Fractal dimensions and random walks on random trees

Abstract

We determine the fractal dimension d f of infinite spherically symmetric random trees (all vertices at distance n from the root have the same degree d n where { d n } are independent random variables). If d n takes the value 3 or 2 with probabilities q n and 1− q n , then d f =( log 2) lim n nq n+1 a.s. We show how d f is closely related to the type of the simple random walks (SRW) on trees. We prove that the SRW is a.s. transient if d f>2 a.s. and a.s. recurrent if d f<2 a.s. and if d f=2 a.s. we obtain a.s. transience or recurrence. We also consider another type of random trees which are corresponding to branching processes in varying environments. In particular, we consider a tree such that the degrees of the vertices at distance n from the root are independent identically distributed (iid) random variables following the distribution of a random variable that takes the value 3 or 2 with probabilities q n and 1− q n respectively. These iid random variables are also independent of the degrees of the vertices of the other generations. We prove that the SRW is a.s. recurrent if and only if d f⩽2 a.s. We also prove for such trees that d f = lim n nq n+1 a.s.