The Directed Edge Reinforced Random Walk: The Ant Mill Phenomenon

We define here a directed edge reinforced random walk on a connected locally finite graph. As the name suggests, this walk keeps track of its past, and gives a bias towards directed edges previously crossed proportional to the exponential of the number of crossings. The model is inspired by the so called Ant Mill phenomenon, in which a group of army ants forms a continuously rotating circle until they die of exhaustion. For that reason we refer to the walk defined in this work as the Ant RW. Our main result justifies this name. Namely, we will show that on any finite graph which is not a tree, and on Z(d) with d >= 2, the Ant RW almost surely gets eventually trapped into some directed circuit which will be followed forever. In the case of Z we show that the Ant RW eventually escapes to infinity and satisfies a law of large number with a random limit which we explicitly identify.