  The spectral dimension measures the dimensionality of a space as witnessed by
a diffusing random walker. Within the causal dynamical triangulations approach
to the quantization of gravity, the spectral dimension exhibits novel
scale-dependent dynamics: reducing towards a value near 2 on sufficiently small
scales, matching closely the topological dimension on intermediate scales, and
decaying in the presence of positive curvature on sufficiently large scales. I
report the first comprehensive scaling analysis of the small-to-intermediate
scale spectral dimension for the test case of the causal dynamical
triangulations of 3-dimensional Einstein gravity. I find that the spectral
dimension scales trivially with the diffusion constant. I find that the
spectral dimension is completely finite in the infinite volume limit, and I
argue that its maximal value is exactly consistent with the topological
dimension of 3 in this limit. I find that the spectral dimension reduces
further towards a value near 2 as this case's bare coupling approaches its
phase transition, and I present evidence against the conjecture that the bare
coupling simply sets the overall scale of the quantum geometry. On the basis of
these findings, I advance a tentative physical explanation for the dynamical
reduction of the spectral dimension observed within causal dynamical
triangulations: branched polymeric quantum geometry on sufficiently small
scales. My analyses should facilitate attempts to employ the spectral dimension
as a physical observable with which to delineate renormalization group
trajectories in the hope of taking a continuum limit of causal dynamical
triangulations at a nontrivial ultraviolet fixed point.
