(2 + 1)-DIMENSIONAL INTERFACE DYNAMICS: MIXING TIME, HYDRODYNAMIC LIMIT AND ANISOTROPIC KPZ GROWTH

Abstract

Proceedings of the International Congress of Mathematicians (ICM 2018), pp. 2733-2758 (2019) No Access(2 + 1)-DIMENSIONAL INTERFACE DYNAMICS: MIXING TIME, HYDRODYNAMIC LIMIT AND ANISOTROPIC KPZ GROWTHFABIO TONINELLIFABIO TONINELLICNRS AND INSTITUT CAMILLE JORDAN, UNIVERSITÉ LYON 1, 43 BD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE, FRANCEhttps://doi.org/10.1142/9789813272880_0158Cited by:1 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: Stochastic interface dynamics serve as mathematical models for diverse time-dependent physical phenomena: the evolution of boundaries between thermodynamic phases, crystal growth, random deposition… Interesting limits arise at large space-time scales: after suitable rescaling, the randomly evolving interface converges to the solution of a deterministic PDE (hydrodynamic limit) and the fluctuation process to a (in general non-Gaussian) limit process. In contrast with the case of (1 + 1)-dimensional models, there are very few mathematical results in dimension (d + 1); d ≥ 2. As far as growth models are concerned, the (2 + 1)-dimensional case is particularly interesting: Dietrich Wolf in 1991 conjectured the existence of two different universality classes (called KPZ and Anisotropic KPZ), with different scaling exponents. Here, we review recent mathematical results on (both reversible and irreversible) dynamics of some (2 + 1)-dimensional discrete interfaces, mostly defined through a mapping to two-dimensional dimer models. In particular, in the irreversible case, we discuss mathematical support and remaining open problems concerning Wolf's conjecture on the relation between the Hessian of the growth velocity on one side, and the universality class of the model on the other. Keywords: Interacting particle systemsDimer modelInterface growthHydrodynamic limitAnisotropic KPZ equationStochastic Heat EquationJELC: primary 82C20, secondary 60J10, secondary 60K35, secondary 82C24 FiguresReferencesRelatedDetailsCited By 1The two-dimensional KPZ equation in the entire subcritical regimeFrancesco Caravenna, Rongfeng Sun and Nikos Zygouras1 May 2020 | The Annals of Probability, Vol. 48, No. 3 Proceedings of the International Congress of Mathematicians (ICM 2018)Metrics History KeywordsInteracting particle systemsDimer modelInterface growthHydrodynamic limitAnisotropic KPZ equationStochastic Heat EquationPDF download

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