Convergence Towards the End Space for Random Walks on Schreier Graphs

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.