Matrix random products with singular harmonic measure

2008/07/07 by Vadim A. Kaimanovich, Kaimanovich, Vadim A., Vincent Le Prince +1 · 3 citations
Mathematics · #28A78 #37D25 #53C35 #60J50 #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0807.1015

openalex publication_date 2008/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Any Zariski dense countable subgroup of SL(d,R) is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new upper estimate for the Hausdorff dimension of the projections of the harmonic measure onto Grassmannians in R^d in terms of the associated differential entropies and differences between the Lyapunov exponents; (2) an explicit construction of random walks with uniformly bounded entropy and Lyapunov exponents going to infinity.

Citations

• Some relations between dimension and Lyapounov exponents
• Dimension, entropy and Lyapunov exponents

Cited by

• Random walks on cocompact Fuchsian and Kleinian groups
• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
• A note on the singularity conjecture for infinite covolume discrete subgroups

Related

• Dimensional properties of the harmonic measure for a random walk on a hyperbolic group
• Dimension conservation of harmonic measures in products of hyperbolic spaces
• Singularity of compound stationary measures
• Random Walks on Homogeneous Spaces by Sparse Solvable Measures
• The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups