This proposal is concerned with the asymptotic enumeration of dimer packings with gaps. More specifically, using work of Fisher and Stephenson as its starting point, it studies how the total number of dimer coverings of the complement of the gaps changes as the gaps are moved around on the lattice graph. The joint correlation of a collection of gaps is a non-negative real number measuring this change, and is the central object of study of this proposal. In earlier work, the proposer proved that the correlation of gaps on the hexagonal lattice is governed, for large separations between the gaps, by a law closely resembling the superposition principle of electrostatics: If each gap is regarded as a point charge of magnitude given by the signed difference between the number of white and black vertices in it (in a fixed white-black coloring of the vertices in which each edge has oppositely colored endpoints), then, for large distances between the gaps, their correlation is proportional to the exponential of the negative of the 2D electrostatic energy of the resulting system of charges. Other previous results concern two naturally defined fields, which the proposer proved approach the electric field in the limit when the lattice spacing approaches zero. In the current project, the proposer presents a program organized in several inter-related groups comprising twenty four specific problems and conjectures. The bulk of this program is aimed at developing further the analogy to phenomena from physics, but the list includes also independent combinatorial problems, such as conjectures on tiling enumeration of new regions and generalizations of classical results on plane partitions.
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 in the study of lubricants) of adsorption of a liquid consisting of diatomic 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). In some of the instances we encounter, the usually more difficult problem of determining quantities exactly turns out in fact to be more tractable, and allows progress in the asymptotic study.
on NSF grant DMS 1101670 This proposal is concerned with the asymptotic enumeration of dimer packings with gaps. More specifically, using classical work of Fisher and Stephenson from the 1960's as its starting point, it studies how the total number of dimer coverings of the complement of the gaps changes as the gaps are moved around on the lattice graph. The joint correlation of a collection of gaps is a non-negative real number measuring this change, and is the central object of study of this proposal. The main goal of the proposal was to continue the study of the correlation of gaps in dimer systems, especially in the new context when a boundary is present, and understand the new effects and how they fit in with physical phenomena. One unexpected finding was that in the presence of a boundary, the behavior we get is paralleled by the steady state heat flow problem in a uniform block of material in which there are heat sources and sinks corresponding to the gaps. The surprising aspect of the finding is that in previous work we developed parallels between the interaction of gaps in dimer systems and another physical phenomenon, that of the interaction of charges in electrostatics (we note that in the bulk, i.e. far away from the boundary, these two physical phenomena are governed by the same equations, and therefore can be considered equivalent). Another direction of study in the proposal is concerned with the concrete finite regions in which we embed the gaps. The goal is to do this in such a way that the resulting regions have a number of tilings that can be determined concretely. Then the regions can be enlarged more and more, and in the limit the formulas we found can be used to determine the correlation that we want to find. In several cases the resulting regions with gaps, and the formulas we obtained for the number of their tilings, turned out to be interesting in their own right. One case that stands out is what can be regarded as a dual of one of the fundamental results in the theory of enumeration of tilings, namely MacMahon's century old classical result on the number of lozenge tilings of a hexagon on the triangular lattice. I have presented the results obtained while holding this grant at several conferences and research seminars, both domestic and international. I have taught a special topics course, touching upon several of these results, in the Spring 2013 semester. By discussing some of these results with colleagues from the Physics and Chemistry departments I have continued to pursue making connections between my work and other fields and disciplines.
