Tree Builder Random Walk beyond uniform ellipticity

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, . . . , right before stepping, the walker adds a random number (possibly zero) Zn of leaves to its current position. We assume that the Zn’s are independent, but, importantly, we do 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 Zn ∼ Ber(p) (thus i.i.d.), the random walk was shown to be ballistic for every p ∈ (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 Zn ∼ Ber(pn), with pn = Θ(n) and γ ∈ (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.