Abstract
There is a natural connection between electrical networks and so called reversible Markov chains. An example for such a chain is the symmetric graph random walk which, in each step, jumps to a randomly chosen graph neighbor at equal probability. This connection is studied here in some detail. As an application, we prove the statement that if such a graph random walk is recurrent, then it is recurrent also on each subgraph. (Although this statement is rather plausible, it is hard to show by different means.) In particular, the graph random walk on a percolation cluster of the planar integer lattice is recurrent.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Doyle PG, Snell JL (1984) Random walks and electric networks. Carus mathematical monographs, vol 22. Mathematical Association of America, Washington
Hughes BD (1995) Random walks and random environments, vol 1. Oxford science publications. The Clarendon Press/Oxford University Press, New York. Random walks
Hughes BD (1996) Random walks and random environments, vol 2. Oxford science publications. The Clarendon Press/Oxford University Press, New York. Random environments
Kesten H, Kozlov MV, Spitzer F (1975) A limit law for random walk in a random environment. Compos Math 30:145–168
Lyons R, Peres Y (2012) Probability on trees. Preliminary version on the Internet at http://mypage.iu.edu/~rdlyons/
Révész P (2005) Random walk in random and non-random environments, 2nd edn. World Scientific, Hackensack
Solomon F (1975) Random walks in a random environment. Ann Probab 3:1–31
Zeitouni O (2004) Random walks in random environment. In: Lectures on probability theory and statistics. Lecture notes in mathematics, vol 1837. Springer, Berlin, pp 189–312
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag London
About this chapter
Cite this chapter
Klenke, A. (2014). Markov Chains and Electrical Networks. In: Probability Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5361-0_19
Download citation
DOI: https://doi.org/10.1007/978-1-4471-5361-0_19
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5360-3
Online ISBN: 978-1-4471-5361-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)