Evolution of the giant component in graphs on orientable surfaces

For a fixed integer g≥0, let Sg(n,m) be a graph chosen uniformly at random from all graphs with n vertices and m edges that are embeddable on the orientable surface Sg of genus g. We prove that the component structure of Sg(n, m) features two phase transitions. The first one is analogous to the emergence of the giant component in the classical Erdős-Rényi random graph G(n, m) at m∼n2 second phase transition occurs at m∼n, when the giant component covers almost all vertices.