Interacting Graphical Models and Related Topics in Statistical Mechanics

9971016本研究主要关注一类相关（非独立）渗透问题，这些问题出现在自旋系统的图形表示中。这些表征包含了原始自旋系统的所有信息，并且在最好的情况下，为相变或相共存提供了明确的信号（渗透）。这种表示最熟悉的例子是q-state Potts模型的随机聚类表示。最近，出现了许多与其他模型相关的新的图形表示。除了它们本身的兴趣之外，对这些问题的研究还有两个重要的原因：(i)这些表示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）这些表示是聚类方法的基础，这是一类现代的蒙特卡罗算法，可以快速模拟这些（潜在）问题。为了实现第（ii）项，以及在大多数情况下实现第(i)项，有必要实现“最好的情况”，即证明统计力学问题中的相变是由图形表示中的渗透相变反映出来的。总的来说，PI将继续研究新的图形表示和聚类算法的发展，着眼于平衡统计力学和其他计算复杂问题的算法构建。本研究可以这样描述：相互作用系统的原始数学描述必然包含大量的多余信息。往好了说，这些信息根本不重要，往坏了说，它会模糊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著相同，但形式却大不相同：许多多余的信息被推到背景中，有趣的特征往往以几何现象的形式戏剧性地表现出来。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（聚类算法）的关键成分。直到最近，只有少数系统承认这样的描述，但这种情况在过去几年中有了很大的改善。在平衡统计力学和其他计算复杂性禁止直接方法的领域，将继续扩大这些方法的适用范围。

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.