476, math.CO/9903134.
  31. Johansson K., Transversal fluctuations for increasing subsequences on the plane, Probab. Theory Related Fields 116 (2000), 445-456, math.PR/9910146.
  32. Johansson K., Discrete polynuclear growth and determinantal processes, Comm. Math. Phys. 242 (2003), 277-329, math.PR/0206208.
  33. Johansson K., Two time distribution in Brownian directed percolation, Comm. Math. Phys., to appear, arXiv:1502.00941.
  34. Kallabis H., Krug J., Persistence of Kardar-Parisi-Zhang interfaces, Europhys. Lett. 45 (1999), 20-25, cond-mat/9809241.
  35. Kardar M., Parisi G., Zhang Y.-C., Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889-892.
  36. Krug J., Meakin P., Halpin-Healy T., Amplitude universality for driven interfaces and directed polymers in random media, Phys. Rev. A 45 (1992), 638-653.
  37. Moreno Flores G., Quastel J., Remenik D., Endpoint distribution of directed polymers in $1+1$ dimensions, Comm. Math. Phys. 317 (2013), 363-380, arXiv:1106.2716.
  38. Peligrad M., Sethuraman S., On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion, ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 245-255, arXiv:0711.0017.
  39. Prähofer M., Spohn H., Scale invariance of the PNG droplet and the Airy process, J. Stat. Phys. 108 (2002), 1071-1106, math.PR/0105240.
  40. Prähofer M., Spohn H., Exact scaling functions for one-dimensional stationary KPZ growth, J. Stat. Phys. 115 (2004), 255-279, cond-mat/0212519.
  41. Quastel J., Introduction to KPZ, in Current Developments in Mathematics, Int. Press, Somerville, MA, 2012, 125-194.
  42. Quastel J., Remenik D., Local behavior and hitting probabilities of the $\text{Airy}_1$ process, Probab. Theory Related Fields 157 (2013), 605-634, arXiv:1201.4709.
  43. Quastel J., Remenik D., Airy processes and variational problems, in Topics in Percolative and Disordered Systems, Springer Proc. Math. Stat., Vol. 69, Springer, New York, 2014, 121-171, arXiv:1301.0750.
  44. Quastel J., Spohn H., The one-dimensional KPZ equation and its universality class, J. Stat. Phys. 160 (2015), 965-984, arXiv:1503.06185.
  45. Sasamoto T., Spatial correlations of the 1D KPZ surface on a flat substrate, J. Phys. A: Math. Gen. 38 (2005), L549-L556, cond-mat/0504417.
  46. Schehr G., Extremes of $N$ vicious walkers for large $N$: application to the directed polymer and KPZ interfaces, J. Stat. Phys. 149 (2012), 385-410, arXiv:1203.1658.
  47. Singha S.B., Persistence of surface fluctuations in radially growing surface, J. Stat. Mech. Theory Exp. 2005 (2005), P08006, 17 pages.
  48. Takeuchi K.A., Statistics of circular interface fluctuations in an off-lattice Eden model, J. Stat. Mech. Theory Exp. 2012 (2012), P05007, 17 pages, arXiv:1203.2483.
  49. Takeuchi K.A., Crossover from growing to stationary interfaces in the Kardar-Parisi-Zhang class, Phys. Rev. Lett. 110 (2013), 210604, 5 pages, arXiv:1301.5081.
  50. 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 (2012), 853-890, arXiv:1203.2530.

Previous article  Next article   Contents of Volume 12 (2016)