Linearly edge-reinforced random walks

In the 1980ies, Diaconis asked whether linearly edge-reinforced random walk on Z^2 is recurrent. This problem is still open. In the talk, I will give an overview on related recent results on linearly edge-reinforced-random walks. In particular, the main ideas for proving recurrence of linearly edge-reinforced-random walks on some two-dimensional graphs will be presented.