On the speed of distance stationary sequences

TL;DR

This paper generalizes the law of large numbers to distance stationary sequences, providing a formula for their speed with applications in random matrices, trees, percolation, and group theory.

Contribution

It introduces a new formula for the speed of distance stationary sequences, extending classical results and enabling diverse applications.

Findings

Provides a general formula for the speed of distance stationary sequences.

Applies the formula to estimate speeds and analyze dimension drops.

Demonstrates applications in random trees, percolation, and group-related random walks.

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.