Cartwright, Donald I.; Kaimanovich, Vadim A.; Woess, Wolfgang
Random walks on the affine group of local fields and of homogeneous trees
Annales de l'institut Fourier, Tome 44 (1994) no. 4 , p. 1243-1288
Zbl 0809.60010 | MR 96f:60121 | 6 citations dans Numdam
doi : 10.5802/aif.1433
URL stable : http://www.numdam.org/item?id=AIF_1994__44_4_1243_0

Le groupe affine d’un corps local 𝔉 agit sur l’arbre 𝕋(𝔉) (l’immeuble de Bruhat-Tits de GL (2,𝔉)) en ayant un point fixe dans l’espace des bouts 𝕋(F). Plus généralement, nous définissons le groupe affine Aff(𝔉) d’un arbre homogène 𝕋 comme le groupe de tous les automorphismes de 𝕋 ayant un point fixe commun dans 𝕋, et établissons les principales propriétés asymptotiques des produits aléatoires dans Aff(𝔉) : (1) la loi des grands nombres et le théorème limite central; (2) la convergence vers 𝕋 et l’existence d’une solution au problème de Dirichlet à l’infini; (3) l’identification de la frontière de Poisson avec 𝕋 donnant une description de l’espace des fonctions μ-harmoniques bornées. Les méthodes utilisées sont étroitement reliées aux propriétés géométriques des arbres homogènes analogues à celles des espaces symétriques de rang un.
The affine group of a local field acts on the tree 𝕋(𝔉) (the Bruhat-Tits building of GL (2,𝔉)) with a fixed point in the space of ends 𝕋(F). More generally, we define the affine group Aff(𝔉) of any homogeneous tree 𝕋 as the group of all automorphisms of 𝕋 with a common fixed point in 𝕋, and establish main asymptotic properties of random products in Aff(𝔉): (1) law of large numbers and central limit theorem; (2) convergence to 𝕋 and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary with 𝕋, which gives a description of the space of bounded μ-harmonic functions. Our methods strongly rely on geometric properties of homogeneous trees as discrete counterparts of rank one symmetric spaces.

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