Random lattices and random sphere packings: typical properties

We review results about the density of typical lattices in $\mathbb R^n$. This density is of the order of $2^{-n}$. We then obtain similar results for random (non-lattice) sphere packings in $\mathbb R^n$: after suitably taking a fraction $\nu$ of centers of spheres in a typical random packing $\sigma$, the resulting packing $\tau$ has density $C(\nu) 2^{-n}$ with a reasonable $C(\nu)$. We obtain estimates of $C(\nu)$.