×

zbMATH — the first resource for mathematics

Harmonic analysis on perturbed Cayley trees. (English) Zbl 1229.47020
J. Funct. Anal. 261, No. 3, 604-634 (2011); corrigendum ibid. 262, No. 10, 4634-4637 (2012).
The author gives an analysis of the mathematical properties of non-homogeneous networks when zero density perturbations are added to homogeneous Cayley trees. For some specific perturbations one is able to write down and, in some cases, solve the secular equation. The Perron Frobenius eigenvector is obtained as the infinite volume limit of the finite volume Perron Frobenius eigenvectors and there is an extensive discussion of its properties. Conditions under which the perturbed graph is recurrent or transient are indicated, too. The whole investigation in this paper is related to the results obtained in [F. Fidaleo, D. Guido and T. Isola, “Bose Einstein condensation on inhomogeneous amenable graphs”, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14, No. 2, 149–197 (2011; Zbl 1223.82012)].

MSC:
47A55 Perturbation theory of linear operators
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
94C15 Applications of graph theory to circuits and networks
05C90 Applications of graph theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Accardi, L.; Ben Ghorbal, A.; Obata, N., Monotone independence, comb graphs and Bose-Einstein condensation, Infin. dimens. anal. quantum probab. relat. top., 7, 419-435, (2004) · Zbl 1073.81057
[2] Bardeen, J.; Cooper, A.N.D.; Schrieffer, J.R., Microscopic theory of superconductivity, Phys. rev., 106, 162-164, (1957)
[3] van den Berg, M.; Dorlas, T.C.; Priezzhev, V.B., The boson gas on a Cayley tree, J. stat. phys., 69, 307-328, (1992) · Zbl 0893.60094
[4] Bratteli, O.; Robinson, D.W., Operator algebras and quantum statistical mechanics II, (1981), Springer Berlin-Heidelberg-New York · Zbl 0463.46052
[5] Burioni, R.; Cassi, D.; Rasetti, M.; Sodano, P.; Vezzani, A., Bose-Einstein condensation on inhomogeneous complex networks, J. phys. B, 34, 4697-4710, (2001)
[6] Feller, W., An introduction to probability theory and its applications II, (1960), John Wiley & Sons New York · Zbl 0138.10207
[7] F. Fidaleo, Harmonic analysis on perturbed Cayley Trees and the Bose Einstein condensation, in preparation. · Zbl 1273.82011
[8] F. Fidaleo, D. Guido, T. Isola, Bose Einstein condensation on inhomogeneous amenable graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top., doi:10.1142/S0219025711004389. · Zbl 1223.82012
[9] F. Fidaleo, D. Guido, T. Isola, work in preparation.
[10] Figà-Talamanca, A.; Picardello, M.A., Harmonic analysis on free groups, (1983), Marcel Dekker, Inc. New York-Basel · Zbl 0536.43001
[11] Jaeck, T.; Pulé, J.; Zagrebnov, V., On the nature of the Bose-Einstein condensation in disordered systems, J. stat. phys., 137, 19-55, (2009) · Zbl 1177.82017
[12] Landau, L.D.; Lifshitz, E.M., Statistical physics, (1980), Pergamon Press Oxford-New York · Zbl 0080.19702
[13] Lenoble, O.; Zagrebnov, V., Bose-Einstein condensation in the Luttinger-sy model, Markov process. related fields, 13, 441-468, (2007) · Zbl 1181.82005
[14] Mohar, B.; Woess, W., A survey on spectra of infinite graphs, Bull. lond. math. soc., 21, 209-234, (1982) · Zbl 0645.05048
[15] Pastur, L.; Figotin, A., Spectra of random and almost-periodic operators, (1992), Springer-Verlag Berlin · Zbl 0752.47002
[16] Royden H, L., Real analysis, (1968), Collier-Mc Millian New York · Zbl 0197.03501
[17] Rudin, W., Real and complex analysis, (1986), Mc Graw-Hill New York
[18] Seneta, E., Nonnegative matrices and Markov chains, (1981), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0471.60001
[19] Serre, J.P., Trees, (1980), Springer-Verlag Berlin-Heidelberg-New York
[20] Silvestrini, P.; Russo, R.; Corato, V.; Ruggiero, B.; Granata, C.; Rombetto, S.; Russo, M.; Cirillo, M.; Trombettoni, A.; Sodano, P., Topology-induced critical current enhancement in Josephson networks, Phys. lett. A, 370, 499-503, (2007)
[21] Takesaki, M., Theory of operator algebras I, (1979), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0990.46034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.