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

摘要

Stochastic interface dynamics serve as mathematical models for diversetime-dependent physical phenomena: the evolution of boundaries betweenthermodynamic phases, crystal growth, random deposition... Interesting limitsarise at large space-time scales: after suitable rescaling, the randomlyevolving 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, thereare very few mathematical results in dimension $(d+1), dge2$. As far as growthmodels are concerned, the $(2+1)$-dimensional case is particularly interesting:D. Wolf conjectured the existence of two different universality classes (calledKPZ and Anisotropic KPZ), with different scaling exponents. Here, we reviewrecent mathematical results on (both reversible and irreversible) dynamics ofsome $(2+1)$-dimensional discrete interfaces, mostly defined through a mappingto two-dimensional dimer models. In particular, in the irreversible case, wediscuss mathematical support and remaining open problems concerning Wolf'sconjecture on the relation between the Hessian of the growth velocity on oneside, and the universality class of the model on the other.