The interaction of gaps in dimer systems and beyond

这一建议涉及带间隙的二聚体填充的渐近计数。更具体地说，它以Fisher和Stephenson的工作为起点，研究了当空位在格图上移动时，空位补的二聚体覆盖总数是如何变化的。缺口集合的联合相关性是衡量这种变化的非负实数，也是本提案研究的中心对象。在早期的工作中，这位提出者证明了，对于间隙之间的大间隔，六角晶格上间隙的相关性受一个与静电学叠加原理非常相似的定律支配：如果每个间隙被视为由其中白色和黑色顶点的数目的符号差给出的数量级的点电荷(其中每条边具有相反颜色的端点的固定的白色-黑色顶点)，那么，对于间隙之间的较大距离，它们的相关性与所产生的电荷系统的2D静电能量的负指数成正比。以前的其他结果涉及两个自然定义的场，作者证明了当晶格间距接近零时，这两个自然定义的场接近极限电场。在目前的项目中，提出者提出了一个由24个具体问题和猜想组成的相互关联的小组组织的计划。本程序的主要目的是进一步发展物理现象的类比，但列表中也包括独立的组合问题，如关于新区域的平铺计数的猜想和关于平面划分的经典结果的推广。这项研究是在组合学的一般领域。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，如电话系统中的网络，以及计算机科学中的算法设计，都处理离散的对象集，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。这一点的一个基本说明是由双原子分子-模型中的二聚体--组成的液体沿晶体表面的吸附的真实世界过程(与润滑剂的研究相关)，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至多一个分子的吸附。在这个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)。在我们遇到的一些例子中，通常更困难的准确确定量的问题实际上变得更容易处理，并使渐近研究取得进展。

This proposal is concerned with the asymptotic enumeration of dimer packings with gaps. 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 gaps changes as the gaps are moved around on the lattice graph. The joint correlation of a collection of gaps 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 gaps on the hexagonal lattice is governed, for large separations between the gaps, by a law closely resembling the superposition principle of electrostatics: If each gap 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 gaps, 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, which 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 twenty four specific problems and conjectures. The bulk of this program is aimed at developing further the analogy to phenomena from physics, but the list includes also independent combinatorial problems, such as conjectures on tiling enumeration of new regions and generalizations of classical results on plane partitions.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 in the study of lubricants) of adsorption of a liquid consisting of diatomic 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). In some of the instances we encounter, the usually more difficult problem of determining quantities exactly turns out in fact to be more tractable, and allows progress in the asymptotic study.