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Two desireable properties of a quantum dynamics for Loop Quantum Gravity(LQG) are that its generators provide an anomaly free representation of theclassical constraint algebra and that physical states which lie in the kernelof these generators encode propagation. A physical state in LQG is expected tobe a sum over graphical $SU(2)$ spin network states. By propagation we meanthat a quantum perturbation at one vertex of a spin network state propagates tovertices which are `many links away' thus yielding a new spin network statewhich is related to the old one by this propagation. A physical state encodespropagation if its spin network summands are related by propagation. Here westudy propagation in an LQG quantization of Smolin's weak coupling limit ofEuclidean Gravity based on graphical $U(1)^3$ `charge' network states. Buildingon our earlier work on anomaly free quantum constraint actions for this system,we analyse the extent to which physical states encode propagation. Inparticular, we show that a slight modification of the constraint actionsconstructed in our previous work leads to physical states which encode robustpropagation. Under appropriate conditions, this propagation merges, seperatesand entangles vertices of charge network states. The `electric' diffeomorphismconstraints introduced in prevous work play a key role in our considerations.The main import of our work is that there are choices of quantum constraintconstructions through LQG methods which are consistent with vigorouspropagation thus providing a counterpoint to Smolin's early observations on thedifficulties of propagation in the context of LQG type operator constructions.Whether the choices considered in this work are physically appropriate is anopen question worthy of further study.

On quantum propagation in Smolin's weak coupling limit of 4d Euclidean Gravity