We consider the coupling of ice-type vertex models to random, planar φ4 quantum-gravity graphs. The well-established techniques for the simulation of dynamical triangulations and their dual φ3 graphs are suitably adapted to the case of four-valent graphs. These methods are combined with a formulation of the loop algorithm for the simulation of the vertex model matter part. We present a preliminary analysis of the dynamical scaling behaviour of the combined algorithm for the case of the 6-vertex model coupled to quantum gravity.
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Weigel, M., & Janke, W. (2002). Algorithmic tools for simulations of vertex models on random graphs. Nuclear Physics B - Proceedings Supplements, 106-107, 986–988. https://doi.org/10.1016/S0920-5632(01)01906-5