Random walks on rank one symmetric spaces of noncompact type

TL;DR

This paper proves fundamental limit theorems for a natural random walk on rank one symmetric spaces of noncompact type, extending classical results to a broader geometric setting using harmonic analysis and algebraic methods.

Contribution

It introduces a unified algebraic framework for random walks on these spaces and establishes their convergence to the heat kernel, a novel result in this geometric context.

Findings

Established a central limit theorem for the random walk.

Proved a local limit theorem and law of large numbers.

Showed convergence of the renormalized walk to the heat kernel.

Abstract

We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural random walk on a symmetric space $M$ of non-compact type and rank one. This class of spaces, which includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes the real hyperbolic space $\mathbb{H}^{n}$. Our approach introduces a unified algebraic framework that generalizes the M\"obius addition, previously used for the constant curvature case, to define the random walk via a non-Euclidean summation of variables. We demonstrate that the renormalized walk converges to the heat kernel associated with the Laplace-Beltrami operator on $M$, which plays the role of the limiting normal law. The proofs leverage the harmonic analysis of spherical functions on symmetric spaces. To the best of our knowledge, these results are new in the context of rank one symmetric spaces.