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) no. 1, pp. 134-150.

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.

Classification : 60B20, 82C22, 60K35
Mots clés : asymmetric simple exclusion process, interacting particle systems, last passage percolation, directed polymers, KPZ

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