Boundary behaviour of -polyharmonic functions on regular trees.

This paper studies the boundary behaviour of -polyharmonic functions for the simple random walk operator on a regular tree, where is complex and , the -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.

Range and Speed of Rotor Walks on Trees.

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 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.