A Furstenberg type formula for the speed of distance stationary sequences

2017/10/02 by Carrasco, Matias, Lessa, Pablo, Paquette, Elliot · 1 citation
#05C81 #34D08 #51M10 #60Dxx #60G55 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1710.00733

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.

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• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank

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