Statistical mechanics of two-dimensional interfaces

对随机表面和随机界面的研究长期以来一直引起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的简单界面的数学技术。下一个最简单的例子，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被称为“阶梯表面”。在这些表面上最简单的测量选择下，对于固定边界的统一测量，PI和Okounkov已经开始使用偏微分方程、分析和热带几何的技术来计算这些表面所采取的大尺度形状。这个模型本质上是随机界面的唯一数学“解决”模型。此外，它还与随机矩阵理论、可积系统、弦理论和Gromov-Witten理论等其他领域有大量的数学联系。由于这些原因，我们有必要更好地理解这个模型，也有必要寻找一般化的方法。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，如盐晶体或钻石，是粗糙的和“随机的”，但在大尺度上，它通常是光滑的和多面的。这些大规模的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一个谜。然而，我们可以制作出在数学意义上易于计算的晶体表面模型，并显示出与真实晶体相同的行为：特别是它们显示出面形和大规模的形状形成。通过研究这些模型，我们希望不仅能理解晶体表面，还能理解大量成分之间的局部相互作用如何发展成大规模行为的一般现象。

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.