Random walks on the affine group of local fields and of homogeneous trees

Cartwright, Donald I. ; Kaimanovich, Vadim A. ; Woess, Wolfgang

Annales de l'Institut Fourier, Tome 44 (1994), p. 1243-1288 / Harvested from Numdam

Résumé

Le groupe affine d’un corps local ${\frak F}$ agit sur l’arbre ${\Bbb T}({\frak F})$ (l’immeuble de Bruhat-Tits de ${\rm GL} (2,{\frak F})$) en ayant un point fixe dans l’espace des bouts $\partial {\Bbb T}(F)$. Plus généralement, nous définissons le groupe affine $\operatorname{Aff}({\frak F})$ d’un arbre homogène ${\Bbb T}$ comme le groupe de tous les automorphismes de ${\Bbb T}$ ayant un point fixe commun dans $\partial {\Bbb T}$, et établissons les principales propriétés asymptotiques des produits aléatoires dans $\operatorname{Aff} ({\frak F})$ : (1) la loi des grands nombres et le théorème limite central; (2) la convergence vers $\partial {\Bbb T}$ et l’existence d’une solution au problème de Dirichlet à l’infini; (3) l’identification de la frontière de Poisson avec $\partial {\Bbb T}$ donnant une description de l’espace des fonctions $\mu $-harmoniques bornées. Les méthodes utilisées sont étroitement reliées aux propriétés géométriques des arbres homogènes analogues à celles des espaces symétriques de rang un.

Abstract

The affine group of a local field acts on the tree ${\Bbb T}({\frak F})$ (the Bruhat-Tits building of ${\rm GL} (2,{\frak F})$) with a fixed point in the space of ends $\partial {\Bbb T}(F)$. More generally, we define the affine group $\operatorname{Aff}({\frak F})$ of any homogeneous tree ${\Bbb T}$ as the group of all automorphisms of ${\Bbb T}$ with a common fixed point in $\partial {\Bbb T}$, and establish main asymptotic properties of random products in $\operatorname{Aff} ({\frak F})$: (1) law of large numbers and central limit theorem; (2) convergence to $\partial {\Bbb T}$ and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary with $\partial {\Bbb T}$, which gives a description of the space of bounded $\mu $-harmonic functions. Our methods strongly rely on geometric properties of homogeneous trees as discrete counterparts of rank one symmetric spaces.

Détail

Publié le : 1994-01-01

MR 96f:60121 | Zbl 0809.60010 | 6 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1433

Bibliographie

[Az] R. Azencott, Espaces de Poisson des Groupes Localement Compacts, Lect. Notes in Math., 148, Springer Berlin, 1970. | MR 58 #18748 | Zbl 0208.15302

[Be] A.F. Beardon, The Geometry of Discrete Groups, Springer New York, 1983. | MR 85d:22026 | Zbl 0528.30001

[Bi] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968. | MR 38 #1718 | Zbl 0172.21201

[CS] D.I. Cartwright and P.M. Soardi, Convergence to ends for random walks on the automorphism group of a tree, Proc. Amer. Math. Soc., 107 (1989), 817-823. | MR 90f:60137 | Zbl 0682.60059

[Ca] J.W.S. Cassels, Local Fields, Cambridge University Press, Cambridge, 1986. | MR 87i:11172 | Zbl 0595.12006

[Ch] F. Choucroun, Arbres, espaces ultramétriques, et bases de structure uniforme, Groupes d'Automorphismes d'Arbres, Thèse, Univ. Paris-Sud, 1993, p. 173-177.

[De] Y. Derriennic, Quelques applications du théorème ergodique sous-additif, Astérisque, 74 (1980), 183-201. | MR 82e:60013 | Zbl 0446.60059

[E1] L. Élie, Étude du renouvellement sur le groupe affine de la droite réelle, Ann. Sci. Univ. Clermont, 65 (1977), 47-62. | Numdam | MR 58 #13364 | Zbl 0384.60010

[E2] L. Élie, Fonctions harmoniques positives sur le groupe affine, Probability Measures on Groups (H. Heyer, ed.), Lect. Notes in Math., 706, Springer, Berlin, 1978, 96-110. | MR 81a:60083 | Zbl 0404.60076

[E3] L. Élie, Comportement asymptotique du noyau potentiel sur les groupes de Lie, Annales Éc. Norm. Sup., 15 (1982), 257-364. | Numdam | MR 85b:60005 | Zbl 0509.60070

[Fe] W. Feller, An Introduction to Probability Theory and its Applications, Vol II, Wiley, New York, 1966. | MR 35 #1048 | Zbl 0138.10207

[Fi] A. Figà-Talamanca, An example in finite harmonic analysis with application to diffusion in compact ultrametric spaces, preprint, Univ. Roma "La Sapienza", 1992.

[FN] A. Figà-Talamanca and C. Nebbia, Harmonic analysis and representation theory for groups acting on homogeneous trees, Cambridge Univ. Press, 1991.

