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After reviewing the basic relevant properties of stationary stochasticprocesses (SSP), defining basic terms and quantities, we discuss the propertiesof the so-called Harrison-Zeldovich like spectra. These correlations, usuallycharacterized exclusively in k-space (i.e. in terms of power spectra P(k)), area fundamental feature of all current standard cosmological models. Examiningthem in real space we note their characteristics to be a {\it negative} powerlaw tail \xi(r) \sim - r^{-4} and a {\it sub-poissonian} normalised variance inspheres \sigma^2(R) \sim R^{-4} \ln R. We note in particular that this latterbehaviour is at the limit of the most rapid decay (\sim R^{-4}) of thisquantity possible for any stochastic distribution (continuous or discrete).This very particular characteristic is usually obscured in cosmology by the useof Gaussian spheres. In a simple classification of all SSP into threecategories, we highlight with the name ``super-homogeneous'' the properties ofthe class to which models like this, with P(0)=0, belong. In statisticalphysics language they are well described as glass-like. They do not have either``scale-invariant'' features, in the sense of critical phenomena, nor fractalproperties. We illustrate their properties with some simple examples, inparticular that of a ``shuffled'' lattice.Comment: 20 pages, 3 postscript figures, corrected some typos and minor  changes to match the accepted version in Physical Review

Glass-like universe: Real-space correlation properties of standard cosmological models