We present here random distributions on $(D+1)$-edge-colored, bipartite graphs with a fixed number of vertices $2p$. These graphs are dual to $D$-dimensional orientable colored complexes. We investigate the behavior of quantities related to those random graphs, such as their number of connected components or the number of vertices of their dual complexes, as $p \to \infty$. The techniques involved in the study of these quantities also yield a Central Limit Theorem for the genus of a uniform map of order $p$, as $p \to \infty$.