Recurrence of edge-reinforced random walk on a two-dimensional graph

Franz Merkl; Silke W. W. Rolles

doi:10.1214/08-aop446

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Recurrence of edge-reinforced random walk on a two-dimensional graph

Franz Merkl·Silke W. W. Rolles

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Abstract

We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from $\mathbb{Z}^2$ by replacing every edge by a sufficiently large, but fixed number of edges in series. We prove that the linearly edge-reinforced random walk on these graphs is recurrent. Furthermore, we derive bounds for the probability that the edge-reinforced random walk hits the boundary of a large box before returning to its starting point.

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Franz Merkl, Silke W. W. Rolles. 2009-09-24. Recurrence of edge-reinforced random walk on a two-dimensional graph. https://doi.org/10.1214/08-aop446

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