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 phi(3) and phi(4) 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.