Random walks on the affine group of local fields and of homogeneous trees
The affine group of a local field acts on the tree (the Bruhat-Tits building of ) with a fixed point in the space of ends . More generally, we define the affine group of any homogeneous tree as the group of all automorphisms of with a common fixed point in , and establish main asymptotic properties of random products in : (1) law of large numbers and central limit theorem; (2) convergence to and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary...