Given (a set of) fundamental models of quantum space time, for instance spin foam models, we aim to understand the large scale physics encoded in these fundamental models. Renormalization and coarse graining address this issue and help to understand how large scale physics depends on parameters in the fundamental models.
I will review recent work on coarse graining and renormalization of spin foam and analogue models, revealing possible large scale phases, depending on parameters of the microscopic models. I will explain how these phases are connected to topological field theories and possible vacua for the theory of quantum gravity, e.g. loop quantum gravity. I outline how these different vacua are connected to different representations of the observable algebra, that is different Hilbert spaces, and how this allows to expand the theory around different vacuum states.