Martin compactifications of affine buildings

Abstract

We carry out an in-depth study of Martin compactifications of affine buildings, from the viewpoint of potential theory and random walks. This work does not use any group action on buildings, although all the results are also stated within the framework of the Bruhat--Tits theory of semisimple groups over non-Archimedean local fields. This choice should allow the use of these building compactifications in intriguing geometric group theory situations, where only lattice actions are available. The resulting compactified spaces use and, at the same time, make it possible to understand geometrically the descriptions of asymptotic behavior of kernels resulting from the non-Archimedean harmonic analysis on affine buildings. Along the paper, we make explicit the most substantial differences with the case of symmetric spaces, namely absence of a group action but existence of precise asymptotics of Green kernels and, of course, no possibility to stand by standard techniques from PDEs.