Attracting edge property for a class of reinforced random walks

Abstract

Using martingale techniques and comparison with the generalized Urn scheme, it is shown that the edge reinforced random walk on a graph of bounded degree, with the weight function $W(k) = k^\rho,\,\rho > 1 $, traverses (crosses) a random attracting edge at all large times. If the graph is a triangle, the above result is in agreement with a conjecture of Sellke.