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An Interlacing Technique for Spectra of Random Walks and Its Application to Finite Percolation Clusters

Author

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  • Florian Sobieczky

    (TU-Graz
    FSU-Jena)

Abstract

A comparison technique for random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the transition matrix other than just the principal eigenvalue. As an application, an upper bound of the expected return probability of a random walk with symmetric transition probabilities is found. In this case, the state space is a random partial graph of a regular graph of bounded geometry and transitive automorphism group. The law of the random edge-set is assumed to be invariant with respect to some transitive subgroup of the automorphism group (‘invariant percolation’). Given that this subgroup is unimodular, it is shown that this invariance strengthens the upper bound of the expected return probability, compared with standard bounds such as those derived from the Cheeger inequality. The improvement is monotone in the degree of the underlying transitive graph.

Suggested Citation

  • Florian Sobieczky, 2010. "An Interlacing Technique for Spectra of Random Walks and Its Application to Finite Percolation Clusters," Journal of Theoretical Probability, Springer, vol. 23(3), pages 639-670, September.
    • Handle: RePEc:spr:jotpro:v:23:y:2010:i:3:d:10.1007_s10959-010-0298-3
    DOI: 10.1007/s10959-010-0298-3
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    References listed on IDEAS

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    1. Haemers, W.H., 1995. "Interlacing eigenvalues and graphs," Other publications TiSEM 35c08207-2c5c-4387-aaf5-2, Tilburg University, School of Economics and Management.
    2. Mark Brown, 1999. "Interlacing Eigenvalues in Time Reversible Markov Chains," Mathematics of Operations Research, INFORMS, vol. 24(4), pages 847-864, November.
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