Abstract
Let Γ be a finitely generated non-elementary Fuchsian group, and let μ be a probability measure with finite support on Γ such that supp μ generates Γ as a semigroup. If Γ contains no parabolic elements we show that for all but a small number of co-compact Γ, the Martin boundaryM of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ. If Γ has cusps, we prove that Γ can be deformed into a group Γ', abstractly isomorphic to Γ, such thatM can be identified with Λ', the limit set of Γ'. Our method uses the identification of Λ with a certain set of infinite reduced words in the generators of Γ described in [15]. The harmonic measure ν (ν is the hitting distribution of random paths in Γ on Λ) is a Gibbs measure on this space of infinite words, and the Poisson boundary of Γ, μ can be identified with Λ, ν.
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Series, C. Martin boundaries of random walks on Fuchsian groups. Israel J. Math. 44, 221–242 (1983). https://doi.org/10.1007/BF02760973
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DOI: https://doi.org/10.1007/BF02760973