Mathematical Sciences: Random Tilings

9500936研究人员研究二维结构，如有限区域的平铺，目的是了解在存在随机性的情况下，边界上的约束如何向内传播。他使用了组合学、动力系统和相互作用的粒子系统等数学领域的方法，以证明量化边界条件缓和随机性的方式的结果。他目前的主要研究对象是暗淡模型，也可以解释为随机平铺模型。晶体通常被认为是有序排列的终极，但在原子尺度上，经常存在结构无序，对物质的整体性质产生可测量的后果。统计物理学家在对这类结构进行建模方面取得了很大进展，通常是通过做出一些简化的假设。其中之一是假设结构是二维的(如果研究晶体表面，这是一个合适的假设)；另一个假设是结构没有边界。研究人员正在放松第二个假设，并研究在某些晶体物质的标准模型中发生了什么，在这些模型中，存在具有特定结构的特定形状的边界，并且所有的随机性都限制在内部。

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.