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

Florian Sobieczky ()
Additional contact information
Florian Sobieczky: TU-Graz

Journal of Theoretical Probability, 2010, vol. 23, issue 3, 639-670

Abstract: 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.

Keywords: Random walks; Random walks on random partial graphs; Percolation; Critical percolation; Heat kernel decay; Return probability; Comparison theorems; 60K37; 60B99; 60D05; 60J10; 60J27 (search for similar items in EconPapers)
Date: 2010
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-010-0298-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:23:y:2010:i:3:d:10.1007_s10959-010-0298-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-010-0298-3

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().