Statistical mechanics of two-dimensional interfaces

Brown University, Providence RI

Investigators

Abstract

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.