[Paper Review] A Furstenberg type formula for the speed of distance stationary sequences

Matías Carrasco, Pablo Lessa+1 more|arXiv (Cornell University)|Oct 2, 2017

Bayesian Methods and Mixture Models19 references3 citations

TL;DR

This paper establishes a Furstenberg-type formula for the speed of distance stationary random sequences, enabling the analysis of linear drift in random walks on Riemannian symmetric spaces. The authors apply this to Poisson-Delaunay random walks in hyperbolic space, proving that the harmonic measure exhibits dimension drop—its Hausdorff dimension is strictly less than $d-1$—as the intensity of the Poisson process tends to zero.

ABSTRACT

We prove a formula for the speed of distance stationary random sequences. A particular case is the classical formula for the largest Lyapunov exponent of an i.i.d. product of two by two matrices in terms of a stationary measure on projective space. We apply this result to Poisson-Delaunay random walks on Riemannian symmetric spaces. In particular, we obtain sharp estimates for the asymptotic behavior of the speed of hyperbolic Poisson-Delaunay random walks when the intensity of the Poisson point process goes to zero. This allows us to prove that a dimension drop phenomena occurs for the harmonic measure associated to these random walks. With the same technique we give examples of co-compact Fuchsian groups for which the harmonic measure of the simple random walk has dimension less than one.

Motivation & Objective

• To derive a general Furstenberg-type formula for the speed of distance stationary sequences using random horofunctions.
• To establish that Poisson-Delaunay random walks on Riemannian symmetric spaces are distance stationary under degree bias.
• To analyze the asymptotic speed of hyperbolic Poisson-Delaunay random walks as the intensity of the Poisson process tends to zero.
• To prove that the harmonic measure of such walks exhibits dimension drop—its Hausdorff dimension is strictly less than $d-1$—for low intensities.
• To construct examples of co-compact Fuchsian groups where the harmonic measure of simple random walks has dimension less than one.

Proposed method

• Define a random horofunction $\xi$ based on the past tail of a distance stationary sequence to capture linear drift.
• Prove that the expected speed $\mathbb{E}[\ell] = -\mathbb{E}[\xi(x_0) - \xi(x_1)]$, generalizing Furstenberg's formula for Lyapunov exponents.
• Use degree bias to show that Poisson-Delaunay random walks are distance stationary, enabling application of the speed formula.
• Establish ergodicity of the walk's speed via a zero-one law, proving almost sure constancy of graph and ambient speeds.
• Derive sharp asymptotic estimates for the ambient speed $\ell_\lambda$ as $\lambda \to 0$ using geometric and probabilistic estimates on Delaunay edges.
• Apply the speed formula and entropy bounds to prove $\dim(\nu_\lambda) \leq h_\lambda / \ell_\lambda$, leading to dimension drop when $h_\lambda / \ell_\lambda \to (d-1)/2$.

Experimental results

Research questions

• RQ1 Can a Furstenberg-type formula be generalized to distance stationary sequences beyond i.i.d. matrix products?
• RQ2 What is the asymptotic behavior of the speed of hyperbolic Poisson-Delaunay random walks as the intensity $\lambda$ of the Poisson process tends to zero?
• RQ3 Does the harmonic measure of Poisson-Delaunay random walks on hyperbolic spaces exhibit dimension drop for low intensities?
• RQ4 For which co-compact Fuchsian groups does the harmonic measure of simple random walks have dimension less than one?
• RQ5 How do the graph speed and ambient speed of Poisson-Delaunay walks relate in symmetric spaces?

Key findings

• The speed of a distance stationary sequence satisfies $\mathbb{E}[\ell] = -\mathbb{E}[\xi(x_0) - \xi(x_1)]$, where $\xi$ is a random horofunction derived from the past tail.
• For i.i.d. $\mathrm{SL}(2,\mathbb{R})$ matrix products, this recovers the classical Furstenberg formula for the top Lyapunov exponent.
• The ambient speed $\ell_\lambda$ of hyperbolic Poisson-Delaunay random walks satisfies $\ell_\lambda \sim \frac{1}{2} \log(\lambda^{-1})$ as $\lambda \to 0$.
• The graph speed of the walk tends to its maximal possible value of 1 as $\lambda \to 0$, resolving an open question from [BPP14].
• The dimension of the harmonic measure $\nu_\lambda$ satisfies $\limsup_{\lambda \to 0} \dim(\nu_\lambda) \leq \frac{d-1}{2}$, proving dimension drop.
• For low-intensity hyperbolic Poisson-Delaunay walks, $\dim(\nu_\lambda) < d-1$ holds almost surely, confirming a dimension drop phenomenon.