Recurrence, transience and degree distribution for the Tree Builder Random Walk

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 $\mathsf{B}\mathsf{e}\mathsf{r}$(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 $Z_n\sim \mathsf{B}\mathsf{e}\mathsf{r}(p_n)$, with $p_n=\mathrm{\Theta }(n^{-\gamma })$ and $\gamma \in (2\mathrm{/}3,1]$. 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.