Reinforced Random Walk on the d-dimensional Integer Lattice

T. Sellke

2008, v.14, Issue 2, 291-308

ABSTRACT

Let $w_k$, $k \geq 0$, be a sequence of positive constants. Consider nearest--neighbor reinforced random walk on the $d$-dimensional integer lattice starting at the origin with edge--weight sequence $\{w_k\}$. In this process, the weight of an edge connecting nearest neighbors in the lattice is $w_k$ if the edge has previously been crossed exactly $k$ times. At integer times, the process jumps to a nearest neighbor of the previous position, with probabilities being proportional to current edge-weights. We show that the individual coordinates return to zero infinitely often if there is positive probability that the range (= set of visited sites) is infinite. If $\sum^\infty_{k=0} w^{-1}_{2k}$ $= \infty$ and $\sum^\infty_{k=0} w^{-1}_{2k+1} = \infty$, then the range is infinite. If both sums are finite, then the process eventually gets stuck crossing the same edge over and over again. If one sum is finite and the other is infinite, then, with probability one, all edges not touching the origin are crossed at most finitely often. Hence, in this last case the process eventually crosses only edges touching the origin, or the distance from the origin diverges over time to infinity. Editor's Note. A draft of this paper dates back to 1994. The paper is one of the first to make any progress on this type of reinforcement problem, and it has motivated a number of other studies of reinforcement models. The original unpublished manuscript has been frequently cited, despite its lack of availability. The opportunity to rectify this situation motivated us to publish a result that may have already entered the mathematical folklore.

Keywords: reinforced random walk,recurrence,transience,phase transition

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