论文标题
极端边界条件和随机瓷砖
Extreme boundary conditions and random tilings
论文作者
论文摘要
标准的统计机械或凝结的物质参数告诉我们，物理系统的批量特性不大于边界条件。大型区域的随机瓷砖提供了反直觉的反例，如著名的“北极圆定理”所示，在二维中为二聚体覆盖物所示。在这些注释中，我在关键现象的背景下讨论了此类示例，及其与1+1D量子粒子模型的关系。所有这些都可以共享一个共同的特征：它们是不均匀的，从某种意义上说，当地密度现在取决于散装的位置。我解释了如何使用变分（或流体动力学）论点，如何处理远距离相关性以及如何发生非琐碎的边缘行为来理解此类问题。尽管所有这些都是在二聚体模型的示例中完成的，但此处介绍的结果具有更大的通用性。从这个意义上讲，二聚体模型是讨论更广泛的方法和结果的机会。 [这些笔记仅需要统计力学的基本知识。]
Standard statistical mechanical or condensed matter arguments tell us that bulk properties of a physical system do not depend too much on boundary conditions. Random tilings of large regions provide counterexamples to such intuition, as illustrated by the famous 'arctic circle theorem' for dimer coverings in two dimensions. In these notes, I discuss such examples in the context of critical phenomena, and their relation to 1+1d quantum particle models. All those turn out to share a common feature: they are inhomogeneous, in the sense that local densities now depend on position in the bulk. I explain how such problems may be understood using variational (or hydrodynamic) arguments, how to treat long range correlations, and how non trivial edge behavior can occur. While all this is done on the example of the dimer model, the results presented here have much greater generality. In that sense the dimer model serves as an opportunity to discuss broader methods and results. [These notes require only a basic knowledge of statistical mechanics.]
