Random walks on free products

Kuhn, M. Gabriella

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

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.

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 $\mu$.

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