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We perform simulations of an absolute value version of the Villain model on φ 3 and φ 4 Feynman diagrams, " thin " 3-regular and 4-regular random graphs. The φ 4 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 φ 3 graphs and again find excellent agreement with the numerical data for single φ 3 graphs. The simulations confirm the picture of a mean field vortex transition which is suggested by the analytical results. Further simulations on φ 5 and φ 6 graphs and of the standard XY model on φ 3 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.