Scaling limits of evolving spanning trees and of random walks on evolving spanning trees

在这个项目中，我们研究了进化均匀生成树（UST）在其他类型的网络（包括Erdös-Renyi图，密集连接扩展图序列和低维环面）上的缩放限制。主要动机来自于对大型稀疏连接网络的建模。树是稀疏连接网络的极端情况。在真实的世界网络中，网络的结构可能会随着时间的推移而改变。该项目的一个重点是关注一个特定的网络动态。这是Aldous-Broder算法，它是生成UST的树值随机过程。随机游走是网络上的一个简单的随机过程，它允许探索网络的结构。在通信网络的上下文中（例如，因特网、WiFi），其可以被理解为从设备发送到设备的消息。在目前的研究中，动态网络模型上的随机游动与静态网络上的随机游动进行了比较。在这个项目中，我们确定的空间和时间尺度上的随机游走有一个扩散的缩放限制。

In this project we study scaling limits of evolving uniform spanning trees (UST) on further classes of networks (including Erdös-Renyi graphs, sequences of densely connected expander graphs and low-dimensional tori). The main motivation comes from modeling large and sparsely connected networks. Trees are the extreme cases of sparsely connected networks. In real world networks, the structure of the network might change over time. One emphasis of the project concerns a particular network dynamics. This is the Aldous-Broder algorithm which is a tree-valued stochastic process that generates the UST. A random walk is a simple stochastic process on a network which allows to explore the structure of the network. In the context of communication networks (e.g.\ internet, wifi) it can be understood as a message sent from device to device. In the current research random walks on dynamic network models are compared with random walks on static networks. In this project we determine the space and time scales on which the random walk has a diffusive scaling limit.