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Algebra and Discrete Mathematics, 2012, Volume 13, Issue 2, Pages 237–272
(Mi adm76)
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This article is cited in 2 scientific papers (total in 2 papers)
RESEARCH ARTICLE
The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters
R. I. Grigorchuka, Ya. S. Krylyukb a Department of Mathematics, Mailstop 3368 Texas A&M University College Station, TX 77843-3368, USA
b Mathematics Department, De Anza College, 21250 Stevens Creek Blvd, Cupertino, CA 95014, USA
Abstract:
It is shown that the KNS-spectral measure of the typical Schreier graph of the action of $3$-generated $2$-group of intermediate growth constructed by the first author in 1980 on the boundary of binary rooted tree
coincides with the Kesten’s spectral measure, and coincides (up to affine transformation of $\mathbb R$)
with the density of states of the corresponding diatomic linear chain.
Jacoby matrix associated with Markov operator of simple random walk on these graphs is computed. It shown shown that KNS and Kesten's spectral measures of the Schreier graph based on the orbit of the point
$1^{\infty}$ are different but have the same support and are absolutely continuous with respect to the Lebesgue measure.
Keywords:
group of intermediate growth, diatomic linear chain, random walk, spectral measure, Schreier graph, discrete Laplacian.
Received: 03.04.2012 Accepted: 03.04.2012
Citation:
R. I. Grigorchuk, Ya. S. Krylyuk, “The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters”, Algebra Discrete Math., 13:2 (2012), 237–272
Linking options:
https://www.mathnet.ru/eng/adm76 https://www.mathnet.ru/eng/adm/v13/i2/p237
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Abstract page: | 383 | Full-text PDF : | 169 | References: | 60 | First page: | 1 |
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