Condensation in random geometric graphs

我们建议的项目是第一个资助期同名项目的延续。在项目的两个部分中，我们探索了当几何随机图以某些不可能事件为条件时产生的凝结现象。该项目的第一部分集中在具有异常多边的图的事件上，在这种情况下，冷凝可以在具有异常高程度的顶点或具有异常高连通性的局部集群的出现中表现出来。在第二部分中，我们的主要目标是将这项工作扩展到更一般的大偏差事件，例如具有异常多三角形，异常长路径或大型连接组件的图。这些事件也基于凝结效应，我们将对此进行更深入的探讨。我们还将使用和扩展最近在分析几何随机图和其他渗透模型中的急剧相变中开发的新方法。

The project we propose is a continuation of the project of the same name in the first funding period. In both parts of the project we explore the condensation phenomena arising when a geometric random graph is conditioned on certain unlikely events. The first part of the project was focused on the event of the graph having exceptionally many edges, in which case the condensation can manifest itself in the emergence of vertices with exceptionally high degree or of localized clusters of exceptionally high connectivity. In the second part our principal aim is to extend this work to more general large deviation events, for example the graph having unusually many triangles, untypically long paths or large connected components. These events are also based on a condensation effect, which we will explore in greater depth. We will also use and extend recent new methods developed in the analysis of sharp phase transitions in geometric random graphs and other percolation models.