Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces

Abstract

We study random walks on the isometry group of a Gromov hyperbolic space or Teichmüller space. We prove that the translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment. While doing this, we recover the central limit theorem of Benoist and Quint for the displacement of a reference point and establish its converse. Also discussed are the corresponding laws of the iterated logarithm. Finally, we prove sublinear geodesic tracking by random walks with finite $(1/2)$-th moment and logarithmic tracking by random walks with finite exponential moment.

arXiv comment61 pages, 5 figures. Accepted by Advances in Mathematics

References in corpus (8)

• Hyperbolicity in Teichmüller space
• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• Sublinear deviation between geodesics and sample paths
• Exponential bounds for random walks on hyperbolic spaces without moment conditions
• Central limit theorems for mapping class groups and $\text{Out}(F_N)$
• Analyticity of the entropy and the escape rate of random walks in hyperbolic groups
• Two extensions of Thurston's spectral theorem for surface diffeomorphisms
• Thin triangles and a multiplicative ergodic theorem for Teichmüller geometry

Cited by in corpus (2)

• Pseudo-Anosovs are exponentially generic in mapping class groups
• Random walks and contracting elements I: Deviation inequality and Limit laws