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Renewal theory for random walks on surface groups

Published online by Cambridge University Press:  12 May 2016

PETER HAÏSSINSKY ,
PIERRE MATHIEU  and
SEBASTIAN MÜLLER
PETER HAÏSSINSKY
Affiliation:
IMT, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France email phaissin@math.univ-toulouse.fr
PIERRE MATHIEU
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France email pierre.mathieu@univ-amu.fr, sebastian.muller@univ-amu.fr
SEBASTIAN MÜLLER
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France email pierre.mathieu@univ-amu.fr, sebastian.muller@univ-amu.fr
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Abstract

We construct a renewal structure for random walks on surface groups. The renewal times are defined as times when the random walks enter a particular type of cone and never leave it again. As a consequence, the trajectory of the random walk can be expressed as an aligned union of independent and identically distributed trajectories between the renewal times. Once having established this renewal structure, we prove a central limit theorem for the distance to the origin under exponential moment conditions. Analyticity of the speed and of the asymptotic variance are natural consequences of our approach. Furthermore, our method applies to groups with infinitely many ends and therefore generalizes classic results on central limit theorems on free groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 155 - 179
Copyright
© Cambridge University Press, 2016 

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