The Villain model on thin graphs

We perform simulations of an absolute value version of the Villain model on φ3 and φ4 Feynman diagrams, “thin” 3-regular and 4-regular random graphs. The φ4 results are in excellent quantitative agreement with the exact calculations by Dorey and Kurzepa for an annealed ensemble of thin graphs, in spite of simulating only a single graph of each size. We also derive exact results for an annealed ensemble of φ3 graphs and again find excellent agreement with the numerical data for single φ3 graphs. The simulations confirm the picture of a mean field vortex transition which is suggested by the analytical results. Further simulations on φ5 and φ6 graphs and of the standard XY model on φ3 graphs confi the universality of these results. The calculations of Dorey and Kurzepa were based on reinterpreting the large orders behaviour of the anharmonic oscillator in a statistical mechanical context so we also discuss briefly the interpretation of singularities in the large orders behaviour in other models as phase transitions.