Reinforced random walks

Edge-reinforced random walk was introduced by Diaconis in the late 1980s. Diaconis asked for which values of d the edge-reinforced random walk on $Z^d$ is recurrent. This question is stillopen for all $d \ge 2$.

In this talk, I will present recent results for edge-reinforced random walk on ladders. These include recurrence results and limit theorems. The analysis is based on 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 analyze this Gibbsian process.

This is joint work with Franz Merkl.