Emergent classical geometries on boundaries of randomly connected tensor networks

It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensornetworks with appropriately chosen tensorsin the thermodynamic limit. With variation of the tensors, the dimensions of the spaces can be freely chosen, and the geometries, which are curved in general, can be varied. We give the explicit solvable examples of emergent flat tori in arbitrary dimensions, and the correspondence from the tensorsto the geometries for general curved cases. The perturbative dynamics in the emergent space is shown to be described by an effective actionwhich is invariant under the spatial diffeomorphismdue to the underlying orthogonal groupsymmetry of the randomly connected tensornetwork. It is also shown that there are various phase transitions among spaces, including extended and point-like ones, under continuous change of the tensors.