Cutpoints and resistance of random walk paths

Itai Benjamini*, Ori Gurel-Gurevich, Oded Schramm

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

The Weizmann Institute of Science, Microsoft Research and Microsoft Research 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].

Original languageAmerican English
Pages (from-to)1122-1136
Number of pages15
JournalAnnals of Probability
Volume39
Issue number3
DOIs
StatePublished - May 2011
Externally publishedYes

Keywords

  • Cutpoints
  • Graph
  • Path
  • Random walk

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