Tree Builder Random Walk beyond uniform ellipticity – 2021

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


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.

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