Statistical mechanics of two-dimensional interfaces

长期以来，对随机表面和随机界面的研究一直是物理学家和数学家的兴趣所在。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被称为“阶梯面”。在这些表面上最简单的测量选择下，固定边界的统一测量，这些表面所采取的大尺度形状已经开始由PI和Okounkov使用偏微分方程组、分析和热带几何的技术来计算。这个模型实质上是唯一的随机界面的数学“解”模型。此外，它还包含了与其他领域的大量数学联系：随机矩阵理论、可积系统、弦理论和Gromov-Witten理论。由于这些原因，我们值得更好地理解这个模型，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，如盐晶体或钻石，是粗糙和随机的，但在大尺度上，它通常是光滑和多面的。这些大尺度特征是如何从构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，我们可以制作在数学意义上计算容易处理的晶体表面模型，并显示出与真实晶体相同的行为：特别是它们显示出刻面和大规模的形状形成。通过研究这些模型，我们希望不仅了解晶体表面，而且了解大量组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.