Once edge-reinforced random walk on a tree

Abstract

We consider a nearest neighbor walk on a regular tree, with transition probabilities proportional to weights or conductances of the edges. Initially all edges have weight 1, and the weight of an edge is increased to c > 1 when the edge is traversed for the first time. After such a change the weight of an edge stays at c forever. We show that such a walk is transient for all values of c ≥ 1, and that the walk moves off to infinity at a linear rate. We also prove an invariance principle for the height of the walk.