Stationary measures and random walks on $\tilde{A}_2$-buildings

TL;DR

This paper studies random walks on affine buildings of type our A_2, proving the uniqueness of stationary measures and convergence properties, extending classical results to more general, possibly exotic, non-discrete settings.

Contribution

It establishes the uniqueness of -stationary measures and convergence of random walks on our A_2 buildings, including exotic non-discrete cases, extending Furstenberg's classical theorems.

Findings

Unique -stationary measure supported on the boundary

Almost sure convergence of the walk to a boundary point

Lyapunov spectrum of the walk is simple

Abstract

We consider a non-elementary group action $G \curvearrowright X$ of a locally compact second countable group $G$ on a possibly exotic non-discrete affine building $X$ of type $\tilde{A}_2$. We prove that if $\mu$ is an admissible symmetric probability measure on $G$, there is a unique $\mu$-stationary measure supported on the chambers of the spherical building at infinity. We use this result to study random walks induced by the $G$-action, and we prove that if $\mu$ has finite second moment, $(Z_n o)$ converges almost surely to a regular point of the boundary and the Lyapunov spectrum of the random walk is simple. Applied to Bruhat-Tits buildings, these results extend some classical theorems due to H.~Furstenberg.