Scale-dependent homogeneity measures for causal dynamical triangulations

Understand

I propose two scale-dependent measures of the homogeneity of the quantum geometry determined by an ensemble of causal triangulations.

• The first measure is volumetric, probing the growth of volume with graph geodesic distance.
• The second measure is spectral, probing the return probability of a random walk with diffusion time.
• Both of these measures, particularly the first, are closely related to those used to assess the homogeneity of our own universe on the basis of galaxy redshift surveys.