Speed of Vertex-Reinforced Jump Process on Galton–Watson Trees

Abstract

We give an alternative proof of the fact that the vertex-reinforced jump process on Galton–Watson tree has a phase transition between recurrence and transience as a function of $$c$$ , the initial local time, see Basdevant et al. (Ann Appl Probab 22(4):1728–1743, 2012). Further, applying techniques in Aidekon (Probab Theory Relat Fields 142(3–4):525–559, 2008), we show a phase transition between positive speed and null speed for the associated discrete-time process in the transient regime.