  We perform simulations of an absolute value version of the Villain model on
phi3 and phi4 Feynman diagrams, ``thin'' 3-regular and 4-regular random graphs.
The phi4 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 phi3 graphs and again find excellent
agreement with the numerical data for single phi3 graphs.
  The simulations confirm the picture of a mean field vortex transition which
is suggested by the analytical results. Further simulations on phi5 and phi6
graphs and of the standard XY model on phi3 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.
