Combinatorial Methods for Random Structures in the Plane

摘要项目负责人：Winkler， Peter项目编号：DMS - 0901475机构：Dartmouth college题目：平面随机结构的组合方法PI将处理平面上的三个随机模型，它们都显示出受益于组合技术应用的趋势。第一个问题与支链聚合物有关；第二种是核心气体；第三，协同渗透。支链聚合物是空间中球形的连接构型；它们的配分函数可以在2维和3维上组合计算。硬核气体是空间中不重叠球体的随机集合；这里PI有一些离散版本的想法，这可能有助于证明固态相的存在。最后，应用于协调渗透的组合技术已经产生了渗透概率的精确表达式，其中一个地点的寿命取决于与该点相交的线相关的事件。这三种模型与生物化学、物理学和计算机科学的应用有关。参与光合作用的复杂的树状蛋白质、血红蛋白和分子可以与随机的数学“支链聚合物”进行比较，以帮助识别未预料到的力或约束的存在。从理论上讲，晶体结构的形成应该类似于硬球体在平面或空间中被压缩时发生的情况——但这个简单的数学模型尚未完全被理解。最后，调度过程（如计算机程序）以避免冲突是通过避免随机漫步中的碰撞来建模的，这是坐标渗透的几种应用之一。PI的方法是使用组合方法——粗略地说，复杂的计数和操作离散对象的方法——来更好地理解这三种模型。

ABSTRACTPrincipal Investigator: Winkler, Peter Proposal Number: DMS - 0901475Institution: Dartmouth CollegeTitle: Combinatorial Methods for Random Structures in the PlaneThe PI will tackle three random models in the plane, all of which show signs of benefiting from the application of combinatorial techniques. The first of these concerns branched polymers; the second, hard-core gases; the third, coordinate percolation. Branched polymers are connected configurations of spheres in space; their partition functions can be computed combinatorially in dimensions 2 and 3. Hard-core gases are random sets of non-overlapping spheres in space; here the PI has some ideas for discrete versions which may help in proving the existence of a solid-state phase. Finally, combinatorial techniques applied to coordinate percolation, in which the life of a site depends on events associated with the lines that cross there, has already yielded an exact expression for percolation probability.The three models are connected to applications in biochemistry, physics and computer science. Complex tree-like forms of proteins, hemoglobins, and molecules involved in photosynthesis can be compared with random mathematical ``branched polymers'' to help discern the presence of unsuspected forces or constraints. Formation of crystal structures should, in theory, resemble what happens to collections of hard spheres when they are compressed on the plane or in space---but this simple mathematical model is not yet fully understood. Finally, scheduling processes (such as computer programs) to avoid conflict is modeled by avoiding collisions in random walks, one of several applications of coordinate percolation. The PI's approach is to use combinatorial methods---rougly speaking, sophisticated means of counting and manipulating discrete objects---to better understand all three models.