The Directed Edge Reinforced Random Walk: The Ant Mill Phenomenon

TL;DR

This paper introduces a directed edge reinforced random walk inspired by the Ant Mill phenomenon, showing that on most graphs it gets trapped in cycles, while on the integer line it escapes to infinity with a predictable behavior.

Contribution

It defines a new model of reinforced random walk motivated by biological phenomena and proves its long-term behavior on various graph structures.

Findings

On finite non-tree graphs, the walk almost surely gets trapped in a directed cycle.

On  with da0a2, the walk escapes to infinity and follows a law of large numbers.

The model provides a mathematical explanation for the Ant Mill phenomenon.

Abstract

We define here a \textit{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 \textit{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 \textit{Ant RW}. Our main result justifies this name. Namely, we will show that on any finite graph which is not a tree, and on $\mathbb Z^d$ with $d\geq 2$, the Ant RW almost surely gets eventually trapped into some directed circuit which will be followed forever. In the case of~$\mathbb Z$ 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.