Number of faces in a random embedding of a complete graph

Any finite graph can be embedded on a surface with sufficiently high genus. Such an embedding can be described (up to homeomorphism) by local rotations at vertices, i.e., a cyclical order of all edges incident to a vertex. A random embedding is then just an embedding with randomly chosen local rotations at each vertex. A genus distribution of some types of graphs is well known. However, our main interest is a distribution of the number of faces of some length. We also restrict the work only on one type of a graph, i.e., a complete graph. The main result is that the number of faces of a fixed length has asymptotically Poisson distribution. We knew there is a close relation between cycles in a random permutation and faces of a random embedding. We found out that they have the same limit distribution, however, we also found differences between these two objects. Relations are described by theoretical calculations of various moments and by simulations of a random embedding and a random permutation.