Symmetric Vertex Models on Planar Random Graphs

We discuss a 4-vertex model on an ensemble of 3-valent (Φ) planar random graphs, which has the effect of coupling the vertex model to 2D quantum gravity. The regular lattice equivalent of the model is the symmetric 8-vertex model on the honeycomb lattice, which can be mapped on to an Ising model in field, as was originally shown by Wu et.al. using generalised weak graph transformation techniques. We note that the Ising equivalence still holds for the 4-vertex model on Φ graphs, and indeed for higher valency symmetric vertex models, which again allows a determination of the critical behaviour of the vertex model. The relations between the vertex weights and Ising model parameters are unrenormalized by the coupling to gravity, as is the equation of the Ising locus for the vertex weights. The generalised weak graph symmetry of the vertex weights which is fundamental to the solution can be understood in the Φ case as a change of integration variable in the matrix integral used to define the model. We note the Ising equivalence can be used to determine the critical behaviour of the spin one Blume-Emery-Griffiths model for a particular subset of couplings on Φ graphs, as on the honeycomb lattice, giving a rare exact solution for a higher spin model.