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

这个项目是在组合数学的一般领域。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，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。 这方面的一个基本说明是由双原子分子组成的液体沿着晶体表面吸附的真实过程（与润滑剂的研究有关），晶体的固定原子形成晶格图案，任何两个相邻位置都能够容纳一个分子，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，分子可以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，通常更困难的确定数量的问题实际上变得更容易处理，并允许渐近研究的进展。本项目涉及带间隙的二聚体填充的精确和渐近计数。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互补的间隙的总数的变化，作为间隙移动周围的格子图。间隙集合的联合相关性是衡量这种变化的非负真实的数，并且是本建议的中心研究对象。在早期的工作中，提出者证明了六边形晶格上间隙的相关性，对于间隙之间的大间隔，由一个非常类似于静电叠加原理的定律控制：如果把每个间隙看作一个点电荷，其大小由其中的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，其中每个边具有相反着色的端点），那么，对于间隙之间的大距离，它们的相关性与所得电荷系统的二维静电能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，其中的提议者证明，接近电场的极限时，晶格间距接近零。在目前的项目中，提案人提出了一个由27个具体问题和建议组成的几个相互关联的小组组织的计划。该计划旨在进一步发展物理现象的类比（特别是材料块中的热流，这对应于间隙与边界的相互作用），并理解与独立组合问题的联系，例如新区域的平铺枚举和平面分区经典结果的推广。

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.