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

@article{AIF_1991__41_2_467_0,
author = {Kuhn, M. Gabriella},
title = {Random walks on free products},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
volume = {41},
number = {2},
year = {1991},
pages = {467-491},
doi = {10.5802/aif.1261},
zbl = {0725.60009},
mrnumber = {93a:43008},
language = {en},
url = {http://www.numdam.org/item/AIF_1991__41_2_467_0}
}

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. http://www.numdam.org/item/AIF_1991__41_2_467_0/

[A] K. Aomoto, Spectral theory on a free group and algebraic curves, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 31 (1984), 297-317. | MR 86m:58127 | Zbl 0583.60068

[AK] K. Aomoto, Y. Kato, Green functions and spectra on free products of cyclic groups, Annales Inst. Fourier, 38-1 (1988), 59-85. | Numdam | MR 89m:58201 | Zbl 0639.60008

[CF-T] C. Cecchini, A. Figá-Talamanca, Projections of uniqueness for Lp(G), Pacific J. of Math., 51 (1974), 34-37. | MR 52 #14849 | Zbl 0252.43007

[CS1] D. I. Cartwright, P. M. Soardi, Harmonic analysis on the free product of two cyclic groups, J. Funct. Anal., 65 (1986), 147-171. | MR 87m:22015 | Zbl 0619.43003

[CS2] D. I. Cartwright, P. M. Soardi, Random walks on free products, quotient and amalgams, Nagoya Math. J., 102 (1986), 163-180. | MR 88i:60120a | Zbl 0592.60052

[CT] J. M. Cohen, A. R. Trenholme, Orthogonal polynomials with a constant recursion formula and an application to harmonic analysis, J. Funct. Anal., 59 (1984), 175-184. | MR 86d:42024 | Zbl 0549.43002

[DM] E. D. Dynkin, M. B. Malyutov, Random walk on groups with a finite number of generators, Sov. Math. Dokl., 2 (1961), 399-402. | Zbl 0214.44101

[DS] N. Dunford, J. T. Schwartz, Linear Operators, Interscience, New York, 1963.

[F-TS] A. Figá-Talamanca, T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, to appear in Memoirs A.M.S. | Zbl 0836.43019

[IP] A. Iozzi, M. Picardello, Spherical functions on symmetrical graphs, Harmonic Analysis, Proceedings Cortona, Italy, Springer Lecture Notes in Math.

[K] G. Kuhn, Anisotropic random walks on the free product of cyclic groups, irreducible representations and indempotents of C*reg(G), preprint. | Zbl 0767.22001

[K-S] G. Kuhn, T. Steger, Restrictions of the special representation of Aut(Trees) to two cocompact subgroups, to appear in Rocky Moutain J. | Zbl 0795.22004

[M-L] Mclaughlin, Random walks and convolution operators on free products, Doctoral Dissertation, New York University.

[S] T. Steger, Harmonic analysis for anisotropic random walks on homogeneous trees, Doctoral Dissertation, Washington University, St. Louis. | Zbl 0836.43019

[T1] A. R. Trenholme, Maximal abelian subalgebras of function algebras associated with free products, J. Funct. Anal., 79 (1988), 342-350. | MR 90c:46073 | Zbl 0665.46049

[T2] A. R. Trenholme, A Green's function for non-homogeneous random walks on free products, Math. Z., 199 (1989), 425-441. | MR 90d:60012 | Zbl 0638.60010

[W1] W. Woess, Nearest neighbour random walks on free products of discrete groups, Boll. U.M.I., 65 B (1986), 961-982. | MR 88i:60120b | Zbl 0627.60012

[W2] W. Woess, Context-free language and random walks on groups, Discrete Math., 64 (1987), 81-87. | MR 88m:60020 | Zbl 0637.60014