JournalsggdVol. 12, No. 3pp. 1069–1121

Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups

  • Lorenz Gilch

    Universität Passau, Germany
  • Sebastian Müller

    Aix-Marseille Université, Marseille, France
  • James Parkinson

    University of Sydney, Australia
Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups cover
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Abstract

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.

Cite this article

Lorenz Gilch, Sebastian Müller, James Parkinson, Limit theorems for random walks on Fuchsian buildings and Kac–Moody groups. Groups Geom. Dyn. 12 (2018), no. 3, pp. 1069–1121

DOI 10.4171/GGD/465