ATTRACTING EDGE PROPERTY FOR A CLASS OF REINFORCED RANDOM WALKS

Vlada Limic

Using martingale techniques and comparison with a Urn-type process, 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 $, 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 (1994).