The Furstenberg Poisson Boundary and CAT(0) Cube Complexes

We show under weak hypotheses that $\mathrm{\partial }X$, the Roller boundary of a finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group $\mathrm{\Gamma }$. In particular, we show that if $\mathrm{\Gamma }$ admits a nonelementary proper action on $X$, and $\mu$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\mu$-stationary measure on $\mathrm{\partial }X$ making it the Furstenberg-Poisson boundary for the $\mu$-random walk on $\mathrm{\Gamma }$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.