The Villain model on thin graphs

We perform simulations of an absolute value version of the Villain model on f³ and f⁴ Feynman diagrams, "thin" 3-regular and 4-regular random graphs. The f⁴ 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 f³ graphs and again find excellent agreement with the numerical data for single f³ graphs. The simulations confirm the picture of a mean field vortex transition which is suggested by the analytical results. Further simulations on f⁵ and f⁶ graphs and of the standard XY model on f³ graphs confirm 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.