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.
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
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Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs (1999). http://www.stat.berkeley.edu/aldous/RWG/book.html
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
Levin, D.A., Peres, Y., Wilmer, E.L.: Markov chains and mixing times. American Mathematical Society, Providence (2006)
Janson S., Łuczak T., Ruciński A.: Random graphs. In: Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (2000)
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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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DOI: https://doi.org/10.1007/s00440-011-0403-2
Mathematics Subject Classification (2000)
- 05C81
- 60J05