Recurrence of Reinforced Random Walk on a Ladder
Thomas Sellke, Purdue University
Abstract
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 δ > 0 to the weight of an edge upon first
crossing, with no reinforcement thereafter. This paper proves recurrence
for all δ > 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has caused the original unpublished manuscript to be frequently cited,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.
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