Random Walks on Directed Covers of Graphs
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 no longer coincide in general. Furthermore, the behavior 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.
KeywordsTrees Random walk Recurrence Transience Upper Collatz–Wielandt number Branching process Rate of escape Asymptotic entropy
Mathematics Subject Classification (2000)05C05 60J10 60F05 60J85
Unable to display preview. Download preview PDF.
- 4.Björklund, M.: Central limit theorems for Gromov hyperbolic groups. J. Theor. Probab. doi: 10.1007/s10959-009-0230-x (2009, to appear)
- 5.Blachère, S., Brofferio, S.: Internal diffusion limited aggregation on discrete groups having exponential growth. Preprint available at arXiv:math/0507582v1 [math.PR] (2005)
- 13.Gantert, N., Müller, S., Popov, S., Vachkovskaia, M.: Survival of branching random walks in random environment. J. Theor. Probab. doi: 10.1007/s10959-009-0227-5 (2009, to appear)
- 18.Lyons, R., Peres, Y.: Probability on Trees and Networks. Current version available at http://mypage.iu.edu/~rdlyons/ (2009, in preparation)
- 26.Zeitouni, O.: Random walks in random environment. In: Lectures on Probability Theory and Statistics. Lecture Notes in Math., vol. 1837, pp. 189–312. Springer, Berlin (2004) Google Scholar