The development of clustering in initially quasi-uniform infinite distributions of point particles evolving purely under their Newtonian self-gravity has been the subject of extensive numerical study in cosmology over the last several decades (see e.g. [1] for a review). This is the case because these “N-body” (particle) simulations of the Newtonian limit are believed to give a very good approximation to the formation of structure  formation in current dark matter dominated models of the universe. The impressive growth in the size of these simulations has led essentially to phenomenological models of the associated dynamics. Analytical understanding, which would be very useful in trying to extend the numerical results and also control for their reliability, remains very limited.

In attempts to progress in this direction it is natural to look to simplified toy models which may provide insight and qualitative understanding. Such models may also be interesting theoretically in a purely statistical mechanics setting, and specifically in the context of the investigation of out of equilibrium dynamics of systems with long-range interactions. An obvious toy model for this full 3d problem is the analogous problem in 1d , i.e., the generalization to an infinite space (static or expanding) of the so-called “sheet model”, which is formulated for finite mass distributions

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