  Let $X$ be a nonelementary CAT(0) cubical complex. We prove that if $X$ is
essential and irreducible, then the contact graph of $X$ (introduced in
\cite{Hagen}) is unbounded and its boundary is homeomorphic to the regular
boundary of $X$ (defined in \cite{Fernos}, \cite{KarSageev}). Using this, we
reformulate the Caprace-Sageev's Rank-Rigidity Theorem in terms of the action
on the contact graph. Let $G$ be a group with a nonelementary action on $X$,
and $(Z_n)$ a random walk corresponding to a generating probability measure on
$G$ with 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 (where $A=\lim \frac{d(Z_no,o)}{n}$ is the drift of the random
walk, and $o\in X$ is an arbitrary basepoint).
