Vertex-reinforced jump processes on trees and finite graphs

We study the continuous time process on the vertices of the b-ary tree which jumps to each nearest neighbor vertex at the rate of the time already spent at that vertex times delta, plus 1, where delta is a positive constant. We show that for fixed b>1, if delta is large enough the process is transient, and if delta is close enough to zero it is recurrent. Related results for some other graphs and trees are also proved.