The Annals of Probability

Cutpoints and resistance of random walk paths

Itai Benjamini, Ori Gurel-Gurevich, and Oded Schramm

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Abstract

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].

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 1122-1136.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281734

Digital Object Identifier
doi:10.1214/10-AOP569

Mathematical Reviews number (MathSciNet)
MR2789585

Zentralblatt MATH identifier
1223.60012

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G50: Sums of independent random variables; random walks

Keywords
Graph random walk path cutpoints

Citation

Benjamini, Itai; Gurel-Gurevich, Ori; Schramm, Oded. Cutpoints and resistance of random walk paths. Ann. Probab. 39 (2011), no. 3, 1122--1136. doi:10.1214/10-AOP569. https://projecteuclid.org/euclid.aop/1300281734


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References

  • [1] Benjamini, I. and Gurel-Gurevich, O. Almost sure recurrence of the simple random walk path. Available at http://arxiv.org/abs/math/0508270.
  • [2] Benjamini, I., Gurel-Gurevich, O. and Lyons, R. (2007). Recurrence of random walk traces. Ann. Probab. 35 732–738.
  • [3] Csáki, E., Földes, A. and Révész, P. On the number of cutpoints of the transient nearest neighbor random walk on the line. J. Theoret. Probab. 23 624–638.
  • [4] James, N., Lyons, R. and Peres, Y. (2008). A transient Markov chain with finitely many cutpoints. In Probability and Statistics: Essays in Honor of David A. Freedman. Inst. Math. Stat. Collect. 2 24–29. Inst. Math. Statist., Beachwood, OH.
  • [5] James, N. and Peres, Y. (1996). Cutpoints and exchangeable events for random walks. Teor. Veroyatnost. i Primenen. 41 854–868.
  • [6] Lawler, G. F. (1996). Cut times for simple random walk. Electron. J. Probab. 1 24 pp. (electronic).
  • [7] Thomassen, C. (1992). Isoperimetric inequalities and transient random walks on graphs. Ann. Probab. 20 1592–1600.