Decorating random quadrangulations

On various regular lattices (simple cubic, body centred cubic, etc) decorating an edge with an Ising spin coupled by bonds of strength L to the original vertex spins and competing with a direct anti-ferromagnetic bond of strength a L can give rise to three transition temperatures for suitable a. The system passes through ferromagnetic, paramagnetic, anti-ferromagnetic and paramagnetic phases respectively as the temperature is increased. For the square lattice on the other hand, multiple decoration is required to see this effect. We note here that a single decoration suffices for the Ising model on planar random quadrangulations (coupled to 2D quantum gravity). Other random bipartite lattices such as the Penrose tiling are more akin to the regular square lattice and require multiple decoration to have any affect.