Ultralocal nature of geometrogenesis

Abstract: In this article we show that the ultralocal state of gravity can be associated with the so-called crumpled phase of gravity, observed e.g. in Causal Dynamical Triangulations. By considering anisotropic scaling present in the Ho\v{r}ava-Lifshitz theory, we prove that in the ultralocal scaling limit ($z \rightarrow 0$) the graph representing connectivity structure of space is becoming complete. In consequence, transition from the ultralocal phase ($z=0$) to the standard relativistic scaling ($z=1$) is implemented by the geometrogensis, similar to the one considered in Quantum Graphity approach. However, the relation holds only for the finite number of nodes $N$ and in the continuous limit ($N\rightarrow \infty$) the complete graph reduces to the set of disconnected points due to the weights $w=1/N$ associated with the links. By coupling Ising spin matter to the considered graph we show that the process of geometrogensis can be associated with critical behavior. Based on both analytical and numerical analysis phase diagram of the system is reconstructed showing that (for a ring graph) symmetry broken phase occurs at $z\in [0, 0.5)$. Finally, cosmological consequences of the considered process of geometrogenesis as well as similarities with the so-called synaptic pruning are briefly discussed.