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Attracting edge property for a class of reinforced random walks. (English) Zbl 1057.60048
The author considers the so-called edge-reinforced random walk with weight function $$W$$ on a connected (unoriented) graph of bounded degree. That is, the walker jumps along the edges, and the probability along a given edge is given by the quotient of the value of $$W$$ of the number of previous jumps along that edge, divided by the sum of all the $$W$$-values of the number of previous jumps along all the adjacent edges of the present site. Here $$W$$ is a positive weight function, which is assumed to be $$W(k)=k^\rho$$ for some $$\rho>1$$.
It was previously known that, for the graph $$\mathbb Z^ d$$, under the condition $$\sum_{k\in\mathbb N}1/W(k)<\infty$$, there is almost surely one (random) attracting edge which is traversed infinitely often. The present paper strengthens this result for arbitrary bounded-degree graphs with $$W(k)=k^\rho$$ for some $$\rho>1$$, by showing that this edge is unique, almost surely. The main tools are martingale techniques and a comparison to the generalized urn scheme. The assumption that $$W(k)=k^\rho$$ is crucial for most of the arguments.

##### MSC:
 60G50 Sums of independent random variables; random walks
##### Keywords:
reinforced random walk; supermartingale; coupling; urn
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##### References:
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