Recurrence of reinforced random walk on a ladder

Consider reinforced random walk on a graph that looks like a doubly infinite ladder. All edges have initial weight 1, and the reinforcement convention is to add delta > 0 to the weight of an edge upon first crossing, with no reinforcement thereafter. This paper proves recurrence for all delta > 0. In so doing, we introduce a more general class of processes, termed multiple-level reinforced random walks.