Universality of slow decorrelation in KPZ growth

Ivan Corwin; Patrik L. Ferrari; Sandrine Péché

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 1, page 134-150
  • ISSN: 0246-0203

Abstract

top
There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z = 3/2, that means one should find a universal space–time limiting process under the scaling of time as tT, space like t2/3X and fluctuations like t1/3 as t → ∞. In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times tT and tT + tν are identical, for any ν < 1. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally/partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to space–time regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applies to a wide variety of models.

How to cite

top

Corwin, Ivan, Ferrari, Patrik L., and Péché, Sandrine. "Universality of slow decorrelation in KPZ growth." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 134-150. <http://eudml.org/doc/272068>.

@article{Corwin2012,
abstract = {There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z = 3/2, that means one should find a universal space–time limiting process under the scaling of time as tT, space like t2/3X and fluctuations like t1/3 as t → ∞. In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times tT and tT + tν are identical, for any ν &lt; 1. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally/partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to space–time regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applies to a wide variety of models.},
author = {Corwin, Ivan, Ferrari, Patrik L., Péché, Sandrine},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {asymmetric simple exclusion process; interacting particle systems; last passage percolation; directed polymers; KPZ},
language = {eng},
number = {1},
pages = {134-150},
publisher = {Gauthier-Villars},
title = {Universality of slow decorrelation in KPZ growth},
url = {http://eudml.org/doc/272068},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Corwin, Ivan
AU - Ferrari, Patrik L.
AU - Péché, Sandrine
TI - Universality of slow decorrelation in KPZ growth
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 1
SP - 134
EP - 150
AB - There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z = 3/2, that means one should find a universal space–time limiting process under the scaling of time as tT, space like t2/3X and fluctuations like t1/3 as t → ∞. In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times tT and tT + tν are identical, for any ν &lt; 1. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally/partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to space–time regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applies to a wide variety of models.
LA - eng
KW - asymmetric simple exclusion process; interacting particle systems; last passage percolation; directed polymers; KPZ
UR - http://eudml.org/doc/272068
ER -

