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Convergence Towards the End Space for Random Walks on Schreier Graphs

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  • Bogdan Stankov

    (PSL Research University)

Abstract

We consider a transitive action of a finitely generated group G and the Schreier graph $$\varGamma $$ Γ defined by this action for some fixed generating set. For a probability measure $$\mu $$ μ 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 $$\varGamma $$ Γ . 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.

Suggested Citation

  • Bogdan Stankov, 2022. "Convergence Towards the End Space for Random Walks on Schreier Graphs," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1412-1422, September.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:3:d:10.1007_s10959-021-01104-6
    DOI: 10.1007/s10959-021-01104-6
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