Attracting edge property for a class of reinforced random walks (Q1431504)

The author considers the so-called edge-reinforced random walk with weight function \(W\) on a connected (unoriented) graph of bounded degree. That is, the walker jumps along the edges, and the probability along a given edge is given by the quotient of the value of \(W\) of the number of previous jumps along that edge, divided by the sum of all the \(W\)-values of the number of previous jumps along all the adjacent edges of the present site. Here \(W\) is a positive weight function, which is assumed to be \(W(k)=k^\rho\) for some \(\rho>1\). It was previously known that, for the graph \(\mathbb Z^ d\), under the condition \(\sum_{k\in\mathbb N}1/W(k)<\infty\), there is almost surely one (random) attracting edge which is traversed infinitely often. The present paper strengthens this result for arbitrary bounded-degree graphs with \(W(k)=k^\rho\) for some \(\rho>1\), by showing that this edge is unique, almost surely. The main tools are martingale techniques and a comparison to the generalized urn scheme. The assumption that \(W(k)=k^\rho\) is crucial for most of the arguments.