Amorphous packings of hard spheres for large space dimension

In a recent paper we derived an expression for the replicated free energy of a liquid of hard spheres based on the hypernetted chain (HNC) free energy functional. An approximate equation of state for the glass and an estimate of the random close packing density were obtained for d = 3. Here we show that the HNC approximation is not needed: the same expression can be obtained from the full diagrammatic expansion of the replicated free energy. Then, we consider the asymptotics of this expression when the space dimension d is very large. In this limit, the entropy of the hard sphere liquid has been computed exactly. Using this solution, we derive asymptotic expressions for the glass transition density and for the random close packing density for hard spheres for large space dimension.