We explore whether the phase diagram of tensor models could feature a
pregeometric, discrete and a geometric, continuum phase for the building blocks
of space. The latter are associated to rank d tensors of size N. We search
for a universal large N scaling limit in a rank-3 model with real tensors
that could be linked to a transition between the two phases. We extend the
conceptual development and practical implementation of the flow equation for
the pregeometric setting. This provides a pregeometric "coarse-graining" by
going from many microscopic to few effective degrees of freedom by lowering
N. We discover several candidates for fixed points of this coarse graining
procedure, and specifically explore the impact of a novel class of interactions
allowed in the real rank-3 model. In particular, we explain how most
universality classes feature dimensional reduction, while one candidate,
involving a tetrahedral interaction, might potentially be of relevance for
three-dimensional quantum gravity.Comment: 23 pages plus appendix and reference
