Universality of slow decorrelation in KPZ growth
Corwin, Ivan ; Ferrari, Patrik L. ; Péché, Sandrine
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 134-150 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Récemment des progrès notables ont été obtenus dans la description des fluctuations « spatiales » limites pour certains modèles de croissance dans la classe d'universalité de Kardar-Parisi-Zhang (KPZ). Grâce à un changement d'échelle temporelle approprié, les fluctuations en espace-temps pour ces modèles sont censées être non triviales. En dimension 1, on conjecture que l'exposant d'échelle dynamique est z = 3/2. Donc si on considère le changement d'échelle suivant: temps ∼tT, espace ∼t2/3X et fluctuations ∼t1/3, on s'attend à obtenir dans la limite t → ∞ un processus universel en espace-temps. Dans cet article on montre, sous des hypothèses assez générales, l'existence du phenomène de decorrélation lente dans des modèles de croissance, c'est-à-dire que les processus spatiaux limites pour des temps tT et tT + tν sont identiques, quelque soit ν < 1. On montre que ces hypothèses sont en particulier satisfaites par certains modèles de percolation de dernier passage, le modèle de croissance polynucléaire et le processus d'exclusion simple complètement/partiellement asymétrique. À l'aide de la decorrélation lente on peut ainsi étendre les résultats sur les fluctuations limites spatiales à des régions de l'espace-temps pour lequelles les fonctions de corrélations sont inconnues. L'approche utilisée dans cet article est basée sur des hypothèses minimales pour la decorrélation lente et donne une preuve simple, intuitive, qui s'applique à une large classe de modèles.

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.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/11-AIHP440
Classification:  60B20,  82C22,  60K35 
@article{AIHPB_2012__48_1_134_0,
     author = {Corwin, Ivan and Ferrari, Patrik L. and P\'ech\'e, Sandrine},
     title = {Universality of slow decorrelation in KPZ growth},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {48},
     year = {2012},
     pages = {134-150},
     doi = {10.1214/11-AIHP440},
     mrnumber = {2919201},
     zbl = {1247.82041},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_1_134_0}
}

Corwin, Ivan; Ferrari, Patrik L.; Péché, Sandrine. Universality of slow decorrelation in KPZ growth. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 134-150. doi : 10.1214/11-AIHP440. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_1_134_0/

[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. | MR 2796514 | Zbl 1222.82070

[2] J. Baik. Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 33 (2006) 205-235. | MR 2225691 | Zbl 1139.33006

[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. | MR 2165575 | Zbl 1086.15022

[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. | MR 2642384 | Zbl 1194.82067

[5] J. Baik and E. Rains. Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 (2000) 523-541. | MR 1788477 | Zbl 0976.82043

[6] G. Ben Arous and I. Corwin. Current fluctuations for TASEP: A proof of the Prähofer-Spohn conjecture. Ann. Probab. 39 (2011) 104-138. | MR 2778798 | Zbl 1208.82036

[7] L. Bertini and G. Giacomin. Stochastic burgers and KPZ equations from particle systems. Comm. Math. Phys. 183 (1997) 571-607. | MR 1462228 | Zbl 0874.60059

[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. | MR 2438811 | Zbl 1187.82084

[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. | Zbl 1136.82321

[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. | MR 2363389 | Zbl 1136.82028

[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. | MR 2444377 | Zbl 1214.82062

[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. | MR 2430639 | Zbl 1201.82030

[13] A. Borodin, P. L. Ferrari and T. Sasamoto. Two speed TASEP. J. Stat. Phys. 137 (2009) 936-977. | MR 2570757 | Zbl 1183.82062

[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. | MR 2415103 | Zbl 1145.82021

[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. | MR 2659279 | Zbl 1197.82078

[16] I. Corwin and J. Quastel. Universal distribution of fluctuations at the edge of the rarefaction fan. Available at arXiv:1006.1338.

[17] I. Corwin and J. Quastel. Renormalization fixed point of the KPZ universality class. Unpublished manuscript. Available at arXiv:1103.3422.

[18] L. C. Evans. Partial Differential Equations, 2nd edition. Grad. Stud. Math. 19. Amer. Math. Soc., Providence, RI, 2010. | Zbl 1194.35001

[19] P. L. Ferrari. Slow decorrelations in KPZ growth. J. Stat. Mech. Theory Exp. 2008 (2008) P07022.

[20] P. L. Ferrari. From interacting particle systems to random matrices. J. Stat. Mech. Theory Exp. 2010 (2010) P10016. | MR 2800495

[21] T. Imamura and T. Sasamoto. Fluctuations of the one-dimensional polynuclear growth model with external sources. Nuclear Phys. B 699 (2004) 503-544. | MR 2098552 | Zbl 1123.82352

[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. | MR 2344715 | Zbl 1136.82029

[23] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. | MR 1737991 | Zbl 0969.15008

[24] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277-329. | MR 2018275 | Zbl 1031.60084

[25] H. Kallabis and J. Krug. Persistence of Kardar-Parisi-Zhang interfaces. Europhys. Lett. 45 (1999) 20-25.

[26] M. Kardar, G. Parisi and Y. C. Zhang. Dynamic scaling of growth interfaces. Phys. Rev. Lett. 56 (1986) 889-892. | Zbl 1101.82329

[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] J. Krug and H. Spohn. Kinetic roughening of growning surfaces. In Solids Far From Equilibrium. C. Godrèche (Ed.). Cambridge Univ. Press, Cambridge, 1991.

[29] T. M. Liggett. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin, 1999. | MR 1717346 | Zbl 0949.60006

[30] T. M. Liggett. Interacting Particle Systems. Springer, Berlin, 2005. Reprint of 1985 original edition. | MR 2108619 | Zbl 0559.60078

[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. | MR 1901953 | Zbl 1015.60093

[32] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 (2002) 1071-1106. | MR 1933446 | Zbl 1025.82010

[33] H. Rost. Non-equilibrium behavior of a many particle process: Density profile and the local equilibrium. Z. Wahrsch. Verw. Gebiete 58 (1981) 41-53. | MR 635270 | Zbl 0451.60097

[34] T. Sasamoto and H. Spohn. Universality of the one-dimensional KPZ equation. Phys. Rev. Lett. 104 (2010) 230602.

[35] T. Seppälänen. Scaling for a one-dimensional directed polymer with constrained endpoints. Available at arXiv:0911.2446.

[36] T. Seppäläinen. Hydrodyanamic scaling, convex duality and asymptotic shapes of growth models. Markov Process. Related Fields 4 (1998) 1-26. | MR 1625007 | Zbl 0906.60082

[37] C. Tracy and H. Widom. Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1994) 151-174. | MR 1257246 | Zbl 0789.35152

[38] C. Tracy and H. Widom. Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 (2008) 815-844. | MR 2386729 | Zbl 1148.60080

[39] C. Tracy and H. Widom. A Fredholm determinant representation in ASEP. J. Stat. Phys. 132 (2008) 291-300. | MR 2415104 | Zbl 1144.82045

[40] C. Tracy and H. Widom. Asymptotics in ASEP with step initial condition. Comm. Math. Phys. 290 (2009) 129-154. | MR 2520510 | Zbl 1184.60036

[41] C. Tracy and H. Widom. Total current fluctuations in the asymmetric simple exclusion processes. J. Math. Phys. 50 (2009) 095204. | MR 2566884 | Zbl 1241.82051

[42] C. Tracy and H. Widom. On ASEP with step Bernoulli initial condition. J. Stat. Phys. 137 (2009) 825-838. | MR 2570751 | Zbl 1188.82043