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

这个建议是有关的渐近计数的格子区域与洞的平铺。更具体地说，使用费舍尔和斯蒂芬森的工作作为其出发点，它研究了当所有剩余部分被平铺时孔的相互作用，通过对所有这种可能的平铺进行某种平均，称为孔的联合相关性。在早期的工作中，提出者证明了三角形晶格上的三角形孔的情况，对于孔之间的大间隔，由一个非常类似于静电叠加原理的定律控制，其中孔对应于大小等于它们包围的每个方向的单位三角形的数量之差的电荷。在当前的项目中，提议者提出了一个研究问题的计划，在一些新的，相互关联的方向上扩展了他以前的工作。其中一个问题涉及到将相关性的叠加原理推广到当孔可以是任意（不一定是连通的）边2的网格三角形的并集时的情况。另一个问题是考虑相关性的更精细的分析，即研究其在单个孔的小位移下的变化。提议者认为这些细微的变化也受叠加原理的支配，类似于静电学中电场的叠加原理。第三个问题与研究边界效应有关。除了上述核心研究计划外，提议者还打算研究由正方形、正六边形和正十二边形平铺平面所确定的格上的某类图的完美匹配的对称类，并确定在考虑对称性的作用时，他在这些图的完美匹配和深入研究的平面划分之间发现的相似性在多大程度上延伸本研究属于组合数学的一般领域。组合数学的目标之一是找到有效的方法来研究如何安排对象的离散集合。离散系统的行为对现代通信极为重要。例如，大型网络的设计，如电话系统中的网络设计，以及计算机科学中的算法设计，都要处理离散的对象集，这就需要使用组合研究。在这个项目中的具体问题是统计物理的二聚体模型的实例。 一个基本的例子是液体吸附的真实过程（与润滑剂的研究有关），由两个原子分子组成-模型中的二聚体-沿着晶体表面，其固定的原子形成晶格图案，任何两个相邻的位置能够容纳一个分子，并且任何给定的晶体原子参与最多一个分子的吸附。在这种情况下的主要问题是所研究的量的渐近行为（具体来说，晶体表面可以被分子覆盖的不同方式的数量）。在我们遇到的一些例子中，通常更困难的确定量的问题最终使渐近研究取得了进展。

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.