Convergence towards the end space for random walks on Schreier graphs

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.

10 pages, 2 figures. Changes from previous version: Added Proposition 2.6. Updated references

References in corpus (2)

• The Liouville property and random walks on topological groups
• Non-triviality of the Poisson boundary of random walks on the group $H(\mathbb{Z})$ of Monod