Geodesic
The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters
Algebra and discrete mathematics, Tome 13 (2012) no. 2, pp. 237-272

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{ADM_2012_13_2_a6,
     author = {R. I. Grigorchuk and Ya. S. Krylyuk},
     title = {The spectral measure of the {Markov} operator related to 3-generated 2-group of intermediate growth and its {Jacobi} parameters},
     journal = {Algebra and discrete mathematics},
     pages = {237--272},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2012_13_2_a6/}
}
TY  - JOUR
AU  - R. I. Grigorchuk
AU  - Ya. S. Krylyuk
TI  - The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters
JO  - Algebra and discrete mathematics
PY  - 2012
SP  - 237
EP  - 272
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2012_13_2_a6/
LA  - en
ID  - ADM_2012_13_2_a6
ER  - 
%0 Journal Article
%A R. I. Grigorchuk
%A Ya. S. Krylyuk
%T The spectral measure of the Markov operator related to 3-generated 2-group of intermediate growth and its Jacobi parameters
%J Algebra and discrete mathematics
%D 2012
%P 237-272
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2012_13_2_a6/
%G en
%F ADM_2012_13_2_a6
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 and discrete mathematics, Tome 13 (2012) no. 2, pp. 237-272. http://geodesic.mathdoc.fr/item/ADM_2012_13_2_a6/