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.
"Recurrence of edge-reinforced random walk on a two-dimensional graph." Ann. Probab. 37 (5) 1679 - 1714, September 2009. https://doi.org/10.1214/08-AOP446