Tree builder random walk

2023. 10. 12. 16:15
János Engländer (Boulder, CO)

We study the tree builder random walk (TBRW): a randomly growing tree, built by a walker as she is walking around the tree. Namely, at each time n, she adds a leaf to her current vertex with probability pn=n where γ∈(0,1], (or some other sequence tending to zero) then moves to a uniform random neighbor on the possibly modified tree.
Thus, the model, depending on one’s point of view, is:
• a self-interacting random walk, in a dynamically evolving environment, or
• an evolving random tree built by the walker itself, as it walks around.
After presenting some results on the behavior of the walker, we show that for γ>2/3 the tree process at its
growth times, can be coupled with the Barabási-Albert preferential attachment tree model.
Hence, our TBRW-model is a local dynamics giving rise to the BA-model. The coupling also implies
that many properties known for the BA-model, such as diameter and degree distribution, can be
directly transferred to our TBRW-model.
This is joint work with G. Iacobelli (Rio de Janeiro), G. Pete (Budapest) and R. Ribeiro (Denver).