Universal spectra of quasirandom objects produced by off-equilibrium space divisions

A simple model of quasirandomness is presented. The model is a self-similar space division: A d-dimensional space is divided successively into smaller subspaces with randomness. Positions of subspaces are fixed. Then centers of subspaces distribute quasirandomly. Two typical cases of the model are discussed. In one case (repulsive case) the power spectrum of centers of subspaces exhibits a $k^{\mathrm{-}\alpha }$ (α≊d) singularity in a broad range of a randomness parameter, where k is the wave number. This spectrum corresponds to a so-called 1/f noise spectrum for a one-dimensional time sequence. The $k^{\mathrm{-}d}$ singularity will be observed in the scattering intensity from a phase-separating system if droplets cannot move freely due to the strong correlation. This singularity does not seem to depend on details of the model, and is universal. We can show the same singularity analytically for a modified model. In the other case (attractive case) the power spectrum exhibits a $k^{\mathrm{-}\alpha }$ (α≊0.4d) singularity in a broad range of the randomness parameter. This singularity is equivalent to the typical singularity observed in the universe. The physical reason for this agreement is discussed.