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。在这个项目的两个部分中，我们探索了几何随机图在某些不太可能的事件的条件下产生的凝聚现象。该项目的第一部分集中在图具有异常多的边的事件上，在这种情况下，凝聚可以通过出现具有异常高程度的顶点或出现具有异常高连通性的局部簇来体现。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，例如具有异常多的三角形、异常长的路径或大的连通分量的图。这些事件也是基于凝聚效应，我们将更深入地探讨这一点。我们还将使用和推广最近发展的新方法来分析几何随机图和其他渗流模型中的尖锐相变。

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.