Random Lattices and Random Sphere Packings: Typical Properties

Abstract

We review results about the density of typical lattices in $R^n.$ They state that such density is of the order of $2^{-n}.$ We then obtain similar results for random packings in $R^n$: after taking suitably a fraction $\nu$ of a typical random packing $\sigma$, the resulting packing $\tau$ has density $C(\nu )2^{-n},$ with a reasonable $C(\nu ).$ We obtain estimates on $C(\nu ).$