  The so-called spectral dimension is a scale-dependent number associated with
both geometries and field theories that has recently attracted much attention,
driven largely though not exclusively by investigations of causal dynamical
triangulations (CDT) and Horava gravity as possible candidates for quantum
gravity. We advocate the use of the spectral dimension as a probe for the
kinematics of these (and other) systems in the region where spacetime curvature
is small, and the manifold is flat to a good approximation. In particular, we
show how to assign a spectral dimension (as a function of so-called diffusion
time) to any arbitrarily specified dispersion relation. We also analyze the
fundamental properties of spectral dimension using extensions of the usual
Seeley-DeWitt and Feynman expansions, and by saddle point techniques. The
spectral dimension turns out to be a useful, robust and powerful probe, not
only of geometry, but also of kinematics.
