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.