**Edge-reinforced random walks**

In 1987, Persi Diaconis introduced edge-reinforced random walks
as a model for a stochastic process in a self-generated stochastic
environment. I spite of its elementary definition, it is hard
to derive properties of these random walks.
In particular, a question asked by Diaconis in the late eighties,
whether the edge reinforced random walk on **Z**^{d},
*d* > 1, is recurrent, i.e. whether it
returns to its starting point almost surely, remains unsolved.

Recently, Silke Rolles and F.M. have proven
that the edge-reinforced random walk on the ladder
**Z** x {1,2} with initial weights *a* > 3/4 is recurrent. The
proof uses a representation of the edge-reinforced random
walk on a finite piece of the ladder as a random walk in a random
environment.
This environment is given by a marginal
of a multi-component Gibbsian process. A transfer operator
technique and entropy estimates from statistical mechanics are
used to analyse this Gibbsian process.
The talk will explain these techniques and show how they
are applied to derive properties of the edge reinforced random walk.

*Franz Merkl*

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