Preprint

Tree Builder Random Walk beyond uniform ellipticity – 2021

with János Englander (Colorado University Boulder) and Giulio Iacobelli (UFRJ)

Abstract

We investigate a self-interacting random walk, whose dynamically evolving environment is a random tree built by the walker itself, as it walks around.At time $n=1,2,\dots$, right before stepping, the walker adds a random number (possibly zero) $Z_n$ of leaves to its current position. We assume that the $Z_n$’s are independent, but, importantly, we do \emph{not} assume that they are identically distributed.
We obtain non-trivial conditions on their distributions under which the random walk is recurrent. This result is in contrast with some previous work in which, under the assumption that $Z_n\sim \mathsf{Ber}(p)$ (thus i.i.d.), the random walk was shown to be ballistic for every $p \in (0,1]$. %We also obtain results on the transience of the walk, and the possibility that it “gets stuck.”
From the perspective of the environment, we provide structural information about the sequence of random trees generated by the model when $Z_n\sim \mathsf{Ber}(p_n)$, with $p_n=\Theta(n^{-\gamma})$ and $\gamma \in (2/3,1]$. We prove that the empirical degree distribution of this random tree sequence converges almost surely to a power-law distribution of exponent $3$, thus revealing a connection to the well known preferential attachment model.

Find the PDF file at

arXiv

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: