On the speed of distance-stationary sequences

Abstract

We prove a formula for the speed of distance-stationary random sequences generalizing the law of large numbers of Karlsson and Ledrappier. A particular case is the classical formula for the largest Lyapunov exponent of i.i.d. matrix products, but our result has applications in various different contexts. In many situations it gives a method to estimate the speed, and in others it allows to obtain results of dimension drop for escape measures related to random walks. We show applications to stationary reversible random trees with conductances, Bernoulli bond percolation of Cayley graphs, and random walks on cocompact Fuchsian groups.

On the speed of distance stationary sequences

“…This was first proved in [CLP17] as far as the author's are aware, and was re-obtained independently in recent work of Petr Kosenko [Kos19]. His result covers all but finitely many cases for the number of sides and polygons per vertex (P, Q), while we estimate the dimension of the boundary measure only for large Q.…”

“…This was first proved in [CLP17] as far as the authors are aware. In particular the harmonic measure ν P,Q is singular with respect to the Lebesgue measure on the circle.…”

Elements of a metric spectral theory

“…In my experience, many people know of one or a few applications, but few have an overview of all the applications. Other applications are found below or in papers listed in the bibliography, for example let us mention a recent Furstenberg-type formula for the drift of random walks on groups [CLP17] in part building on [KaL06,KaL11]. It is also the case that the last two decades have seen identifications and understanding of horofunctions for various classes of metric spaces.…”

The fundamental inequality for cocompact Fuchsian groups

“…In this case, the hyperbolic metric and the word metric on G are quasi-isometric to each other, and hence distortion arguments do not work. So far, the only known examples are the recent ones from [32] and [11], where the singularity of hitting measure is proven for cocompact Fuchsian groups whose fundamental domain is a regular polygon (except for a finite number of cases with few sides). These examples form a countable family.…”

“…If m is even, the above construction yields the standard presentation of a hyperelliptic Fuchsian group of genus 𝑔 = 𝑚 2 ; if m is odd, we also obtain a discrete cocompact group of genus 𝑔 = 𝑚−1 2 . To compare with [11] and [32], the authors of [11] use percolation to obtain a formula for the drift of the random walk, and then they obtain an asymptotic lower bound for the drift as the number of sides tends to ∞. Kosenko [32] obtains, in the regular case, explicit lower bounds for the translation lengths using hyperbolic geometry without resorting to approximation by percolation.…”