Large Finite Random Structures

该项目使用概率方法研究有限随机结构的行为，因为它们的尺寸越来越大，重点是五个特定的主题。两个是来自理论组合学：非均匀随机映射;和最大公共子结构的设置，如偏序或叶标记树。一个是来自统计物理学：无序系统的平均场模型，特别是第一通道渗流及其相关的渗流函数。另一个是生物学：纯随机宏观进化模型，可以同时处理现存物种的系统发育树和灭绝物种的时间序列。最后一个主题是随机亲和图，它试图抽象真实的世界图的结构的一个方面，例如人类社交网络的结构（"小世界”）或万维网链接。从概念上讲，该项目旨在作为随机结构的抽象数学理论之间的桥梁，该理论侧重于不切实际的简单模型的困难特性，以及工程类型的复杂网络（互联网的物理和超文本链接;交通网络）的例子，其模型利用了特定结构的细节。人们希望，这五个特定的模型的研究将导致新的数学方法在这些更适用的领域。

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.