Ising spins on thin graphs

Abstract

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 φ 3 and φ 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.

Frustration effects in antiferromagnets on planar random graphs

“…Consequently, one expects mean-field phase transitions for the ferromagnetic models. These are indeed found for the Erdös-Rényi and k-regular (or thin) random graphs 2,4,7,8,9,10 , transforming them into convenient alternatives to treatments on the Bethe lattice or the complete graph, not encumbered with boundary effects. More generally, degree distributions with divergent moments can lead to interesting deviations from mean-field behaviour 11 .…”

“…The existence of oddlength loops on many of the graphs discussed leads to severe geometric frustration with the possibility of altogether precluding the onset of a long-range ordered phase. Interestingly, however, this problem for random graphs has received very little attention to date 8 , such that the behaviour of, e.g., the Ising antiferromagnet for the various cases is unclear. The effect of tuning the amount of frustration on a regular lattice has been considered in Ref.…”

Relaxation in graph coloring and satisfiability problems

“…Related models have been studied by Baillie, Johnston, and coworkers (e.g. [24,25]), who considered Potts models on φ n -model Feynman diagrams. They found similarities between models on φ 3 and φ 4 graphs and Bethe lattices, and showed that mean-field theories work well for describing both ferromagnetic and antiferromagnetic models on Feynman diagrams.…”

Vertex models on Feynman diagrams

“…generic, non-planar) random graphs [1,2]. In particular, one finds mean-field like behaviour for the spin models due to the locally-tree-like structure of the graphs.…”

“…Such saddle point methods in graph theory been independently derived from a probabilistic viewpoint by Whittle in [5]. The topic of the current paper has also been presaged by the same author in [6], where the main concern was understanding the statistics of random directed graphs using complex integrals rather than the real integrals of [1,2,5]. We take a rather different tack here, where our aim is to investigate vertex models on random graphs in their own right, so the focus is on the matter living on the graphs rather than the graphs themselves.…”