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) no. 4, pp. 1243-1288.

Le groupe affine d’un corps local $𝔉$ agit sur l’arbre $𝕋\left(𝔉\right)$ (l’immeuble de Bruhat-Tits de $\mathrm{GL}(2,𝔉)$) en ayant un point fixe dans l’espace des bouts $\partial 𝕋\left(F\right)$. Plus généralement, nous définissons le groupe affine $Aff\left(𝔉\right)$ d’un arbre homogène $𝕋$ comme le groupe de tous les automorphismes de $𝕋$ ayant un point fixe commun dans $\partial 𝕋$, et établissons les principales propriétés asymptotiques des produits aléatoires dans $Aff\left(𝔉\right)$ : (1) la loi des grands nombres et le théorème limite central; (2) la convergence vers $\partial 𝕋$ et l’existence d’une solution au problème de Dirichlet à l’infini; (3) l’identification de la frontière de Poisson avec $\partial 𝕋$ 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.

The affine group of a local field acts on the tree $𝕋\left(𝔉\right)$ (the Bruhat-Tits building of $\mathrm{GL}(2,𝔉)$) with a fixed point in the space of ends $\partial 𝕋\left(F\right)$. More generally, we define the affine group $Aff\left(𝔉\right)$ of any homogeneous tree $𝕋$ as the group of all automorphisms of $𝕋$ with a common fixed point in $\partial 𝕋$, and establish main asymptotic properties of random products in $Aff\left(𝔉\right)$: (1) law of large numbers and central limit theorem; (2) convergence to $\partial 𝕋$ and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary with $\partial 𝕋$, 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.

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