After reviewing some basic relevant properties of stationary stochastic processes (SSP), we discuss the properties of the so‐called Harrison‐Zeldovich like spectra of mass density perturbations. These correlations are a fundamental feature of all current standard cosmological models. Examining them in real space we note they imply a sub‐poissonian normalized variance in spheres σ2M(R)∼R−4lnR. In particular this latter behavior is at the limit of the most rapid decay (∼ R−4) of this quantity possible for any stochastic distribution (continuous or discrete). In a simple classification of all SSP into three categories, we highlight with the name “super‐homogeneous” the properties of the class to which models like this, with P(0) = 0, belong. In statistical physics language they are well described as lattice or glass‐like. We illustrate their properties through two simple examples: (i) the “shuffled” lattice and the One Component Plasma at thermal equilibrium.
Gabrielli, A., Joyce, M., & Sylos Labini, F. (2003). Real Space Statistical Properties of Standard Cosmological Models. In MODELING OF COMPLEX SYSTEMS: Seventh Granada Lectures (pp.188-194) [10.1063/1.1571311].