Dimer-mediated interaction of gaps in lattice graphs

本文研究了带洞格图的二聚覆盖的渐近计数问题。更具体地说，使用Fisher和斯蒂芬森的工作作为其出发点，它研究了如何二聚体覆盖的补洞的总数的变化，作为孔被转移周围的格子图。孔集合的联合相关性是测量这种变化的非负真实的数，并且是本提议的研究的中心对象。在早期的工作中，提出者证明，对于孔之间的大间隔，六边形晶格上孔的相关性由一个非常类似于静电叠加原理的定律控制：如果把每个空穴看作一个点电荷，其大小由空穴中白色顶点和黑色顶点的数目之差给出（在顶点的固定的白-黑着色中，其中每个边缘具有相反颜色的端点），那么，对于孔之间的大距离，它们的相关性与所得电荷系统的2D静电能量的负值的指数成比例。其他先前的结果涉及两个自然定义的领域，提案人证明接近电场的限制时，晶格间距接近零。在目前的项目中，提案人提出了一个由20多个具体问题和建议组成的几个相互关联的小组组织的程序。该计划的大部分旨在进一步发展静电学的类比，但该列表也包括独立的组合问题，如新区域的平铺枚举和卡片洗牌研究中出现的问题。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 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.