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The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters. (English) Zbl 1297.20025

Summary: 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 Kesten’s spectral measure, and coincides (up to affine transformation of \(\mathbb R\)) with the density of states of the corresponding diatomic linear chain.
Jacobi matrix associated with Markov operator of simple random walk on these graphs is computed. It is 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.

MSC:

20E08 Groups acting on trees
37A50 Dynamical systems and their relations with probability theory and stochastic processes
20F05 Generators, relations, and presentations of groups
60G50 Sums of independent random variables; random walks
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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