Asymptotic Enumeration of Perfect Matchings of Lattice Graphs

这个项目涉及某些格图的完美匹配的渐近和精确计数(或者，在等价的语言中，对与这些图对偶的区域的平铺的计数)，这些匹配被证明与组合学中的几个重要问题(生成树的计数、平面划分、交替符号矩阵的计数)密切相关，并且以二聚体模型的名义出现在统计物理中。具体地说，受Fisher和Stephenson引入的单体-单体关系的启发，并基于作者发现的沿其对称轴带有三角形孔洞的某些六角形区域的瓦片的精确计数(推广了MacMahon关于平面划分的计数定理)，当这些孔不一定在区域的对称轴上时，作者继续扩展他关于瓦片的渐近计数的工作。这应该会对Fisher和Stephenson关于单体-单体关联的旋转不变性的一个重要猜想带来有用的见解。此外，作者还研究了另外三个问题。首先，提出者试图扩展论点，使他能够从他发现的将平面划分的十个对称类中的八个与剩余的三个恒等式相关联的四个类似恒等式中直接证明一个恒等式。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列举的，并将使为所有十种情况寻找组合证明的任务接近完成。其次，通过考虑改进下界的问题，作者继续了他在三维二聚体问题上的工作，使用了产生符号枚举的Gessel-Viennot和Kasteleyn定理的三维扩展。第三，利用他的完美匹配互补定理的推广，对阿兹特克钻石的周期加权进行了分类，得到了简单的乘积计数公式，从而对Elkies，Kuperberg，Larsen和Propp，B.Y.Yang，Stanley和他的几个结果和建议给出了统一的观点。组合学的目标之一是找到有效的方法来研究离散的对象集合如何排列。离散系统的行为对于现代通信来说是极其重要的。例如，大型网络的设计，如那些发生在电话系统中的网络，以及计算机科学中的算法设计，都涉及离散的对象集，这利用了组合研究。这个项目中的具体问题是统计物理的二聚体模型的例子。这方面的一个基本说明是液体在现实世界中的吸附过程(与润滑剂的研究有关)，液体由两个原子分子-模型中的二聚体--沿着晶体的表面吸附，其固定的原子形成晶格图案，任何两个相邻的位置都可以容纳一个分子，并且任何给定的晶体原子最多只参与一个分子的吸附。这一背景下的主要问题是所研究的量的渐近行为(具体地说，分子覆盖晶体表面的不同方式的数量)，但在目前的上下文中，通常更困难的确定量的问题准确地允许渐近研究的进展。

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.