  In this work, we investigate the 4d path integral for Euclidean quantum
gravity on a hypercubic lattice, as given by the Spin Foam model by Engle,
Pereira, Rovelli, Livine, Freidel and Krasnov (EPRL-FK). To tackle the problem,
we restrict to a set of quantum geometries that reflects the large amount of
lattice symmetries. In particular, the sum over intertwiners is restricted to
quantum cuboids, i.e. coherent intertwiners which describe a cuboidal geometry
in the large-$j$ limit.
  Using asymptotic expressions for the vertex amplitude, we find several
interesting properties of the state sum. First of all, the value of coupling
constants in the amplitude functions determines whether geometric or
non-geometric configurations dominate the path integral. Secondly, there is a
critical value of the coupling constant $\alpha$, which separates two phases.
In both phases, the diffeomorphism symmetry appears to be broken. In one, the
dominant contribution comes from highly irregular, in the other from highly
regular configurations, both describing flat Euclidean space with small quantum
fluctuations around them, viewed in different coordinate systems. On the
critical point diffeomorphism symmetry is nearly restored, however.
  Thirdly, we use the state sum to compute the physical norm of kinematical
states, i.e. their norm in the physical Hilbert space. We find that states
which describe boundary geometry with high torsion have exponentially
suppressed physical norm. We argue that this allows one to exclude them from
the state sum in calculations.
