  The spectral dimension has proven to be a very informative observable to
understand the properties of quantum geometries in approaches to quantum
gravity. In loop quantum gravity and its spin foam description, it has not been
possible so far to calculate the spectral dimension of spacetime. As a first
step towards this goal, here we determine the spacetime spectral dimension in
the simplified spin foam model restricted to hypercuboids. Using Monte Carlo
methods we compute the spectral dimension for state sums over periodic spin
foam configurations on infinite lattices. For given periodicity, i.e. number of
degrees of freedom, we find a range of scale where an intermediate spectral
dimension between 0 and 4 can be found, continuously depending on the parameter
of the model. Under an assumption on the statistical behaviour of the Laplacian
we can explain these results analytically. This allows us to take the
thermodynamic limit of large periodicity and find a phase transition from a
regime of effectively 0-dimensional to 4-dimensional spacetime. At the point of
phase transition, dynamics of the model are scale invariant which can be seen
as restoration of diffeomorphism invariance of flat space. Considering the
spectral dimension as an order parameter for renormalization we find a
renormalization group flow to this point as well. Being the first instance of
an emergence of 4-dimensional spacetime in a spin foam model, the properties
responsible for this result seem to be rather generic. We thus expect similar
results for more general, less restricted spin foam models.
