  Simplicial complexes are increasingly used to study complex system structure
and dynamics including diffusion, synchronization and epidemic spreading. The
spectral dimension of the graph Laplacian is known to determine the diffusion
properties at long time scales. Using the renormalization group here we
calculate the spectral dimension of the graph Laplacian of two classes of
non-amenable $d$ dimensional simplicial complexes: the Apollonian networks and
the pseudo-fractal networks. We analyse the scaling of the spectral dimension
with the topological dimension $d$ for $d\to \infty$ and we point out that
randomness such as the one present in Network Geometry with Flavor can diminish
the value of the spectral dimension of these structures.
