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On the eigenspaces of lamplighter random walks and percolation clusters on graphs

Author(s): Franz Lehner
Journal: Proc. Amer. Math. Soc. 137 (2009), 2631-2637.
MSC (2000): Primary 43A05, 47B80, 60K35
Posted: March 17, 2009
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Abstract: We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete orthonormal basis consisting of finitely supported eigenfunctions.

References:

1.: Laurent Bartholdi and Wolfgang Woess, Spectral computations on lamplighter groups and Diestel-Leader graphs, J. Fourier Anal. Appl. 11 (2005), no. 2, 175-202. MR 2131635 (2006e:20052)
2.: Rostislav I. Grigorchuk, Peter Linnell, Thomas Schick, and Andrzej Żuk, On a question of Atiyah, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 9, 663-668. MR 1797748 (2001m:57050)
3.: Franz Lehner, Markus Neuhauser, and Wolfgang Woess, On the spectrum of lamplighter groups and percolation clusters, Math. Ann. 342 (2008), no. 1, 69-89. MR 2415315

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Additional Information:

Franz Lehner
Affiliation: Institut für Mathematische Strukturtheorie, Steyrergasse 30, A-8010 Graz, Austria
Email: lehner@finanz.math.tu-graz.ac.at

DOI: 10.1090/S0002-9939-09-09869-4
PII: S 0002-9939(09)09869-4
Keywords: Wreath product, percolation, random walk, spectral measure, point spectrum, eigenfunctions
Received by editor(s): July 9, 2008
Posted: March 17, 2009
Communicated by: Marius Junge
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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