Ising (anti-)ferromagnet on dynamical triangulations and quadrangulations

We write down matrix models for Ising spins with zero external field on the vertices of dynamical triangulated random surfaces (DTRS) and dynamically quadrangulated random surfaces (DQRS) and compare these with the standard matrix model approach which places the spins on the dual $\phi^3$ and $\phi^4$ graphs. We show that the critical temperatures calculated in the DTRS and DQRS models agree with those deduced from duality arguments in the standard approach. Using the DQRS model we observe that the Ising antiferromagnet still undergoes a phase transition to a Neel (checkerboard) ordered ground state which is absent because of frustration in the other cases.