Large Finite Random Structures

该项目使用概率方法研究有限随机结构在其尺寸越来越大时的行为，重点关注五个特定主题。两个来自理论组合学：非均匀随机映射；以及设置中的最大公共子结构，如偏序或叶标记树。一种来自统计物理：无序系统的平均场模型，特别是首次通过渗流及其相关渗流函数的平均场模型。另一种来自生物学：纯粹随机的宏观进化模型，可以同时处理现存物种和灭绝物种的时间序列的系统发育树。最后一个主题是随机亲和图，它试图抽象现实世界图结构的一个方面，如人类社会网络(小世界)或WWW链接的结构。从概念上讲，该项目旨在成为随机结构抽象数学理论和复杂网络工程类型示例(互联网的物理和超文本链接；交通网络)之间的桥梁，随机结构抽象数学理论侧重于不现实的简单模型的困难性质，其模型利用特定结构的细节。希望对这五个特定模型的研究将导致在这些更多应用领域有用的新的数学方法。

The project studies, using probabilistic methods, the behavior of finite random structures as their size grows larger and larger, focusing on five specific topics. Two are from theoretical combinatorics: non-uniform random mappings; and largest common substructures in settings such as partial orders or leaf-labeled trees. One is from statistical physics: mean-field models of disordered systems, in particular of first-passage percolation and its associated percolation function. Another is from biology: models of purely random macroevolution which can handle simultaneously phylogenetic trees on extant species and time series of extinct species. The final topic is random affinity graphs, which seek to abstract one aspect of the structure of real world graphs such as those for human social networks (``small worlds") or WWW links.Conceptually, the project is intended as a bridge between abstract mathematical theory of random structures, which focuses on difficult properties of unrealistically simple models, and the engineering type of examples of complex networks (the physical and hypertext links of the Internet; transportation networks) whose models exploit the details of the particular structure. It is hoped that study of these five particular models will lead to novel mathematical methodology useful in these more applied areas.