  We consider a finite-dimensional, locally finite CAT(0) cube complex X
admitting a co-compact properly discontinuous countable group of automorphisms
G. We construct a natural compact metric space B(X) on which G acts by
homeomorphisms, the action being minimal and strongly proximal. Furthermore,
for any generating probability measure on G, B(X) admits a unique stationary
measure, and when the measure has finite logarithmic moment, it constitutes a
compact metric model of the Poisson boundary. We identify a dense G-delta
subset of B(X) on which the action of G is Borel-amenable, and describe the
relation of these two spaces to the Roller boundary. Our construction can be
used to give a simple geometric proof of Property A for the complex. Our
methods are based on direct geometric arguments regarding the asymptotic
behavior of half-spaces and their limiting ultrafilters, which are of
considerable independent interest. In particular we analyze the notions of
median and interval in the complex, and use the latter in the proof that B(X)
is the Poisson boundary via the strip criterion developed by V. Kaimanovich.