[Fu] H. Furstenberg, Random walks and discrete subgroups of Lie groups, Advances in Probability and Related Topics, Vol. I (P. Ney, ed.), M. Dekker, New York, 1971, p. 1-63. | MR 44 #1794 | Zbl 0221.22008

[Ge] P. Gérardin, Harmonic functions on buildings of reductive split groups, Operator Algebras and Group Representations 1, Pitman, Boston, 1984, p. 208-221. | MR 86f:22020 | Zbl 0536.22018

[Gre] U. Grenander, Stochastic groups, Ark. Mat., 4 (1961), 189-207. | MR 26 #798 | Zbl 0141.14904

[G1] A.K. Grincevičjus, A central limit theorem for the group of linear transformations of the real axis, Soviet Math. Doklady, 15 (1974), 1512-1515. | Zbl 0326.60021

[G2] A.K. Grincevičjus, Limit theorems for products of random linear transformations of a straight line (in Russian), Lithuanian Math. J., 15 (1975), 61-77. | MR 54 #1333 | Zbl 0362.60018

[Gro] M. Gromov, Hyperbolic groups, Essays in Group Theory (S.M. Gersten, ed.), Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263. | MR 89e:20070 | Zbl 0634.20015

[Gui] F. Guimier, Simplicité du spectre de Liapounoff d'un produit de matrices aléatoires sur un corps ultramétrique, C. R. Acad. Sc. Paris, 309 (1989), 885-888. | MR 91d:28030 | Zbl 0696.60009

[Gu] Y. Guivarc'H, Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, Astérisque, 74 (1980), 47-98. | MR 82g:60016 | Zbl 0448.60007

[GKR] Y. Guivarc'H, M. Keane and B. Roynette, Marches Aléatoires sur les Groupes de Lie, Lecture Notes in Math., 624, Springer Berlin-Heidelberg-New York, 1977. | MR 58 #24454 | Zbl 0367.60081

[dH] P. De La Harpe, Free groups in linear groups, L'Enseignement Math., 29 (1983), 129-144. | MR 84i:20050 | Zbl 0517.20024

[K1] V.A. Kaimanovich, An entropy criterion for maximality of the boundary of random walks on discrete groups, Soviet Math. Doklady, 31 (1985), 193-197. | MR 86m:60025 | Zbl 0611.60060

[K2] V.A. Kaimanovich, Lyapunov exponents, symmetric spaces and multiplicative ergodic theorem for semi-simple Lie groups, J. Soviet Math., 47 (1989), 2387-2398. | MR 89m:22006 | Zbl 0696.22012

[K3] V.A. Kaimanovich, Poisson boundaries of random walks on discrete solvable groups, Probability Measures on Groups X (H. Heyer, ed.), Plenum Press, New York, 1991, p. 205-238. | MR 94m:60014 | Zbl 0823.60006

[K4] V.A. Kaimanovich, Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy, Harmonic Analysis and Discrete Potential Theory (M.A. Picardello, ed.), Plenum Press, New York, 1992, p. 145-180.

[K5] V.A. Kaimanovich, Boundaries of random walks revisited, in preparation, Univ. Edinburgh.

[Ki] J.F.C. Kingman, The ergodic theory of subadditive processes, J. Royal Stat. Soc., Ser. B, 30 (1968), 499-510. | MR 40 #8114 | Zbl 0182.22802

[Ma] G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups (Erg. Math. Grenzgeb., 3. Folge, Band 17), Springer, New York, 1989.

[Mo] S.A. Molchanov, Martin boundaries for invariant Markov processes on a solvable group, Theory of Probability and Appl., 12 (1967), 310-314. | Zbl 0308.60044

[Ne] C. Nebbia, On the amenability and the Kunze-Stein property for groups acting on a tree, Pacific J. Math., 135 (1988), 371-380. | MR 90f:43004 | Zbl 0671.43003

[Rag] M.S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362. | MR 81f:60016 | Zbl 0415.28013

[Rau] A. Raugi, Fonctions harmoniques et théorèmes limites pour les marches aléatoires sur les groupes, Mém. Bull. Soc. Math. France, 54 (1977), 5-118. | Numdam

[Sch] G. Schlichting, Polynomidentitäten und Permutationsdarstellungen lokalkompakter Gruppen, Inventiones Math., 55 (1979), 97-106. | MR 81d:22006 | Zbl 0426.22007

[S1] J.-P. Serre, Local Fields, Springer, New York, 1979.

[S2] J.-P. Serre, Trees, Springer, New York, 1980.

[Sp] F. Spitzer, Principles of Random Walk, (2nd ed.), Springer, New York, 1976. | MR 52 #9383 | Zbl 0359.60003

[SW] P.M. Soardi and W. Woess, Amenability, unimodularity, and the spectral radius of random walks on infinite graphs, Math. Zeitschr., 205 (1990), 471-486. | MR 91m:43002 | Zbl 0693.43001

[Ti] J. Tits, Sur le groupe des automorphismes d'un arbre, Essays in Topology and Related Topics (mémoires dédiés a G. de Rham), Springer, Berlin, 1970, p. 188-211. | MR 45 #8582 | Zbl 0214.51301

[Tr] V.I. Trofimov, Automorphism groups of graphs as topological groups, Math. Notes, 38 (1985), 717-720. | MR 87d:05091 | Zbl 0596.05033

[W1] W. Woess, Boundaries of random walks on graphs and groups with infinitely many ends, Israel J. Math., 68 (1989), 271-301. | MR 91e:60222 | Zbl 0723.60009

[W2] W. Woess, Random walks on infinite graphs and groups: a survey on selected topics, Bull. London Math. Soc., 26 (1994), 1-60. | MR 94i:60081 | Zbl 0830.60061

[W3] W. Woess, The Martin boundary of random walks on the automorphism group of a homogeneous tree, print Univ. Milano.