A central limit theorem for random walks on horospherical products of Gromov hyperbolic spaces

Abstract

We develop a metric on the horospherical product of a finite number of proper Gromov hyperbolic spaces and study the asymptotic behavior of random walks with respect to this metric. Our setting includes horospherical products of a finite number of tree-like spaces and rank one symmetric spaces, lamplighter groups over $\mathbb{Z}$, Diestel–Leader graphs, and Sol groups. Under finite second moment and non-zero drift conditions, we establish a central limit theorem for the displacement of a random walk on affine groups of these metric spaces. Along the way, we prove some geometric properties of these spaces.