Statistical mechanics of two-dimensional interfaces

随机表面和随机界面的研究长期以来一直吸引着物理学家和数学家的兴趣。然而，直到最近才出现了理解一维以上简单界面的数学技术。下一个最简单的情况，即三维空间中的二维界面，已经相当困难了。我们研究的二维界面称为“阶梯表面”。根据最简单的选择措施对这些表面，统一措施的固定边界，大规模的形状采取这些表面已开始制定出的PI和Okounkov，使用技术从偏微分方程，分析和热带几何。这个模型基本上是随机界面的唯一数学“解决”模型。此外，它包含了大量的数学连接与其他领域：随机矩阵理论，可积系统，弦理论和Gromov-Witten理论。由于这些原因，更好地理解这个模型是值得的，也值得寻找推广。我们正在研究晶体表面的数学模型。在原子尺度上，晶体的表面，如盐晶体或金刚石，是粗糙和“随机”的，但在大尺度上，它通常是光滑和有小面的。这些大尺度特征是如何从组成晶体的原子的微观相互作用中产生的，在很大程度上仍然是个谜。然而，我们可以制造出在数学意义上易于计算的晶体表面模型，并显示出与真实的晶体相同的行为：特别是它们显示出小面和大尺度形状的形成。通过研究这些模型，我们不仅希望了解晶体表面，而且希望了解大量组分之间的局部相互作用如何发展成大尺度行为的一般现象。

The study of random surfaces and random interfaces has long held the interest of physicists and mathematicians. Only recently, however, have there arisen mathematical techniques for understanding simple interfaces of more than one dimension. The next simplest case,that of two dimensional interfaces in three space,is already quite difficult. The two-dimensional interfaces we study are called 'stepped surfaces'. Under the simplest choice of measure on these surfaces, the uniform measure for a fixed boundary, the large-scale shapes taken by these surfaces has begun to be worked out by the PI and Okounkov, using techniques from PDEs, analysis and tropical geometry. This model is essentially the only mathematically 'solved' model of random interfaces. Moreover it contains a great deal of mathematical connections with other areas: to random matrix theory, integrable systems, string theory and Gromov-Witten theory. For these reasons it is worth understanding this model better, and also worth looking for generalizations.We are studying mathematical models of crystal surfaces. On an atomic scale, the surface of a crystal, such as a salt crystal or diamond, is rough and 'random', but at large scales it is typically smooth and facetted. How these large scale features arise from the microscopic interactions of the atoms comprising the crystal is, to a large extent, still mysterious. However we can make models of crystal surfaces which are computationally tractable in a mathematical sense, and display the same behavior as real crystals: in particular they display facetting and large-scale shape formation. By studying these models we hope to gain understanding not just of crystal surfaces but of the general phenomenon of how local interactions among a large number of constituents can develop into large-scale behavior.