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Hard spheres are ubiquitous in condensed matter: they have been used asmodels for liquids, crystals, colloidal systems, granular systems, and powders.Packings of hard spheres are of even wider interest, as they are related toimportant problems in information theory, such as digitalization of signals,error correcting codes, and optimization problems. In three dimensions thedensest packing of identical hard spheres has been proven to be the FCClattice, and it is conjectured that the closest packing is ordered (a regularlattice, e.g, a crystal) in low enough dimension. Still, amorphous packingshave attracted a lot of interest, because for polydisperse colloids andgranular materials the crystalline state is not obtained in experiments forkinetic reasons. We review here a theory of amorphous packings, and moregenerally glassy states, of hard spheres that is based on the replica method:this theory gives predictions on the structure and thermodynamics of thesestates. In dimensions between two and six these predictions can be successfullycompared with numerical simulations. We will also discuss the limit of largedimension where an exact solution is possible. Some of the results we presenthere have been already published, but others are original: in particular weimproved the discussion of the large dimension limit and we obtained newresults on the correlation function and the contact force distribution in threedimensions. We also try here to clarify the main assumptions that are beyondour theory and in particular the relation between our static computation andthe dynamical procedures used to construct amorphous packings.

Mean field theory of hard sphere glasses and jamming