The Annals of Probability

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

Franz Merkl and Silke W. W. Rolles

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We consider a linearly edge-reinforced random walk on a class of two-dimensional graphs with constant initial weights. The graphs are obtained from ℤ2 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.

Article information

Ann. Probab., Volume 37, Number 5 (2009), 1679-1714.

First available in Project Euclid: 21 September 2009

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Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

Reinforced random walk recurrence hitting probabilities


Merkl, Franz; Rolles, Silke W. W. Recurrence of edge-reinforced random walk on a two-dimensional graph. Ann. Probab. 37 (2009), no. 5, 1679--1714. doi:10.1214/08-AOP446.

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