Ising spins on thin graphs

The Ising model on "thin" graphs (standard Feynman diagrams) displays several interesting properties. For ferromagnetic couplings there is a mean-field phase transition at the corresponding Bethe-lattice transition point. For antiferromagnetic couplings the replica trick gives some evidence for a spin-glass phase. In this paper we investigate both the ferromagnetic and antiferromagnetic models with the aid of simulations. We confirm the Bethe-lattice values of the critical points for the ferromagnetic model on f³ and f⁴ graphs and examine the putative spin-glass phase in the antiferromagnetic model by looking at the overlap between replicas in a quenched ensemble of graphs. We also compare the Ising results with those for higher-state Potts models and Ising models on "fat" graphs, such as those used in 2D gravity simulations.