References

top
  1. [1] G. Amir, I. Corwin and J. Quastel. Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Comm. Pure Appl. Math.64 (2011) 466–537. Zbl1222.82070MR2796514
  2. [2] J. Baik. Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J.33 (2006) 205–235. Zbl1139.33006MR2225691
  3. [3] J. Baik, G. Ben Arous and S. Péché. Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab.33 (2005) 1643–1697. Zbl1086.15022MR2165575
  4. [4] J. Baik, P. L. Ferrari and S. Péché. Limit process of stationary TASEP near the characteristic line. Comm. Pure Appl. Math.63 (2010) 1017–1070. Zbl1194.82067MR2642384
  5. [5] J. Baik and E. Rains. Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys.100 (2000) 523–541. Zbl0976.82043MR1788477
  6. [6] G. Ben Arous and I. Corwin. Current fluctuations for TASEP: A proof of the Prähofer–Spohn conjecture. Ann. Probab.39 (2011) 104–138. Zbl1208.82036MR2778798
  7. [7] L. Bertini and G. Giacomin. Stochastic burgers and KPZ equations from particle systems. Comm. Math. Phys.183 (1997) 571–607. Zbl0874.60059MR1462228
  8. [8] A. Borodin and P. L. Ferrari. Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab.13 (2008) 1380–1418. Zbl1187.82084MR2438811
  9. [9] A. Borodin, P. L. Ferrari and M. Prähofer. Fluctuations in the discrete TASEP with periodic initial configurations and the Airy1process. Int. Math. Res. Papers 1 (2007) rpm002. Zbl1136.82321
  10. [10] A. Borodin, P. L. Ferrari, M. Prähofer and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys.129 (2007) 1055–1080. Zbl1136.82028MR2363389
  11. [11] A. Borodin, P. L. Ferrari and T. Sasamoto. Transition between Airy1 and Airy2 processes and TASEP fluctuations. Comm. Pure Appl. Math.61 (2008) 1603–1629. Zbl1214.82062MR2444377
  12. [12] A. Borodin, P. L. Ferrari and T. Sasamoto. Large time asymptotics of growth models on space-like paths II: PNG and parallel TASEP. Comm. Math. Phys.283 (2008) 417–449. Zbl1201.82030MR2430639
  13. [13] A. Borodin, P. L. Ferrari and T. Sasamoto. Two speed TASEP. J. Stat. Phys.137 (2009) 936–977. Zbl1183.82062MR2570757
  14. [14] A. Borodin and S. Péché. Airy kernel with two sets of parameters in directed percolation and random matrix theory. J. Stat. Phys.132 (2008) 275–290. Zbl1145.82021MR2415103
  15. [15] I. Corwin, P. L. Ferrari and S. Péché. Limit processes for TASEP with shocks and rarefaction fans. J. Stat. Phys.140 (2010) 232–267. Zbl1197.82078MR2659279
  16. [16] I. Corwin and J. Quastel. Universal distribution of fluctuations at the edge of the rarefaction fan. Available at arXiv:1006.1338. 
  17. [17] I. Corwin and J. Quastel. Renormalization fixed point of the KPZ universality class. Unpublished manuscript. Available at arXiv:1103.3422. 
  18. [18] L. C. Evans. Partial Differential Equations, 2nd edition. Grad. Stud. Math. 19. Amer. Math. Soc., Providence, RI, 2010. Zbl1194.35001
  19. [19] P. L. Ferrari. Slow decorrelations in KPZ growth. J. Stat. Mech. Theory Exp. 2008 (2008) P07022. 
  20. [20] P. L. Ferrari. From interacting particle systems to random matrices. J. Stat. Mech. Theory Exp. 2010 (2010) P10016. MR2800495
  21. [21] T. Imamura and T. Sasamoto. Fluctuations of the one-dimensional polynuclear growth model with external sources. Nuclear Phys. B699 (2004) 503–544. Zbl1123.82352MR2098552
  22. [22] T. Imamura and T. Sasamoto. Dynamical properties of a tagged particle in the totally asymmetric simple exclusion process with the step-type initial condition. J. Stat. Phys.128 (2007) 799–846. Zbl1136.82029MR2344715
  23. [23] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys.209 (2000) 437–476. Zbl0969.15008MR1737991
  24. [24] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys.242 (2003) 277–329. Zbl1031.60084MR2018275
  25. [25] H. Kallabis and J. Krug. Persistence of Kardar–Parisi–Zhang interfaces. Europhys. Lett.45 (1999) 20–25. 
  26. [26] M. Kardar, G. Parisi and Y. C. Zhang. Dynamic scaling of growth interfaces. Phys. Rev. Lett.56 (1986) 889–892. Zbl1101.82329
  27. [27] J. Krug, H. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray and C. Sire. Persistence exponents for fluctuating interfaces. Phys. Rev. E 56 (1997) 2702. 
  28. [28] J. Krug and H. Spohn. Kinetic roughening of growning surfaces. In Solids Far From Equilibrium. C. Godrèche (Ed.). Cambridge Univ. Press, Cambridge, 1991. 
  29. [29] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999. Zbl0949.60006MR1717346
  30. [30] T. M. Liggett. Interacting Particle Systems. Springer, Berlin, 2005. Reprint of 1985 original edition. Zbl0559.60078MR2108619
  31. [31] M. Prähofer and H. Spohn. Current fluctuations for the totally asymmetric simple exclusion process. In In and Out of Equilibrium 185–204. Progr. Probab. 51. Birkhäuser, Boston, MA, 2002. Zbl1015.60093MR1901953
  32. [32] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys.108 (2002) 1071–1106. Zbl1025.82010MR1933446
  33. [33] H. Rost. Non-equilibrium behavior of a many particle process: Density profile and the local equilibrium. Z. Wahrsch. Verw. Gebiete58 (1981) 41–53. Zbl0451.60097MR635270
  34. [34] T. Sasamoto and H. Spohn. Universality of the one-dimensional KPZ equation. Phys. Rev. Lett. 104 (2010) 230602. Zbl1204.35137
  35. [35] T. Seppälänen. Scaling for a one-dimensional directed polymer with constrained endpoints. Available at arXiv:0911.2446. 
  36. [36] T. Seppäläinen. Hydrodyanamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields4 (1998) 1–26. Zbl0906.60082MR1625007
  37. [37] C. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys.159 (1994) 151–174. Zbl0789.35152MR1257246
  38. [38] C. Tracy and H. Widom. Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys.279 (2008) 815–844. Zbl1148.60080MR2386729
  39. [39] C. Tracy and H. Widom. A Fredholm determinant representation in ASEP. J. Stat. Phys.132 (2008) 291–300. Zbl1144.82045MR2415104
  40. [40] C. Tracy and H. Widom. Asymptotics in ASEP with step initial condition. Comm. Math. Phys.290 (2009) 129–154. Zbl1184.60036MR2520510
  41. [41] C. Tracy and H. Widom. Total current fluctuations in the asymmetric simple exclusion processes. J. Math. Phys. 50 (2009) 095204. Zbl1241.82051MR2566884
  42. [42] C. Tracy and H. Widom. On ASEP with step Bernoulli initial condition. J. Stat. Phys.137 (2009) 825–838. Zbl1188.82043MR2570751

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.