Dimer-mediated interaction of gaps in lattice graphs

Indiana University, Bloomington IN

Investigators

Abstract

This proposal is concerned with the asymptotic enumeration of dimer coverings of lattice graphs with holes. 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 holes changes as the holes are shifted around on the lattice graph. The joint correlation of a collection of holes 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 holes on the hexagonal lattice is governed, for large separations between the holes, by a law closely resembling the superposition principle of electrostatics: If each hole 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 holes, 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 that 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 two dozen specific problems and conjectures. The bulk of this program is aimed at developing further the analogy to electrostatics, but the list includes also independent combinatorial problems, such as conjectures on tiling enumeration of new regions and problems arising in the study of card shufflings. 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.