  At criticality, discrete quantum-gravity models are expected to give rise to
continuum spacetime. Recent progress has established the functional
renormalization group method in the context of such models as a practical tool
to study their critical properties and to chart their phase diagrams. Here, we
apply these techniques to the multi-matrix model with $ABAB$-interaction
potentially relevant for Lorentzian quantum gravity in 3 dimensions. We
characterize the fixed-point structure and phase diagram of this model, paving
the way for functional RG studies of more general multi-matrix or tensor models
encoding causality and subjecting the technique to another strong test of its
performance in discrete quantum gravity by comparing to known results.
