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Linear cover time is exponentially unlikely
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  • Published: 06 December 2011

Linear cover time is exponentially unlikely

  • Itai Benjamini1,
  • Ori Gurel-Gurevich2 &
  • Ben Morris3 

Probability Theory and Related Fields volume 155, pages 451–461 (2013)Cite this article

  • 241 Accesses

  • 3 Citations

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Abstract

We show that the probability that a simple random walk covers a finite, bounded degree graph in linear time is exponentially small. We conjecture that the same holds for any simple graph.

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Abbreviations

B v (r):

The ball of radius r around v

A v (r) = B v (r) \ B v (r − 1):

The annulus of radius r around v

τ v (r):

The hitting time of A v (r)

\({\ell^v_t}\) :

The number of visits to v before time t

ℓ v(r):

The number of visits to v before time τ v (r)

τ cov(S):

The cover time of S

τ cov :

The cover time of the graph

\({\tau_{\rm cov}^*(S)}\) :

The time to cover and exit S

\({\tau_{\rm cov}^*}\) :

The time to cover the graph and exit \({B_{X_{\tau_{\rm cov}}}(2r)}\)

References

  1. Aldous D.J.: Threshold limits for cover times. J. Theor. Probab. 4, 197–211 (1991). doi:10.1007/BF01047002

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs (1999). http://www.stat.berkeley.edu/aldous/RWG/book.html

  3. Feige U.: A tight lower bound on the cover time for random walks on graphs. Random Struct. Algorithms 6(4), 433–438 (1995). doi:10.1002/rsa.3240060406

    Article  MathSciNet  MATH  Google Scholar 

  4. Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. American Mathematical Society, Providence (2006)

  5. Janson S., Łuczak T., Ruciński A.: Random graphs. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Weizmann Institute of Science, 76100, Rehovot, Israel

    Itai Benjamini

  2. University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Ori Gurel-Gurevich

  3. University of California, Davis, CA, 95616, USA

    Ben Morris

Authors
  1. Itai Benjamini
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  2. Ori Gurel-Gurevich
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  3. Ben Morris
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Corresponding author

Correspondence to Ori Gurel-Gurevich.

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Cite this article

Benjamini, I., Gurel-Gurevich, O. & Morris, B. Linear cover time is exponentially unlikely. Probab. Theory Relat. Fields 155, 451–461 (2013). https://doi.org/10.1007/s00440-011-0403-2

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  • Received: 13 September 2011

  • Revised: 18 November 2011

  • Published: 06 December 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0403-2

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Mathematics Subject Classification (2000)

  • 05C81
  • 60J05
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