Once reinforced walk on a regular tree

Abstract:

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 this walk is transient for all values of c > 1, and moreover that it has a positive speed. We also prove an invariance principle for the height of the walk. This is a joint work with Rick Durrett and Harry Kesten.