Random and frozen states in complex triangulations

Triangulations of complex surfaces with different genera are studied within a statistical mechanics framework where an energy is associated to deviations from an ideal, ordered ground state. We observe that the complexity of the embedding surface strongly affects the properties of the triangulations. At high temperatures the 'random states' have degree distributions that broaden with the surface genus. At low temperatures the 'frozen states' can reach a higher degree of order with increasing genus. The dynamics between disordered and ordered states is also affected by the surface genus. High genus triangulations start from more disordered states at high temperatures but they quench faster into more ordered states than the low genus counterparts. However, the ground state is never reached because at low temperatures the relaxation dynamics slows down into a glassy kind of behavior. Topological frustration can also play a very important role when the surface genus forces the average degree to be a fractional number.