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

### Description Phase 2

### Description Phase 1

In this project we want to study scaling limits of evolving spanning trees on different graphs (in particular, on d-dimensional tori, where d ≥ 2) and work towards tools for the construction and analytical characterization of diffusions on evolving continuum trees. 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 the study of the scaling limit of a particular network dynamics. This is the Aldous-Broder chain which is a tree-valued Markov chain generating uniform spanning trees.

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. Recently, random walks were studied on dynamic network models and compared with random walks on static networks. The new feature of this project is the construction of scaling limits of random walks on evolving sparsely connected graphs.

### Preprints/Publications

Andreas Greven, Frank den Hollander, Anton Klimovsky, Anita Winter: The grapheme-valued Wright-Fisher diffusion with mutation (08/2023)

Gabriel Berzunza Ojeda, Anita Winter: Convergence of the pruning processes of stable Galton-Watson trees and other ℝ-trees (04/2023)

Josué Nussbaumer, Viet Chi Tran, Anita Winter: Algebraic two-level measure trees (07/2022)

Josué Nussbaumer, Anita Winter: The algebraic α-Ford tree under evolution (06/2020)