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