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LetXbe a nonelementary\mathrm{CAT}(0)cubical complex. We prove that ifXis essential and irreducible, then the contact graph ofX(introduced by Hagen (2014)) is unbounded and its boundary is homeomorphic to the regular boundary ofX(defined by Fernós (2018) and Kar–Sageev (2016)). Using this, we reformulate the Caprace–Sageev’s rank-rigidity theorem in terms of the action on the contact graph. LetGbe a group with a nonelementary action onX, and let (Z_{n})be a random walk corresponding to a generating probability measure onGwith finite second moment. Using this identification of the boundary of the contact graph, we prove a central limit theorem for (Z_{n}), namely that\frac{d(Z_{n} o,o)-nA}{\sqrt{n}}converges in law to a non-degenerate Gaussian distribution (A\hspace{-0.7pt}=\hspace{-0.7pt}\lim_{n\to\infty}\hspace{-0.7pt}\frac{d(Z_{n}o,o)}{n}is the drift of the random walk, ando\in Xis an arbitrary basepoint). 

Abstract A finite-dimensional CAT(0) cube complexXis equipped with several well-studied boundaries. These include theTits boundary$$\partial _TX$$ ${\partial }_{T}X$(which depends on the CAT(0) metric), theRoller boundary$${\partial _R}X$$ ${\partial }_{R}X$(which depends only on the combinatorial structure), and thesimplicial boundary$$\partial _\triangle X$$ ${\partial }_{▵}X$(which also depends only on the combinatorial structure). We use a partial order on a certain quotient of$${\partial _R}X$$ ${\partial }_{R}X$to define a simplicial Roller boundary$${\mathfrak {R}}_\triangle X$$ $R_{▵}X$. Then, we show that$$\partial _TX$$ ${\partial }_{T}X$,$$\partial _\triangle X$$ ${\partial }_{▵}X$, and$${\mathfrak {R}}_\triangle X$$ $R_{▵}X$are all homotopy equivalent,$$\text {Aut}(X)$$ $\text{Aut}\left(X\right)$-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes. 

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.