Limit Shapes from a Combinatorial Viewpoint

Yale University, New Haven CT

Investigators

Abstract

How does water turn into ice? How does a collection of iron atoms gain or lose magnetic properties with changes in temperature? "Statistical mechanics" is the branch of mathematics and physics which deals with understanding such systems: systems consisting of large numbers of identical objects, interacting locally. The main goals of statistical mechanics are to describe large-scale behavior and phase transitions through mathematical methods. One of the most interesting and important types of behavior is when an external force, such as an imposed boundary condition or other constraint, results in a non-homogeneity in the resulting ensemble. A simple example of this is the water/ice transition under a pressure gradient induced by gravity. The principal investigator proposes to study mathematical models of such behavior in a variety of basic settings, in hopes of understanding their common features and to be able to predict similar behavior in other potentially more complex systems. The notion of limit shape in probability describes the property of large random systems to settle into a fixed, nonrandom state in the limit of large system size. Typically limit shapes arise due to a combination of entropic considerations and energy minimization. Several diverse areas where limit shapes have been shown to arise are in the theory of random graphs with subgraph density constraints, random tilings with imposed boundary conditions, and random configurational models such as "square ice." These examples are all studied through variational formulations. By studying their common features the PI hopes to learn more about the limit shape phenomenon in general.