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We explore whether the phase diagram of tensor models could feature apregeometric, discrete and a geometric, continuum phase for the building blocksof space. The latter are associated to rank $d$ tensors of size $N$. We searchfor a universal large $N$ scaling limit in a rank-3 model with real tensorsthat could be linked to a transition between the two phases. We extend theconceptual development and practical implementation of the flow equation forthe pregeometric setting. This provides a pregeometric "coarse-graining" bygoing from many microscopic to few effective degrees of freedom by lowering$N$. We discover several candidates for fixed points of this coarse grainingprocedure, and specifically explore the impact of a novel class of interactionsallowed in the real rank-3 model. In particular, we explain how mostuniversality classes feature dimensional reduction, while one candidate,involving a tetrahedral interaction, might potentially be of relevance forthree-dimensional quantum gravity.

Towards background independent quantum gravity with tensor models