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

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

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 4, page 1243-1288
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Cartwright, Donald I., Kaimanovich, Vadim A., and Woess, Wolfgang. "Random walks on the affine group of local fields and of homogeneous trees." Annales de l'institut Fourier 44.4 (1994): 1243-1288. <http://eudml.org/doc/75096>.

@article{Cartwright1994,
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.},
author = {Cartwright, Donald I., Kaimanovich, Vadim A., Woess, Wolfgang},
journal = {Annales de l'institut Fourier},
keywords = {random walks; law of large numbers; central limit theorem; harmonic functions; asymptotic properties of random products; Dirichlet problem; Poisson boundary; geometric properties of homogeneous trees},
language = {eng},
number = {4},
pages = {1243-1288},
publisher = {Association des Annales de l'Institut Fourier},
title = {Random walks on the affine group of local fields and of homogeneous trees},
url = {http://eudml.org/doc/75096},
volume = {44},
year = {1994},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 4
SP - 1243
EP - 1288
AB - 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.
LA - eng
KW - random walks; law of large numbers; central limit theorem; harmonic functions; asymptotic properties of random products; Dirichlet problem; Poisson boundary; geometric properties of homogeneous trees
UR - http://eudml.org/doc/75096
ER -

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Citations in EuDML Documents

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  1. Wolfgang Woess, Heat diffusion on homogeneous trees (Note on a paper by G. Medolla and A. G. Setti)
  2. Sara Brofferio, The Poisson boundary of random rational affinities
  3. Sara Brofferio, Wolfgang Woess, Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs
  4. Sara Brofferio, How a centred random walk on the affine group goes to infinity
  5. Sara Brofferio, Dariusz Buraczewski, Ewa Damek, On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case
  6. Bruno Schapira, Poisson boundary of triangular matrices in a number field

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