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.
Editor's Note. A draft of this paper was written in 1994. The paper is one of the first to make any progress on this type of reinforcement problem. It has motivated a substantial number of new and sometimes quite difficult studies of reinforcement models in pure and applied probability. The persistence of interest in models related to this has caused the original unpublished manuscript to be frequently cited, despite its lack of availability and the presence of errors. The opportunity to rectify this situation has led us to the somewhat unusual step of publishing a result that may have already entered the mathematical folklore.
"Recurrence of Reinforced Random Walk on a Ladder." Electron. J. Probab. 11 301 - 310, 2006. https://doi.org/10.1214/EJP.v11-313