Asymptotic Enumeration of Perfect Matchings of Lattice Graphs

该项目关注的是某些格图完美匹配的渐近和精确计数（或者，在一个等价的语言中，这些图的对偶区域的平铺计数），这些图与组合学中的几个重要问题（生成树的计数，平面划分，交替符号矩阵）密切相关，并且以二聚体模型的形式出现在统计物理中。 具体地说，由Fisher和斯蒂芬森引入的单体-单体相关性激发，并基于提议者发现的某些六边形区域沿其对称轴沿着具有三角形孔的平铺的精确计数（它推广了MacMahon关于计算平面划分的定理），在洞不一定在对称轴上的情况下，本地区 这应该带来有益的洞察到一个重要的猜想，费舍尔和斯蒂芬森关于旋转不变性的单体单体相关。此外，提案人研究了三个额外的问题。 首先，提出者追求扩展的论点，使他能够直接证明一个身份，从一组四个类似的身份，他发现有关八的十个对称类的平面分区，其余三个身份。 这将有助于解释仍然神秘的事实，即所有十种情况都是由简单的乘积公式枚举的，并且将接近完成为所有十种情况寻找组合证明的任务。 第二，提议者继续他的工作在三维二聚体的问题，考虑提高下界的问题，采用扩展到三维的Gessel-Viennot和Kasteleyn定理，产生签署枚举。 第三，提出者使用他的完美匹配的互补定理的推广来分类的周期加权的阿兹特克钻石，导致简单的产品计数公式，从而给出了一个统一的观点，几个结果的Elkies，Kuperberg，Larsen和Propp，B。Y.杨，斯坦利和提议者。这项研究是在一般领域的组合。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（具体地说，分子可以覆盖晶体表面的不同方式的数量），但在目前的情况下，通常更困难的问题是确定量，这使得渐近研究取得了进展。

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.