Asymptotic Enumeration of Tilings of Lattice Regions With Holes: A Finer Analysis Under Various Boundary Conditions

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This proposal is concerned with the asymptotic enumeration of tilings of lattice regions with holes. More specifically, using work of Fisher and Stephenson as its starting point, it studies the interaction of the holes when all the leftover portion is tiled, via a certain averaging over all such possible tilings, called the joint correlation of the holes. In earlier work, the proposer proved that the case of triangular holes on the triangular lattice is governed, for large separations between the holes, by a law closely resembling the superposition principle of electrostatics, where holes correspond to charges of magnitude equal to the difference between the number of unit triangles of each orientation they enclose. In the current project, the proposer presents a program of research problems that extend his previous work in some new, interrelated directions. One of these problems concerns generalizing the superposition principle for correlation to the case when the holes can be arbitrary (not necessarily connected) unions of lattice triangles of side two. Another problem considers a finer analysis of the correlation, namely the study of its variation under small displacements of individual holes. The proposer conjectures that these finer changes are also governed by a superposition principle, analogous to the superposition principle for the electric field in electrostatics. A third problem is concerned with studying boundary effects. Besides the core research program outlined above, the proposer also intends to study the symmetry classes of perfect matchings of a certain family of graphs on the lattice determined by the tiling of the plane by squares, regular hexagons and regular dodecagons, and determine to what degree a parallel he found between the perfect matchings of these graphs and the intensively studied plane partitions extends when considering the action of symmetry groups.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 (speciffically, 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 to allow progress in the asymptotic study.