Linearly edge-reinforced random walks, part I

We consider a linearly edge-reinforced random walk

on a class of two-dimensional graphs with constant

initial weights. The graphs are obtained

from Z^2 by replacing every edge by a sufficiently large, but fixed

number of edges in series.

We prove that a 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.

Part I will also include an overview on the history of the model.

In part II, some more details about the proofs will be explained.