Mathematical Sciences: Random Tilings

数学科学：随机平铺

• 批准号: 9500936
• 负责人: Sara Billey
• 金额: $ 11万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500936&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Random; Tilings

9500936 Propp Abstract The investigator studies two-dimensional structures such as tilings of finite regions with the goal of understanding how constraints at the boundary propagate inward in the presence of randomness. He uses methods from the mathematical fields of combinatorics, dynamical systems, and interacting particle systems in order to prove results that quantify the way boundary conditions temper randomness. His main objects of study are currently dimer models, which can also be interpreted as random tiling models. Crystals are ordinarily thought of as the ultimate in orderly arrangement, but at an atomic scale there is often structural disorder with measurable consequences for the bulk properties of matter. Statistical physicists have made much progress towards modeling these sorts of structure, usually by making some simplifying assumptions. One of these is the supposition that the structure is two-dimensional (an appropriate assumption if one is studying surfaces of crystals); another is the supposition that the structure has no boundary. The investigator is relaxing this second assumption and studying what happens in certain standard models of crystalline matter in which there is a boundary of a specified shape having specified structure, and in which all the randomness is confined to the interior.

9500936 PROPP摘要研究者研究了二维结构，例如有限区域的瓷砖，目的是了解在存在随机性的情况下边界向内传播的约束。 他使用组合学，动力学系统和相互作用粒子系统的数学字段中的方法，以证明量化边界调节温度随机性的结果。 他的主要研究对象目前是二聚体模型，也可以解释为随机平铺模型。 通常将晶体视为有序排列的最终，但是在原子规模上，通常存在结构性疾病，对物质的整体特性产生可衡量的后果。 统计物理学家通常通过做出一些简化的假设，在建模这些结构方面取得了长足的进步。 其中之一是结构是二维的假设（如果一个人正在研究晶体的表面，则适当的假设）；另一个是结构没有边界的假设。 研究者正在放松第二个假设，并研究在某些标准的晶体物质中发生的情况，在某些标准模型中，有指定形状具有指定结构的指定形状的边界，其中所有随机性都局限于内部。