(2 + 1)-DIMENSIONAL INTERFACE DYNAMICS: MIXING TIME, HYDRODYNAMIC LIMIT AND ANISOTROPIC KPZ GROWTH
(2 + 1)-DIMENSIONAL INTERFACE DYNAMICS: MIXING TIME, HYDRODYNAMIC LIMIT AND ANISOTROPIC KPZ GROWTH
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DOI：
10.1142/9789813272880_0158
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发表时间：
2017-11
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通讯作者：
F. Toninelli
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文献类型：
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作者：
F. Toninelli
随机界面动力学可作为各种随时间变化的物理现象的数学模型：热力学相之间的边界演化、晶体生长、随机沉积…有趣的极限出现在大的时空尺度上：经过适当的重新标度，随机演化的界面收敛到确定性的PDE(流体动力学极限)的解，涨落过程收敛到(一般为非高斯的)极限过程。与$(1+1)$维模型相比，在维$(d+1)，d2$上得到的数学结果很少。就增长模型而言，$(2+1)$维的情形特别有趣：D.Wolf猜想存在两个不同的普适类(称为KPZ和各向异性KPZ)，具有不同的标度指数。在这里，我们回顾了最近关于一些$(2+1)$维离散界面(包括可逆和不可逆)动力学的数学结果，这些结果大多是通过到二维二聚体模型的映射来定义的。特别地，在不可逆的情况下，我们讨论了Wolf关于增长速度的黑森猜想与模型普适类之间的关系的数学支持和尚待解决的问题。
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 $(d+1), d\ge2$. As far as growth models are concerned, the $(2+1)$-dimensional case is particularly interesting: D. Wolf 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.