  In several approaches of non-perturbative quantum gravity, a major
outstanding problem is to obtain results valid at the infinite lattice
refinement limit. Working with Lorentzian simplicial quantum gravity, we
compute light ray fluctuation probabilities in 3D and 4D across different
lattices. In a simplified refined box model with the Einstein-Hilbert action,
numerical results show that lattice refinement does not simply suppress or
simply enhance light ray fluctuations, but actually drives very wide and very
narrow light probability distributions towards intermediate ones. A comparison
across lattices and across couplings reveals numerical hints at a lattice
refinement fixed point associated with a universality class of couplings. The
results fit the intuition that quantum spacetime fluctuations reflected by
light ray fluctuations start wild microscopically and become mild
macroscopically. The refined box model is limited by the assumption of a rigid
frame at all scales. The present results suggest further studies around the
zero-coupling limit to relax the simplifying assumptions of the model.
