# ANNALES DE L'INSTITUT FOURIER

Random walks on free products
Annales de l'Institut Fourier, Volume 41 (1991) no. 2, pp. 467-491.

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 $\mu$ be the probability measure which takes the value ${p}_{j}/{n}_{j}$ on each element of ${G}_{{n}_{j}+1}\setminus \left\{e\right\}$. In this paper we shall describe the point spectrum of $\mu$ in ${C}_{\mathrm{reg}}^{*}\left(G\right)$ 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 $\mu$ in ${C}_{\mathrm{reg}}^{*}\left(G\right)$ in several cases. A family of irreducible representations of $G$, parametrized on the continuous spectrum of $\mu$, is here presented. Finally, we shall get a decomposition of the regular representation of $G$ by means of the Green function of $\mu$ and the decomposition is into irreducibles if and only if there are no true eigenspaces for $\mu$.

Soit $G={*}_{j=1}^{q+1}{G}_{{n}_{j}+1}$ le produit libre de $q+1$ groupes finis d’ordre ${n}_{j}+1$, et $\mu$ la probabilité prenant la valeur ${p}_{j}/{n}_{j}$ sur chaque élément de ${G}_{{n}_{j}+1}\setminus \left\{e\right\}$. Nous décrivons ici le spectre ponctuel de $\mu$ sur ${C}_{\mathrm{reg}}^{*}\left(G\right)$. 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 $\mu$, cette décomposition étant irréductible si, et seulement si, $\mu$ n’a pas de sous-espace propre.

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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},
volume = {41},
number = {2},
year = {1991},
doi = {10.5802/aif.1261},
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mrnumber = {93a:43008},
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url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1261/}
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Kuhn, M. Gabriella. Random walks on free products. Annales de l'Institut Fourier, Volume 41 (1991) no. 2, pp. 467-491. doi : 10.5802/aif.1261. https://aif.centre-mersenne.org/articles/10.5802/aif.1261/

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