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 Zd, 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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