Limiting measure for random-geometric-graph XY measures

Determine whether the random-geometric-graph XY measures converge to a limiting measure on the domain Ω as the number of sampled points tends to infinity and the graph spacing tends to zero.

Background

The paper studies the classical XY model on random geometric graphs whose vertices are independently sampled from a bounded domain Ω and whose interaction range decreases as the sample size increases. The main results establish quenched long-range order in a low-temperature regime, but they do not identify a continuum limiting probability measure for the associated spin configurations. Establishing convergence of the random-graph XY measures would clarify the continuum behavior underlying the discrete models and relate the long-range-order results to a limiting statistical-mechanical system on Ω.