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This paper studies the boundary behaviour of λ -polyharmonic functions for the simple random walk operator on a regular tree, where λ is complex and | λ | > ρ , the ℓ 2 -spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved, and a non-tangential Fatou theorem is proved. [ABSTRACT FROM AUTHOR]

We prove a law of large numbers for the range of rotor walks with random initial configuration on regular trees and on Galton–Watson trees. We also show the existence of the speed for such rotor walks. More precisely, we show that on the classes of trees under consideration, even in the case when the rotor walk is recurrent, the range grows at linear speed. [ABSTRACT FROM AUTHOR]

We consider random walks on non-amenable Baumslag-Solitar groups BS(p, q) and describe their Poisson-Furstenberg boundary. The latter is a probabilistic model for the long-time behaviour of the random walk. In our situation, we identify it in terms of the space of ends of the Bass-Serre tree and the real line using Kaimanovich’s strip criterion. [ABSTRACT FROM AUTHOR]