On the transience of processes defined on Galton–Watson trees

Abstract

We introduce a simple technique for proving the transience of certain processes defined on the random tree G generated by a supercritical branching process. We prove the transience for once-reinforced random walks on G, that is, a generalization of a result of Durrett, Kesten and Limic [Probab. Theory Related Fields 122 (2002) 567-592]. Moreover, we give a new proof for the transience of a family of biased random walks defined on G. Other proofs of this fact can be found in [Ann. Probab. 16 (1988) 1229-1241] and [Ann. Probab. 18 (1990) 931-958] as part of more general results. A similar technique is applied to a vertex-reinforced jump process. A by-product of our result is that this process is transient on the 3-ary tree. Davis and Volkov [Probab. Theory Related Fields 128 (2004) 42-62] proved that a vertex-reinforced jump process defined on the b-ary tree is transient if b ≥ 4 and recurrent if b = 1. The case b = 2 is still open.

Phase transition for the Once-reinforced random walk on $\mathbb{Z}^{d}$-like trees

“…When a is large enough, this probability is small compared to the number of points in the sphere of radius 2 k+1 and this enables us to conclude recurrence. The proof for transience is more subtle, adapting a very nice technique due to Collevecchio [3]. The idea is to consider the walk after it has reached level 2 kn 0 , for some constant integer n 0 , and observe the number Z k of children at level 2 (k+1)n 0 that are hit before the walk goes back to the ancestor at level 2 kn 0 −1 (assuming that the walk eventually comes back to this ancestor).…”

“…Sellke [20] proved that the ORRW is almost surely recurrent on the ladder Z × {1, ..., d} for a ∈ (1, (d − 1)/(d − 2)), see also [22]. In contrast with the ERRW, Durrett, Kesten and Limic [9] showed that the ORRW is transient on regular trees for any a > 1, which was later generalized to any supercritical tree by Collevecchio [3]. Until now, there was no example of graph on which the ORRW exhibits a phase transition and, among the results available so far, there is no good indication that a phase transition occurs.…”

Phase diagram for once-reinforced random walks on trees with exponential weighting scheme

“…Dai (2005) showed that this walk is also transient a.s. on supercritical Galton-Watson trees. Collevecchio (2006) gave a unified approach to show transience on supercritical Galton-Watson trees, which can be applied to some other stochastic processes. On the other hand, the oncereinforced random walk on the ladder Z × {0, 1} is recurrent, which is proved by Sellke (2006).…”

Reversibility of a simple random walk on periodic trees

“…(In the rest of this paper, we shall write LPP for short for Lyons, Pemantle, Peres. ) If T has the distribution of a Galton-Watson tree with offspring distribution {p k } k∈N , then T is infinite with positive probability when kp k > 1, and in this case simple random walk on T is transient almost surely given that T is infinite (see Collevecchio (2006) for a short proof of this fact). In LPP (1995a), it was shown that the speed of simple random walk on T is a constant number almost surely, and an explicit formula was given for the speed.…”