  We propose a new method to characterize the different phases observed in the
non-perturbative numerical approach to quantum gravity known as Causal
Dynamical Triangulation. The method is based on the analysis of the eigenvalues
and the eigenvectors of the Laplace-Beltrami operator computed on the
triangulations: it generalizes previous works based on the analysis of
diffusive processes and proves capable of providing more detailed information
on the geometric properties of the triangulations. In particular, we apply the
method to the analysis of spatial slices, showing that the different phases can
be characterized by a new order parameter related to the presence or absence of
a gap in the spectrum of the Laplace-Beltrami operator, and deriving an
effective dimensionality of the slices at the different scales. We also propose
quantities derived from the spectrum that could be used to monitor the running
to the continuum limit around a suitable critical point in the phase diagram,
if any is found.
