Vertex-reinforced jump processes on trees and finite graphs

Abstract

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 δ, plus 1, where δ is a positive constant. We show that for fixed b > 1, if δ is large enough the process is transient, and if δ is close enough to zero it is recurrent. Related results for some other graphs and trees are also proved.