# Convergence towards the end space for random walks on Schreier graphs

## TL;DR

This paper studies how random walks on Schreier graphs of group actions tend to converge to the space of ends, revealing new boundary behaviors especially for Thompson's group F with finite first moment measures.

## Contribution

It demonstrates convergence to the end space for transitive group actions and establishes non-trivial Poisson boundaries for certain measures on Thompson's group F.

## Key findings

- Random walks on Schreier graphs with finite first moment converge to the space of ends.
- For Thompson's group F, specific measures induce non-trivial Poisson boundaries.
- Moment conditions are essential, as shown by counterexamples.

## Abstract

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the induced random walk is transient, it converges towards the space of ends of $\Gamma$. As a corollary we obtain that for a probability measure with a finite first moment on Thompson's group $F$, the support of which generates $F$ as a semigroup, the induced random walk on the dyadic numbers has a non-trivial Poisson boundary. Some assumption on the moment of the measure is necessary as follows from an example by Juschenko and Zheng.