Convergence Towards the End Space for Random Walks on Schreier Graphs

Bogdan Stankov ()
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Bogdan Stankov: PSL Research University

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 1412-1422

Abstract: 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.

Keywords: Random walks on groups; Poisson boundary; Schreier graph; Thompson’s group F; Primary classes: 05C81; 60B15; 60J50; Secondary classes: 05C25; 20F65; 60J10 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01104-6

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