Martin compactifications of affine buildings

TL;DR

This paper studies Martin compactifications of affine buildings using potential theory and random walks, providing new insights into their asymptotic behavior without relying on group actions.

Contribution

It offers a novel approach to affine building compactifications through potential theory, independent of group actions, with implications for geometric group theory.

Findings

Explicit descriptions of asymptotic behavior of kernels

Green kernel asymptotics established

Compactifications applicable in geometric group theory

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.