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  1. (, Ergodic Theory and Dynamical Systems)
    We show under weak hypotheses that$$\unicode[STIX]{x2202}X$$, 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$$\unicode[STIX]{x1D6E4}$$. In particular, we show that if$$\unicode[STIX]{x1D6E4}$$admits a non-elementary proper action on$$X$$, and$$\unicode[STIX]{x1D707}$$is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a$$\unicode[STIX]{x1D707}$$-stationary measure on$$\unicode[STIX]{x2202}X$$making it the Furstenberg–Poisson boundary for the$$\unicode[STIX]{x1D707}$$-random walk on$$\unicode[STIX]{x1D6E4}$$. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties. 
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