Random Walks on Directed Covers of Graphs

TL;DR

This paper extends the theory of directed covers of finite graphs to infinite graphs, analyzing random walk behavior, growth rates, and entropy, revealing new phenomena and providing explicit formulas.

Contribution

It introduces a comprehensive classification of random walks on infinite directed covers, including recurrence, transience, and entropy, with new formulas and examples.

Findings

Recurrence is equivalent to the almost sure extinction of a branching process.

The rate of escape and asymptotic entropy are explicitly calculated.

Asymptotic entropy is positive if and only if the walk is transient.

Abstract

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates do not longer coincide in general. Furthermore, the behaviour of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.