Thin Absolute Villains

TL;DR

This paper uses simulations of the Villain model on various random graphs to confirm a mean field vortex transition, showing excellent agreement with exact analytical results and exploring universality across different graph types.

Contribution

It provides the first numerical confirmation of the mean field vortex transition in the Villain model on random graphs, validating analytical predictions and extending understanding of universality.

Findings

Excellent agreement between simulations and exact calculations.

Confirmation of mean field vortex transition.

Universality of results across different graph types.

Abstract

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.