Interacting Graphical Models and Related Topics in Statistical Mechanics

9971016这项研究主要涉及一类相关的(非独立的)渗流问题，它是作为自旋系统的图形表示而出现的。这些表示包含原始自旋系统的所有信息，在最好的情况下，为相变或相共存提供确定的信号(渗流)。这种表示的最熟悉的例子是Q态Potts模型的随机簇表示。最近，出现了许多与其他模型相关联的新图形表示法。除了它们的内在兴趣，对这些问题的研究也很重要，原因有两个：(I)这些表示法提供了一个强大的分析工具，用来研究潜在的统计力学问题。(Ii)表示法是群集法的基础，群集法是一种现代的蒙特卡罗算法，允许快速模拟这些(潜在的)问题。要实现第(Ii)项，以及大部分第(I)项，必须达到“最好的情况”，即证明统计力学问题中的相变反映在图形表示中的渗流相变。总的来说，PI将继续研究新的图形表示和集群算法的发展，着眼于平衡统计力学和其他计算复杂问题的算法构建。这项研究可以描述为：相互作用系统的原始数学描述必然涉及大量多余的信息。往好了说，这些信息根本无关紧要，往坏了说，它可能会掩盖作为研究对象的微妙影响。图形表示提供了另一种描述。尽管描述的数学内容与原作相同，但形式却截然不同：许多多余的信息被推到背景中，有趣的特征往往相当戏剧性地表现为几何现象。更重要的是，这些几何表示是统计物理中使用的一类计算机算法(集群算法)的关键组成部分。直到最近，只有少数几个系统承认了这种描述，但在过去几年里，情况已经有了很大改善。在扩大这些方法的适用范围方面将继续取得进展，既包括在平衡统计力学领域，也包括在计算复杂而无法采用直接方法的其他领域。

9971016This research principally concerns a class of correlated (non-independent) percolation problems that arise as graphical representations of spin-systems. These representations contain all of the information of the original spin-system and, in the best of cases, provide a definitive signal (percolation) for phase transitions or phase coexistence. The most familiar example of such a representation is the random cluster representation for the q-state Potts model. Recently, a number of new graphical representations associated with other models have emerged. In addition to their intrinsic interest, the study of these problems is important for two reasons: (i) The representations provide a powerful analytical tool with which to study the underlying statistical mechanics problem. (ii) The representations are the basis of the cluster methods, a modern class of Monte Carlo algorithms that allow for the rapid simulation of these (underlying) problems. To realize item (ii) and, for the most part item (i) as well, it is essential to achieve "the best of cases" i.e. demonstrate that the phase transition in the statistical mechanics problem is mirrored by a percolation transition in the graphical representation. In general, the PI will continue research on the development of new graphical representations and cluster algorithms with an eye towards the construction of algorithms in equilibrium statistical mechanics and other problems that are computationally complex.This research may be described as follows: The primitive mathematical descriptions of interacting systems necessarily involves a great deal of superfluous information. At best this information is simply of no importance and at worst, it can obscure the subtle effects that are the object of study. Graphical representations present an alternative description. Although the mathematical content of the description is identical to the original, the form is quite different: Much of the superfluous information is pushed into the background and the interesting features are often exhibited rather dramatically as geometric phenomena. Of further importance is that these geometric representations are the key ingredient in a class of computer algorithms (cluster algorithms) that are used in statistical physics. Until recently, only a handful of systems have admitted such descriptions but the situation has improved a great deal in the last few years. Progress will continue in expanding the scope of the applicability of these methods both in equilibrium statistical mechanics an in other area where computational complexity forbids the straightforward approach.