Open Access
May 2011 Cutpoints and resistance of random walk paths
Itai Benjamini, Ori Gurel-Gurevich, Oded Schramm
Ann. Probab. 39(3): 1122-1136 (May 2011). DOI: 10.1214/10-AOP569


We construct a bounded degree graph G, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also prove that the expected number of cutpoints of any transient Markov chain is infinite. This answers two questions of James, Lyons and Peres [A Transient Markov Chain With Finitely Many Cutpoints (2007) Festschrift for David Freedman].

Additionally, we consider a simple random walk on a finite connected graph G that starts at some fixed vertex x and is stopped when it first visits some other fixed vertex y. We provide a lower bound on the expected effective resistance between x and y in the path of the walk, giving a partial answer to a question raised in [Ann. Probab. 35 (2007) 732–738].


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Itai Benjamini. Ori Gurel-Gurevich. Oded Schramm. "Cutpoints and resistance of random walk paths." Ann. Probab. 39 (3) 1122 - 1136, May 2011.


Published: May 2011
First available in Project Euclid: 16 March 2011

zbMATH: 1223.60012
MathSciNet: MR2789585
Digital Object Identifier: 10.1214/10-AOP569

Primary: 60D05 , 60G50

Keywords: cutpoints , graph , path , Random walk

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 3 • May 2011
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