I have been reading up on some approaches to quantum gravity apart from string theory. The popular conception of loop quantum gravity is that it says that space is actually physically discrete at some sufficiently small scale, where the lattice spacings are presumably given by the Planck length.
It seems that at least in the case of loop quantum gravity, that this is a misconception and the discreteness is just referring to certain eigenvalues of area in a way which is somewhat familiar from elementary quantum mechanics (see here for example).
Is there some approach where space is actually a graph or where spacetimes are actually chopped up into lattices such that for example the Universe is a cosmological spacetime which can be chopped up into a lattice?
(I don't mean just doing quantum gravity on a flat lattice by analogy with lattice Yang-Mills theory).
Edit: After some more searching, it looks like this kind of theory tends to go by the name of 'induced gravity' or 'emergent gravity'. The details of how it all works are about as confused as they are in LQG unfortunately.