  Markopoulou and Smolin have argued that the low energy limit of LQG may
suffer from a conflict between locality, as defined by the connectivity of spin
networks, and an averaged notion of locality that emerges at low energy from a
superposition of spin network states. This raises the issue of how much
non-locality, relative to the coarse grained metric, can be tolerated in the
spin network graphs that contribute to the ground state. To address this
question we have been studying statistical mechanical systems on lattices
decorated randomly with non-local links. These turn out to be related to a
class of recently studied systems called small world networks. We show, in the
case of the 2D Ising model, that one major effect of non-local links is to
raise the Curie temperature. We report also on measurements of the spin-spin
correlation functions in this model and show, for the first time, the impact of
not only the amount of non-local links but also of their configuration on
correlation functions.
