Geodesic
Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups
Lorenz Gilch1 ; Sebastian Müller2 ; James Parkinson3
1Universität Passau, Germany
2Aix-Marseille Université, Marseille, France
3University of Sydney, Australia
Groups, geometry, and dynamics, Tome 12 (2018) no. 3, pp. 1069-1121

Voir la notice de l'article provenant de la source EMS Press

DOI

In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac–Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in [23]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building.
DOI : 10.4171/ggd/465
Classification : 60-XX, 20-XX, 51-XX
Mots-clés : Random walk, limit theorems, Fuchsian building, Kac-Moody group, Cannon automaton

Lorenz Gilch  1   ; Sebastian Müller  2   ; James Parkinson  3

1 Universität Passau, Germany
2 Aix-Marseille Université, Marseille, France
3 University of Sydney, Australia 
Lorenz Gilch; Sebastian Müller; James Parkinson. Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups. Groups, geometry, and dynamics, Tome 12 (2018) no. 3, pp. 1069-1121. doi: 10.4171/ggd/465
@article{10_4171_ggd_465,
     author = {Lorenz Gilch and Sebastian M\"uller and James Parkinson},
     title = {Limit theorems for random walks on {Fuchsian} buildings and {Kac{\textendash}Moody} groups},
     journal = {Groups, geometry, and dynamics},
     pages = {1069--1121},
     year = {2018},
     volume = {12},
     number = {3},
     doi = {10.4171/ggd/465},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/465/}
}
TY  - JOUR
AU  - Lorenz Gilch
AU  - Sebastian Müller
AU  - James Parkinson
TI  - Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups
JO  - Groups, geometry, and dynamics
PY  - 2018
SP  - 1069
EP  - 1121
VL  - 12
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4171/ggd/465/
DO  - 10.4171/ggd/465
ID  - 10_4171_ggd_465
ER  - 
%0 Journal Article
%A Lorenz Gilch
%A Sebastian Müller
%A James Parkinson
%T Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups
%J Groups, geometry, and dynamics
%D 2018
%P 1069-1121
%V 12
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/465/
%R 10.4171/ggd/465
%F 10_4171_ggd_465

Cité par Sources :