Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T10:41:10.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing Norms of Group-Invariant Transition Operators

Published online by Cambridge University Press:  12 September 2008

Laurent Saloff-Coste  and
Wolfgang Woess
Laurent Saloff-Coste
Affiliation:
CNRS, Statistique et Probabilités, Université Paul Sabatier, 31062 Toulouse Cedex, France
Wolfgang Woess
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider transition operators P on a countable set, which are reversible, irreducible and invariant under a group G of permutations of X with compact point stabilizers. We relate the computation of the spectral radius (norm) of P with the spectral radii of certain matrices defined over the factor set G\X. In various cases, this allows easy computation of the norm of P.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 2 , June 1996 , pp. 161 - 178
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Soardi, P. M. and Woess, W. (1990) Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z 205 471–486.CrossRefGoogle Scholar
[2]Salvatori, M. (1992) On the norms of group-invariant transition operators on graphs. J. Theor. Probab 5 563–576.CrossRefGoogle Scholar
[3]Woess, W. (1991) Topological groups and infinite graphs. Discrete Math 95 373–384.CrossRefGoogle Scholar
[4]Seneta, E. (1981) Non-Negative Matrices and Markov Chains (2nd ed.). Springer-Verlag.CrossRefGoogle Scholar
[5]Kersting, G. (1974) Strong ratio limit property and R -recurrence of reversible Markov chains. Z. Wahrscheinlichkeitsth. verw. Geb 30 343–356.CrossRefGoogle Scholar
[6]Gerl, P. (1988) Random walks on graphs with a strong isoperimetric inequality. J. Theor. Probab 1 171–188.CrossRefGoogle Scholar
[7]Kanai, M. (1985) Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds. J. Math. Soc. Japa 37 391–413.Google Scholar
[8]de la Peña, J. A. and Takane, M. (1992) The spectral radius of the Galois covering of a finite graph. Linear Algebra Appl 160 175–188.CrossRefGoogle Scholar
[9]Sy, P. W. and Sunada, T. (1992) Discrete Schrödinger operators on a graph. Nagoya Math. J 125 141–150.CrossRefGoogle Scholar
[10]Brooks, R. (1981) The fundamental group and the spectrum of the Laplacian. Comm. Math. Helv 56 581–598.CrossRefGoogle Scholar
[11]Schlichting, G. (1979) Polynomidentitäten und Permutationsdarstellungen lokalkompakter Gruppen. Invent. Math 55 97–106.CrossRefGoogle Scholar
[12]Trofimov, V. I. (1985) Automorphism groups of graphs as topological groups. Math. Note 38 171–720.Google Scholar
[13]Leptin, H. (1968) On locally compact groups with invariant means. Proc. Amer. Math. Soc 19 489–494.CrossRefGoogle Scholar
[14]Berg, Ch. and Christensen, J. P. R. (1974) On the relation between amenability of locally compact groups and the norms of convolution operators. Math. Ann 208 149–153.CrossRefGoogle Scholar
[15]Faraut, J. (1975) Moyennabilité et normes d'opérateurs de convolution. Analyse Harmonique sur les Groupes de Lie. Led. Notes in Math, 497, Springer, pp. 153–163.CrossRefGoogle Scholar
[16]Nebbia, C. (1988) On the amenability and the Kunze–Stein property for groups acting on trees. Pacific J. Math 135 371–380.CrossRefGoogle Scholar
[17]Mohar, B. and Woess, W. (1989) A survey on spectra of infinite graphs. Bull. London Math. Soc 21 209–234.CrossRefGoogle Scholar
[18]Woess, W. (1985) Random walks and periodic continued fractions. Adv. in Appl. Probab 17 67–84.CrossRefGoogle Scholar
[19]Gerl, P. (1984) Continued fraction methods for random walks on ℕ and on trees. In: Probability Measures on Groups (ed. Heyer, H.), Led. Notes in Math. 1064, Springer, pp. 131–146.CrossRefGoogle Scholar
[20]Cartwright, I. and Młotkowski, W. (1994) Harmonic analysis for groups acting on triangle buildings. J. Austral. Math. Soc. Ser. 56 345–383.CrossRefGoogle Scholar