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

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.