Randomized Geodesic Flow on Hyperbolic Groups

TL;DR

This work develops a probabilistic analogue of geodesic flow for Gromov-hyperbolic groups by harnessing random walks. It defines a harmonic Bowen–Margulis–Sullivan measure $\Theta$ on $\partial^2 G$ through three equivalent constructions $\Theta_1,\Theta_2,\Theta_3$, with $\Theta_3$ arising from bi-infinite random-walk trajectories and forming the basis for a discrete randomized geodesic flow given by a $\mathbb{Z}$-shift. The authors prove ergodicity of the $G$-action on $\partial^2 G$ with respect to $\Theta$, and establish strong statistical properties: exponential mixing of all orders and a functional central limit theorem for the flow. The technical core combines bi-infinite random-walk deviations from geodesics, sharp Green-function and Ancona-type estimates, and a Bonk–Schramm embedding to translate group geometry into hyperbolic space, enabling boundary density computations and ergodicity results. Overall, the paper provides a robust probabilistic framework for geodesic-flow-like dynamics in hyperbolic groups, yielding precise dynamical and statistical behavior inaccessible in purely deterministic settings.

Abstract

Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $Θ$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We provide three different but related constructions of $Θ$: 1) by moving the base-point along a quasigeodesic ray 2) by moving the base-point along random walk trajectories 3) directly as a push-forward under the boundary map to $\partial^2 G$ of a measure inherited from studying all bi-infinite random walk trajectories (with no restriction on base-point) on $G^\mathbb{Z}$. Of these, the third construction is the most involved and needs new techniques. It relies on developing a framework where we can treat bi-infinite random walk trajectories as analogs of bi-infinite geodesics on complete simply connected negatively curved manifolds. Geodesic flow on a hyperbolic group is typically not well-defined due to non-uniqueness of geodesics. We circumvent this problem in the random walk setup by considering \emph{all} trajectories. We thus get a well-defined discrete flow that we call the \emph{randomized geodesic flow}, given by the $\mathbb{Z}-$shift on bi-infinite random walk trajectories. The $\mathbb{Z}-$shift is the random analog of the time one map of the geodesic flow. As an analog of ergodicity of the geodesic flow on a closed negatively curved manifold, we establish ergodicity of the $G$-action on $(\partial^2G, Θ)$. As a consequence of our construction, we prove that the randomized geodesic flow is exponentially mixing of all orders and establish a functional CLT.

Figures (4)

• Figure 1: Comparing the setup in BF (on the left) to our setup (on the right).
• Figure 2: Possible positions of the images of $p$ and $k$ in the tripod under the $(1,c)$-quasi isometry.
• Figure 3: This is an illustration of the embedding of a random walk started in $a$ into $\mathbb{H}^2$ under the Bonk-Schramm embedding. In the image, all paths converging to a point in $B$ and staying outside $B_{n}(0)$ have to hit each of the barriers $H_1,...,H_4$ at some time.
• Figure 4: Showing ergodicity in $\partial^2 G$ requiring a detour via the $G$ and $\mathbb{Z}$-quotients of $G^{\mathbb{Z}}$.

Theorems & Definitions (68)

• Theorem 2.1
• Definition 2.2
• Theorem 2.3: Double Ergodicity
• Definition 2.4
• Definition 2.5
• Theorem 2.6: Exponential Mixing of all orders
• Theorem 2.7: Functional CLT
• Definition 3.1
• Theorem 3.2
• Definition 3.3