  We consider infinite random casual Lorentzian triangulations emerging in
quantum gravity for critical values of parameters. With each vertex of the
triangulation we associate a Hilbert space representing a bosonic particle
moving in accordance with standard laws of Quantum Mechanics. The particles
interact via two-body potentials decaying with the graph distance. A
Mermin--Wagner type theorem is proven for infinite-volume reduced density
matrices related to solutions to DLR equations in the Feynman--Kac (FK)
representation.
