The affine group of a local field acts on the tree (the Bruhat-Tits building of ) with a fixed point in the space of ends . More generally, we define the affine group of any homogeneous tree as the group of all automorphisms of with a common fixed point in , and establish main asymptotic properties of random products in : (1) law of large numbers and central limit theorem; (2) convergence to and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary with , which gives a description of the space of bounded -harmonic functions. Our methods strongly rely on geometric properties of homogeneous trees as discrete counterparts of rank one symmetric spaces.
Le groupe affine d’un corps local agit sur l’arbre (l’immeuble de Bruhat-Tits de ) en ayant un point fixe dans l’espace des bouts . Plus généralement, nous définissons le groupe affine d’un arbre homogène comme le groupe de tous les automorphismes de ayant un point fixe commun dans , et établissons les principales propriétés asymptotiques des produits aléatoires dans : (1) la loi des grands nombres et le théorème limite central; (2) la convergence vers et l’existence d’une solution au problème de Dirichlet à l’infini; (3) l’identification de la frontière de Poisson avec donnant une description de l’espace des fonctions -harmoniques bornées. Les méthodes utilisées sont étroitement reliées aux propriétés géométriques des arbres homogènes analogues à celles des espaces symétriques de rang un.
@article{AIF_1994__44_4_1243_0, author = {Cartwright, Donald I. and Kaimanovich, Vadim A. and Woess, Wolfgang}, title = {Random walks on the affine group of local fields and of homogeneous trees}, journal = {Annales de l'Institut Fourier}, pages = {1243--1288}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {44}, number = {4}, year = {1994}, doi = {10.5802/aif.1433}, zbl = {0809.60010}, mrnumber = {96f:60121}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1433/} }
TY - JOUR AU - Cartwright, Donald I. AU - Kaimanovich, Vadim A. AU - Woess, Wolfgang TI - Random walks on the affine group of local fields and of homogeneous trees JO - Annales de l'Institut Fourier PY - 1994 SP - 1243 EP - 1288 VL - 44 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1433/ DO - 10.5802/aif.1433 LA - en ID - AIF_1994__44_4_1243_0 ER -
%0 Journal Article %A Cartwright, Donald I. %A Kaimanovich, Vadim A. %A Woess, Wolfgang %T Random walks on the affine group of local fields and of homogeneous trees %J Annales de l'Institut Fourier %D 1994 %P 1243-1288 %V 44 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1433/ %R 10.5802/aif.1433 %G en %F AIF_1994__44_4_1243_0
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, Volume 44 (1994) no. 4, pp. 1243-1288. doi : 10.5802/aif.1433. https://aif.centre-mersenne.org/articles/10.5802/aif.1433/
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