Dimer systems with gaps and their connections with statistical physics, plane partitions, and alternating sign matrices

这个项目属于组合学的一般领域。组合学的目标之一是找到研究离散对象集合如何排列的有效方法。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，比如那些出现在电话系统中的网络，以及计算机科学中的算法设计，处理离散的对象集，这就利用了组合研究。这个项目中的具体问题是统计物理的二聚体模型的实例。一个基本的例子是现实世界的过程（与润滑油的研究有关），由双原子分子组成的液体（模型中的二聚体）沿着晶体表面吸附，其固定的原子形成晶格模式，任意两个相邻的位置都能容纳一个分子，并且任何给定的晶体原子最多只能吸附一个分子。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，通常比较困难的确定数量的问题实际上是比较容易处理的，并且允许在渐近研究中取得进展。本文研究带间隙二聚体填料的精确和渐近枚举问题。更具体地说，它以Fisher和Stephenson的工作为出发点，研究了随着空位在晶格图上的移动，空位补的二聚体覆盖的总数是如何变化的。间隙集合的联合相关是衡量这种变化的非负实数，是本文研究的中心对象。在早期的工作中，提出者证明了六角形晶格上的间隙的相关性，对于间隙之间的大间隔，是由一个非常类似于静电叠加原理的定律控制的：如果将每个间隙视为一个点电荷，其大小由其中白色和黑色顶点的数量之间的符号差异给出（在固定的白-黑着色的顶点中，每个边缘具有相反颜色的端点），那么，对于间隙之间的较大距离，它们的相关性与所得电荷系统的二维静电能量的负指数成正比。先前的其他结果涉及两个自然定义的场，提出者证明了当晶格间距趋近于零时电场的极限。在当前的项目中，提议者提出了一个由几个相互关联的小组组成的程序，包括27个具体问题和猜想。本课程旨在进一步发展与物理现象的类比（特别是材料块中的热流，这与边界的相互作用相对应），并了解与独立组合问题的联系，例如关于新区域的平铺枚举的猜想和平面分区的经典结果的推广。

This project 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.This project is concerned with the exact and 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 two-dimensional 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 seven specific problems and conjectures. This program is aimed at developing further the analogy to phenomena from physics (in particular to heat flow in a block of material, which turns out to correspond to the interaction of gaps with boundary) and also understanding the connections with independent combinatorial problems, such as conjectures on tiling enumeration of new regions and generalizations of classical results on plane partitions.