Characterising cosmic inhomogeneity with anomalous diffusion
  Dark matter (DM) clustering at the present epoch is investigated from a
fractal viewpoint in order to determine the scale where the self-similar
scaling property of the DM halo distribution transits to homogeneity. Methods
based on well-established `counts-in-cells' as well as new methods based on
anomalous diffusion and random walks are investigated. Both are applied to DM
halos of the biggest N-Body simulation in the `Dark Sky Simulations' (DS)
catalogue and an equivalent randomly distributed catalogue. Results based on
the smaller `Millennium Run' (MR) simulation are revisited and improved. It is
found that the MR simulation volume is too small and prone to bias to reliably
identify the onset of homogeneity. Transition to homogeneity is defined when
the fractal dimension of the clustered and random distributions cannot be
distinguished within the associated uncertainties. The `counts-in-cells' method
applied to the DS simulation then yields a homogeneity scale roughly consistent
with previous work ($\sim 150$$h^{-1}$Mpc). The characteristic length-scale for
anomalous diffusion to behave homogeneously is found to be at about
250$h^{-1}$Mpc. The behaviour of the fractal dimensions for a halo catalogue
with the same two-point function as the original but with shuffled Fourier
phases is investigated. The methods based on anomalous diffusion are shown to
be sensitive to the phase information, whereas the `counts-in-cells' methods
are not.
