Random walks on free products

Kuhn, M. Gabriella

doi:10.5802/aif.1261

Kuhn, M. Gabriella

Annales de l'Institut Fourier, Tome 41 (1991) no. 2, pp. 467-491.

Résumé
Abstract

Soit $G = *_{j = 1}^{q + 1} G_{n_{j} + 1}$ le produit libre de $q + 1$ groupes finis d’ordre $n_{j} + 1$ , et $μ$ la probabilité prenant la valeur $p_{j} / n_{j}$ sur chaque élément de $G_{n_{j} + 1} ∖ {e}$ . Nous décrivons ici le spectre ponctuel de $μ$ sur $C_{reg}^{*} (G)$ . On montre en particulier que ce spectre ponctuel apparaît pour certains choix des nombres $n_{j}$ , et les espaces propres correspondants dans $l^{2}$ sont décrits. Enfin, on obtient une décomposition de la représentation régulière de $G$ à l’aide de la fonction de Green de $μ$ , cette décomposition étant irréductible si, et seulement si, $μ$ n’a pas de sous-espace propre.

Let $G = *_{j = 1}^{q + 1} G_{n_{j} + 1}$ be the product of $q + 1$ finite groups each having order $n_{j} + 1$ and let $μ$ be the probability measure which takes the value $p_{j} / n_{j}$ on each element of $G_{n_{j} + 1} ∖ {e}$ . In this paper we shall describe the point spectrum of $μ$ in $C_{reg}^{*} (G)$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $n_{j}$ . We also compute the continuous spectrum of $μ$ in $C_{reg}^{*} (G)$ in several cases. A family of irreducible representations of $G$ , parametrized on the continuous spectrum of $μ$ , is here presented. Finally, we shall get a decomposition of the regular representation of $G$ by means of the Green function of $μ$ and the decomposition is into irreducibles if and only if there are no true eigenspaces for $μ$ .

MR Zbl

DOI : 10.5802/aif.1261

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     author = {Kuhn, M. Gabriella},
     title = {Random walks on free products},
     journal = {Annales de l'Institut Fourier},
     pages = {467--491},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {2},
     year = {1991},
     doi = {10.5802/aif.1261},
     mrnumber = {93a:43008},
     zbl = {0725.60009},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1261/}
}

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Kuhn, M. Gabriella. Random walks on free products. Annales de l'Institut Fourier, Tome 41 (1991) no. 2, pp. 467-491. doi : 10.5802/aif.1261. https://www.numdam.org/articles/10.5802/aif.1261/

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