This project is concerned with the asymptotic and exact enumeration of perfect matchings of certain lattice graphs (or, in an equivalent language, enumeration of tilings of the regions dual to these graphs) that turn out to be closely related to several important problems in combinatorics (enumeration of spanning trees, plane partitions, alternating sign matrices), and that appear in statistical physics in the guise of the dimer model. Specifically, motivated by the monomer-monomer correlation introduced by Fisher and Stephenson, and based on the exact enumeration found by the proposer of the tilings of certain hexagonal regions with triangular holes along their symmetry axes (which generalizes MacMahon's theorem on counting plane partitions), the proposer pursues extending his work on the asymptotic enumeration of tilings in the situation when the holes are not necessarily on the symmetry axis of the region. This should bring useful insight into an important conjecture of Fisher and Stephenson concerning the rotational invariance of the monomer-monomer correlation. Furthermore, the proposer studies three additional problems. First, the proposer pursues extending the arguments that allowed him to prove directly one identity from a set of four similar identities he found relating eight of the ten symmetry classes of plane partitions to the remaining three identities. This would help explaining the still mysterious fact that all ten cases are enumerated by simple product formulas and would bring close to completion the task of finding combinatorial proofs for all ten cases. Second, the proposer continues his work on the three dimensional dimer problem by considering the question of improving the lower bound, employing extensions to three dimensions of the Gessel-Viennot and Kasteleyn theorems that yield signed enumerations. And third, the proposer uses a generalization of his complementation theorem for perfect matchings to classify the periodic weightings of the Aztec diamond that lead to simple product enumeration formulas, thus giving a unified perspective on several results of Elkies, Kuperberg, Larsen and Propp, B. Y. Yang, Stanley and the proposer.
This research is in the general area of Combinatorics. One of the goals of Combinatorics is to find efficient methods of studying how discrete collections of objects can be arranged. The behavior of discrete systems is extremely important to modern communications. For example, the design of large networks, such as those occurring in telephone systems, and the design of algorithms in computer science deal with discrete sets of objects, and this makes use of combinatorial research. The specific problems in this project are instances of the dimer model of statistical physics. A basic illustration of this is the real-world process (relevant to the study of of lubricants) of adsorption of a liquid, consisting of two-atom molecules --- the dimers in the model --- along the surface of a crystal, whose fixed atoms form a lattice pattern, with any two neighboring positions capable of holding one molecule, and any given crystal atom being involved in the adsorption of at most one molecule. The main issue in this setting is the asymptotic behavior of the quantities that are studied (specifically, the number of different ways the surface of the crystal can be covered by molecules), but it turns out in the present context that the usually more difficult problem of determining quantities exactly allows progress in the asymptotic study.
